# The symplectic isotopy problem for rational cuspidal curves

Marco Golla, Laura Starkston

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Marco Golla, Laura Starkston. The symplectic isotopy problem for rational cuspidal curves. Compositio Mathematica, 2022, 158 (7), pp.1595-1682. 10.1112/S0010437X2200762X. hal-02186132

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# The symplectic isotopy problem for rational cuspidal curves

Marco Golla, Laura Starkston

<figure><img src="image_1.png" /></figure>

To cite this version:

Marco Golla, Laura Starkston. The symplectic isotopy problem for rational cuspidal curves. Compo-sitio Mathematica, 2022, 158 (7), pp.1595-1682. 10.1112/S0010437X2200762X . hal-02186132

HAL Id: hal-02186132

https://hal.science/hal-02186132v1

Submitted on 24 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
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COMPOSITIO MATHEMATICA

The symplectic isotopy problem for rational
cuspidal curves

Marco Golla and Laura Starkston

Compositio Math. **158** (2022), 1595–1682.

doi:10.1112/S0010437X2200762X

<figure><img src="image_4.png" /></figure>

FOUNDATION
COMPOSITIO
MATHEMATICA

<figure><img src="image_5.png" /></figure>

LONDON
MATHEMATICAL
SOCIETY
EST. 1865

<figure><img src="image_6.png" /></figure>
<figure><img src="image_3.png" /></figure>

COMPOSITIO MATHEMATICA

The symplectic isotopy problem for rational
cuspidal curves

Marco Golla and Laura Starkston

Compositio Math. **158** (2022), 1595–1682.

doi:10.1112/S0010437X2200762X

<figure><img src="image_4.png" /></figure>

FOUNDATION
COMPOSITIO
MATHEMATICA

<figure><img src="image_5.png" /></figure>

LONDON
MATHEMATICAL
SOCIETY
EST. 1865

<figure><img src="image_6.png" /></figure>
<figure><img src="image_7.png" /></figure>

# The symplectic isotopy problem for rational cuspidal curves

Marco Golla and Laura Starkston

## ABSTRACT

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology.

## 1. Introduction

In this article, we take up an extensive study of singular curves in the symplectic category. We focus on rational (genus-zero) curves, taking advantage of the singularities to obtain low-genus curves with high degree. We primarily study irreducible rational cuspidal curves, but also consider reducible configurations with rational cuspidal components. Rational cuspidal curves are a source of rich complexity in algebraic geometry [KP17, PP17, Pal19, PP20]. We use the term *cusp* to refer to any locally irreducible singularity, but we focus on cusps locally modeled on $\{x^p = y^q\}$ where $p$ and $q$ are relatively prime (the link is a $(p, q)$-torus knot). To the best of the authors' knowledge, prior work on singular symplectic curves has been restricted to nodes, simple cusps ((2, 3)-cusps), and tacnodes (simple tangencies between two branches). Rational cuspidal curves provide an effective class to work with in the symplectic category because pseudoholomorphic curves are most powerful in the rational (genus-zero) case. We give obstruction results determining which rational cuspidal curves are realizable symplectically in the complex projective plane, as well as isotopy classification results proving uniqueness of realization up to symplectic isotopy. Both of these problems (existence and uniqueness) are difficult even in the complex algebraic category, and in the symplectic category, uniqueness has not even been proven for smooth curves of degree greater than 17 (our results apply to singular curves of arbitrary degree).

Complex plane curves have a distinguished history in algebraic geometry: from Zariski's examples of singular sextics with distinct fundamental groups in their complements [Zar29], to

Received 27 December 2019, accepted in final form 24 May 2022, published online 16 September 2022.

2020 Mathematics Subject Classification 57K43, 14H50, 57R52, 14H20 (primary).

Keywords: singular, symplectic, isotopy, rational cuspidal, plane curve.

LS was supported by NSF Grant No. 1501728, 1904074, and 2042345 and a Sloan Fellowship.

© 2022 The Author(s). This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. *Compositio Mathematica* is © Foundation Compositio Mathematica.
<figure><img src="image_7.png" /></figure>

# The symplectic isotopy problem for rational cuspidal curves

Marco Golla and Laura Starkston

## ABSTRACT

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology.

## 1. Introduction

In this article, we take up an extensive study of singular curves in the symplectic category. We focus on rational (genus-zero) curves, taking advantage of the singularities to obtain low-genus curves with high degree. We primarily study irreducible rational cuspidal curves, but also consider reducible configurations with rational cuspidal components. Rational cuspidal curves are a source of rich complexity in algebraic geometry [KP17, PP17, Pal19, PP20]. We use the term *cusp* to refer to any locally irreducible singularity, but we focus on cusps locally modeled on $\{x^p = y^q\}$ where $p$ and $q$ are relatively prime (the link is a $(p, q)$-torus knot). To the best of the authors' knowledge, prior work on singular symplectic curves has been restricted to nodes, simple cusps ((2, 3)-cusps), and tacnodes (simple tangencies between two branches). Rational cuspidal curves provide an effective class to work with in the symplectic category because pseudoholomorphic curves are most powerful in the rational (genus-zero) case. We give obstruction results determining which rational cuspidal curves are realizable symplectically in the complex projective plane, as well as isotopy classification results proving uniqueness of realization up to symplectic isotopy. Both of these problems (existence and uniqueness) are difficult even in the complex algebraic category, and in the symplectic category, uniqueness has not even been proven for smooth curves of degree greater than 17 (our results apply to singular curves of arbitrary degree).

Complex plane curves have a distinguished history in algebraic geometry: from Zariski's examples of singular sextics with distinct fundamental groups in their complements [Zar29], to

Received 27 December 2019, accepted in final form 24 May 2022, published online 16 September 2022.

2020 Mathematics Subject Classification 57K43, 14H50, 57R52, 14H20 (primary).

Keywords: singular, symplectic, isotopy, rational cuspidal, plane curve.

LS was supported by NSF Grant No. 1501728, 1904074, and 2042345 and a Sloan Fellowship.

© 2022 The Author(s). This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. *Compositio Mathematica* is © Foundation Compositio Mathematica.
many more recent results continuing to the present from line arrangements to cuspidal curves (see, for example, [Hir83, ZL83, Har86, ACC03, ACCM05, Ryb11, KP17]). Algebraic geometers have built powerful tools to tackle these problems: braid monodromy [Moi81, Lib89], Alexander invariants [Lib82], the log minimal model program [Miy01], and Miyaoka's inequalities [Miy84]. For example, it was conjectured that every rational cuspidal curve has at most four cusps. This has recently been proven by Koras and Palka [KP22] using the almost minimal model program [Pal19] (see [ZO96, Ton05, Pio07] for previous progress on this problem and [PP17, PP20] for more recent developments). However, a complete classification of singular plane curves is still far out of reach in this rich subject. Beyond questions of which singular curves exist, serious study has been devoted to asking how many planar realizations there are of a given singular curve up to automorphisms of the plane, diffeomorphisms of their complements, or isomorphisms of the fundamental groups of their complements. Distinct realizations (under one of these equivalence relations) are often called 'Zariski pairs' and it is unknown in general which curves have Zariski pairs.

Compared with algebraic geometry, the development of tools in symplectic geometry has occurred relatively recently. The classification of any type of symplectic planar curve, at first glance, is completely intractable because the space of symplectic curves is a relatively poorly understood infinite-dimensional moduli space. By contrast, the space of complex algebraic curves of a particular degree with particular singularities is cut out by finitely many discriminant polynomials in finitely many complex variables. However, Gromov's theory of pseudoholomorphic curves brought some hope that some information about moduli spaces of symplectic curves could be understood. In particular, Gromov classified smooth symplectic surfaces in the complex projective plane in degrees one and two up to symplectic isotopy. Although further work extended this to classify smooth symplectic surfaces up to degree 17 [She00, Sik03, ST05], in degrees greater than or equal to 18 the question remains open.

**QUESTION 1.1** (Symplectic isotopy problem). Is every smooth symplectic surface in $\mathbb{CP}^2$ symplectically isotopic to a complex curve?

This question is equivalent to asking, is there a unique symplectic isotopy class of symplectic surfaces in each degree? The equivalence results from the fact that the moduli space of smooth complex curves of fixed degree is connected. (The space of all degree-$d$ curves is parameterized by the coefficients of the defining polynomial, and singular curves have positive complex dimension.) Because of the difficulty of the classification for smooth symplectic curves, there has been somewhat limited work on singular symplectic curves [McD92a, Bar99, IS99, She04, Fra05]. Nevertheless, the study of singular symplectic curves in $\mathbb{CP}^2$ has significant ramifications for the study of all symplectic 4-manifolds via branched covering constructions [Aur00, Aur06].

Our first result addresses curves with a single cusp singularity. Such curves exist in every degree $d \ge 3$, so we are not bound by the uniform degree constraints encountered in attempts to answer Question 1.1. Note that the degrees and singularities for possible complex curves of this type were classified in [FLMN07].

**THEOREM 1.2.** In $\mathbb{CP}^2$, every symplectic rational unicuspidal curve whose unique singularity is the cone on a torus knot is symplectically isotopic to a complex curve, and has a unique symplectic isotopy class.

As a corollary to our results, we see there are no Zariski pairs of rational unicuspidal curves with one Puiseux pair. To the best of the authors' knowledge, the literature does not contain the statement that there are no Zariski pairs for such complex algebraic curves. However, experts
many more recent results continuing to the present from line arrangements to cuspidal curves (see, for example, [Hir83, ZL83, Har86, ACC03, ACCM05, Ryb11, KP17]). Algebraic geometers have built powerful tools to tackle these problems: braid monodromy [Moi81, Lib89], Alexander invariants [Lib82], the log minimal model program [Miy01], and Miyaoka's inequalities [Miy84]. For example, it was conjectured that every rational cuspidal curve has at most four cusps. This has recently been proven by Koras and Palka [KP22] using the almost minimal model program [Pal19] (see [ZO96, Ton05, Pio07] for previous progress on this problem and [PP17, PP20] for more recent developments). However, a complete classification of singular plane curves is still far out of reach in this rich subject. Beyond questions of which singular curves exist, serious study has been devoted to asking how many planar realizations there are of a given singular curve up to automorphisms of the plane, diffeomorphisms of their complements, or isomorphisms of the fundamental groups of their complements. Distinct realizations (under one of these equivalence relations) are often called 'Zariski pairs' and it is unknown in general which curves have Zariski pairs.

Compared with algebraic geometry, the development of tools in symplectic geometry has occurred relatively recently. The classification of any type of symplectic planar curve, at first glance, is completely intractable because the space of symplectic curves is a relatively poorly understood infinite-dimensional moduli space. By contrast, the space of complex algebraic curves of a particular degree with particular singularities is cut out by finitely many discriminant polynomials in finitely many complex variables. However, Gromov's theory of pseudoholomorphic curves brought some hope that some information about moduli spaces of symplectic curves could be understood. In particular, Gromov classified smooth symplectic surfaces in the complex projective plane in degrees one and two up to symplectic isotopy. Although further work extended this to classify smooth symplectic surfaces up to degree 17 [She00, Sik03, ST05], in degrees greater than or equal to 18 the question remains open.

**QUESTION 1.1** (Symplectic isotopy problem). Is every smooth symplectic surface in $\mathbb{CP}^2$ symplectically isotopic to a complex curve?

This question is equivalent to asking, is there a unique symplectic isotopy class of symplectic surfaces in each degree? The equivalence results from the fact that the moduli space of smooth complex curves of fixed degree is connected. (The space of all degree-$d$ curves is parameterized by the coefficients of the defining polynomial, and singular curves have positive complex dimension.) Because of the difficulty of the classification for smooth symplectic curves, there has been somewhat limited work on singular symplectic curves [McD92a, Bar99, IS99, She04, Fra05]. Nevertheless, the study of singular symplectic curves in $\mathbb{CP}^2$ has significant ramifications for the study of all symplectic 4-manifolds via branched covering constructions [Aur00, Aur06].

Our first result addresses curves with a single cusp singularity. Such curves exist in every degree $d \ge 3$, so we are not bound by the uniform degree constraints encountered in attempts to answer Question 1.1. Note that the degrees and singularities for possible complex curves of this type were classified in [FLMN07].

**THEOREM 1.2.** In $\mathbb{CP}^2$, every symplectic rational unicuspidal curve whose unique singularity is the cone on a torus knot is symplectically isotopic to a complex curve, and has a unique symplectic isotopy class.

As a corollary to our results, we see there are no Zariski pairs of rational unicuspidal curves with one Puiseux pair. To the best of the authors' knowledge, the literature does not contain the statement that there are no Zariski pairs for such complex algebraic curves. However, experts
believe this might follow from the negativity conjecture of [PP17, PP20] (which would prove the statement for arbitrary rational cuspidal curves of log general type), together with some analysis of direct classifications in the cases of log Kodaira dimension at most one.

A great deal of complexity arises when we consider rational curves with more than one cusp. Although there are basic restrictions relating the degree to the genus of the torus knots appearing as the links of the singularities, there are many combinations of cusp singularity types which cannot be realized by complex algebraic curves even though they satisfy these basic adjunctive requirements. We initiate exploration of multi-cuspidal curves in the symplectic setting with low-degree rational cuspidal curves. The corresponding problem in algebraic geometry has a long history. Complex algebraic rational cuspidal curves in degrees at most five were completely classified [Nam84] (see [Moe08] for a more modern exposition). The classification of which rational cuspidal curves of degree six can be realized complex algebraically was completed by Fenske [Fen99].

Here we look at symplectic curves up to degree five. Although we do show in § 7.3 that our methods can say something for curves of degree six, even those with significantly different properties than those appearing in degree five, we do not venture into the combinatorics to give a complete classification. In fact, already in degree five there is a great deal of complexity which could indicate how the symplectic category relates to the complex one. The techniques we develop to classify the variety of curves in degree five provide a model for how one could approach many other cases in higher degrees. In degree 5, there are 19 different possible combinations of different cusp types which satisfy the basic degree-genus restrictions. Of these 19, we show 9 cannot embed symplectically into any symplectic manifold, 8 embed into $\mathbb{CP}^2$ with a unique symplectic isotopy class, and the remaining 2 more only embed symplectically and relatively minimally into $\mathbb{CP}^2#4\overline{\mathbb{CP}}^2$. Comparing with known results on such algebraic curves we obtain the following theorem.

**THEOREM 1.3.** In $\mathbb{CP}^2$, every symplectic rational cuspidal curve of degree at most five is symplectically isotopic to a complex curve and has a unique symplectic isotopy class. In particular, there are no Zariski pairs for rational cuspidal curves of degree up to five.

As the first version of this paper appeared, the result has been extended to degrees six and seven by the first author and Kütle [GK22].

As mentioned above, our techniques in fact obtain much stronger results than classifications in $\mathbb{CP}^2$. For each of the rational cuspidal curves we consider, we in fact classify the existence and uniqueness of minimal symplectic embeddings into any closed symplectic 4-manifold (up to symplectomorphism and symplectic isotopy of the pair). Classifications up to symplectomorphism of the pair reduce to ambient symplectic isotopy statements in the case that the ambient manifold is $\mathbb{CP}^2$ by Gromov's result that the space of symplectomorphisms of $(\mathbb{CP}^2, \omega_{FS})$ is homotopy equivalent to $PU(3)$ [Gro85, MS12], so in particular is path-connected.

To each singular symplectic curve $C$, we associate a contact 3-manifold $(Y_C, \xi_C)$ which appears on the boundary of a concave neighborhood of the curve. Our classifications of symplectic embeddings of the curves provide classifications of all symplectic fillings of the contact manifolds associated with our curves. Thus, we obtain new examples of contact manifolds which have no (strong) symplectic fillings, unique symplectic fillings, and non-unique but finitely many symplectic fillings that are classified. Previous complete classifications of symplectic fillings have primarily been restricted to lens spaces [Lis08, PV10], Seifert fibered spaces [OO05, Sta15], and torus bundles [GL16]. Our contact 3-manifolds are more general graph manifolds which do not fall in any of these previous classes.
believe this might follow from the negativity conjecture of [PP17, PP20] (which would prove the statement for arbitrary rational cuspidal curves of log general type), together with some analysis of direct classifications in the cases of log Kodaira dimension at most one.

A great deal of complexity arises when we consider rational curves with more than one cusp. Although there are basic restrictions relating the degree to the genus of the torus knots appearing as the links of the singularities, there are many combinations of cusp singularity types which cannot be realized by complex algebraic curves even though they satisfy these basic adjunctive requirements. We initiate exploration of multi-cuspidal curves in the symplectic setting with low-degree rational cuspidal curves. The corresponding problem in algebraic geometry has a long history. Complex algebraic rational cuspidal curves in degrees at most five were completely classified [Nam84] (see [Moe08] for a more modern exposition). The classification of which rational cuspidal curves of degree six can be realized complex algebraically was completed by Fenske [Fen99].

Here we look at symplectic curves up to degree five. Although we do show in § 7.3 that our methods can say something for curves of degree six, even those with significantly different properties than those appearing in degree five, we do not venture into the combinatorics to give a complete classification. In fact, already in degree five there is a great deal of complexity which could indicate how the symplectic category relates to the complex one. The techniques we develop to classify the variety of curves in degree five provide a model for how one could approach many other cases in higher degrees. In degree 5, there are 19 different possible combinations of different cusp types which satisfy the basic degree-genus restrictions. Of these 19, we show 9 cannot embed symplectically into any symplectic manifold, 8 embed into $\mathbb{CP}^2$ with a unique symplectic isotopy class, and the remaining 2 more only embed symplectically and relatively minimally into $\mathbb{CP}^2#4\overline{\mathbb{CP}}^2$. Comparing with known results on such algebraic curves we obtain the following theorem.

**THEOREM 1.3.** In $\mathbb{CP}^2$, every symplectic rational cuspidal curve of degree at most five is symplectically isotopic to a complex curve and has a unique symplectic isotopy class. In particular, there are no Zariski pairs for rational cuspidal curves of degree up to five.

As the first version of this paper appeared, the result has been extended to degrees six and seven by the first author and Kütle [GK22].

As mentioned above, our techniques in fact obtain much stronger results than classifications in $\mathbb{CP}^2$. For each of the rational cuspidal curves we consider, we in fact classify the existence and uniqueness of minimal symplectic embeddings into any closed symplectic 4-manifold (up to symplectomorphism and symplectic isotopy of the pair). Classifications up to symplectomorphism of the pair reduce to ambient symplectic isotopy statements in the case that the ambient manifold is $\mathbb{CP}^2$ by Gromov's result that the space of symplectomorphisms of $(\mathbb{CP}^2, \omega_{FS})$ is homotopy equivalent to $PU(3)$ [Gro85, MS12], so in particular is path-connected.

To each singular symplectic curve $C$, we associate a contact 3-manifold $(Y_C, \xi_C)$ which appears on the boundary of a concave neighborhood of the curve. Our classifications of symplectic embeddings of the curves provide classifications of all symplectic fillings of the contact manifolds associated with our curves. Thus, we obtain new examples of contact manifolds which have no (strong) symplectic fillings, unique symplectic fillings, and non-unique but finitely many symplectic fillings that are classified. Previous complete classifications of symplectic fillings have primarily been restricted to lens spaces [Lis08, PV10], Seifert fibered spaces [OO05, Sta15], and torus bundles [GL16]. Our contact 3-manifolds are more general graph manifolds which do not fall in any of these previous classes.
M. GOLLA AND L. STARKSTON

One question we had early on in our investigations, was whether symplectic embeddability of a certain type of rational cuspidal curve could distinguish the complex projective plane from a potential exotic or fake copy of $CP^2$. We found that this cannot be the case.

**THEOREM 1.4.** If $X$ is a rational homology $CP^2$ that contains a rational cuspidal curve, then $X$ is symplectomorphic to $CP^2$.

A similar question was raised by Chen [Che18], who asked which symplectic 4-manifolds with the same rational homology as $CP^2$ could be split along a contact-type hypersurface into two pieces, one of which was a rational homology ball.

We also prove a number of existence and uniqueness results for reducible configurations whose components are rational. These results play a role in proving the results for irreducible rational cuspidal curves. We summarize here some examples of these results.

**THEOREM 1.5.** Any configuration of symplectic conics and lines of total degree at most five either has a unique symplectic isotopy class or is obstructed in § 5.2. There is a unique symplectic isotopy class for each of the infinitely many configurations composed of one rational degree-d curve $C$ with a singular point with multiplicity sequence $[d-1]$ together with a line which is tangent to $C$ of order $d$.

Two more complicated configurations of two conics with additional lines appear in Figures 11 and 12 and with further techniques we prove they also have unique symplectic isotopy classes.

To prove these results and relate them to each other, we develop a symplectic theory of birational geometry for pairs $(X^4, C)$ where $X$ is a symplectic 4-manifold and $C$ is a (potentially singular) symplectic curve. We apply this, along with pseudoholomorphic curve techniques, to give symplectic classifications of many reducible configurations of curves in $CP^2$. We define symplectic proper transforms of curves, and see that the outcome is only well-defined *up to symplectic isotopy* (meaning a related through a smoothly varying equisingular family of symplectic curves, not necessarily related by an ambient isotopy). Because we use symplectic blow-ups and blow-downs throughout the paper, most of our statements are true only up to symplectic isotopy. The theory of symplectic birational geometry is based in fundamental work of McDuff on symplectic blow-up and blow-down. We develop this theory for ‘log’ pairs $(X^4, C^2)$ to study singular symplectic curves and their isotopy classifications. Although symplectic geometry does not have the full strength of algebraic geometry’s log minimal model program, by combining this symplectic birational geometry with pseudoholomorphic techniques and topological tools, we are able to prove many new results that have not been addressed with pseudoholomorphic curves alone.

*Organization.* The paper is organized as follows. Section 2 defines singular symplectic curves and their equivalences, provides background on complex curve singularities and their resolutions, defines the contact manifold associated to a singular curve, and reviews previous topological obstructions to rational cuspidal curves. Section 3 develops the tools utilizing pseudoholomorphic curves and birational geometry that we constantly use throughout the paper. In particular, §§ 3.1 and 3.4 contain the crucial definitions of symplectic proper transform, and of birational derivation and birational equivalence of configurations of curves, respectively. Section 4 proves Theorem 1.4. Section 5 studies symplectic isotopy problems for reducible configurations, which will be used to study the isotopy problem for rational cuspidal curves. Sections 6 and 7 give the proofs of Theorems 1.2 and 1.3, respectively. In § 8 we give an example, communicated to us by Stepan Orevkov, of a symplectic rational (non-cuspidal) curve that is not isotopic to any complex curve. The appendix contains a result about rational homology ball symplectic fillings of lens spaces that we use in the proof of Theorem 1.2.
One question we had early on in our investigations, was whether symplectic embeddability of a certain type of rational cuspidal curve could distinguish the complex projective plane from a potential exotic or fake copy of $CP^2$. We found that this cannot be the case.

**THEOREM 1.4.** If $X$ is a rational homology $CP^2$ that contains a rational cuspidal curve, then $X$ is symplectomorphic to $CP^2$.

A similar question was raised by Chen [Che18], who asked which symplectic 4-manifolds with the same rational homology as $CP^2$ could be split along a contact-type hypersurface into two pieces, one of which was a rational homology ball.

We also prove a number of existence and uniqueness results for reducible configurations whose components are rational. These results play a role in proving the results for irreducible rational cuspidal curves. We summarize here some examples of these results.

**THEOREM 1.5.** Any configuration of symplectic conics and lines of total degree at most five either has a unique symplectic isotopy class or is obstructed in § 5.2. There is a unique symplectic isotopy class for each of the infinitely many configurations composed of one rational degree-d curve $C$ with a singular point with multiplicity sequence $[d-1]$ together with a line which is tangent to $C$ of order $d$.

Two more complicated configurations of two conics with additional lines appear in Figures 11 and 12 and with further techniques we prove they also have unique symplectic isotopy classes.

To prove these results and relate them to each other, we develop a symplectic theory of birational geometry for pairs $(X^4, C)$ where $X$ is a symplectic 4-manifold and $C$ is a (potentially singular) symplectic curve. We apply this, along with pseudoholomorphic curve techniques, to give symplectic classifications of many reducible configurations of curves in $CP^2$. We define symplectic proper transforms of curves, and see that the outcome is only well-defined *up to symplectic isotopy* (meaning a related through a smoothly varying equisingular family of symplectic curves, not necessarily related by an ambient isotopy). Because we use symplectic blow-ups and blow-downs throughout the paper, most of our statements are true only up to symplectic isotopy. The theory of symplectic birational geometry is based in fundamental work of McDuff on symplectic blow-up and blow-down. We develop this theory for ‘log’ pairs $(X^4, C^2)$ to study singular symplectic curves and their isotopy classifications. Although symplectic geometry does not have the full strength of algebraic geometry’s log minimal model program, by combining this symplectic birational geometry with pseudoholomorphic techniques and topological tools, we are able to prove many new results that have not been addressed with pseudoholomorphic curves alone.

*Organization.* The paper is organized as follows. Section 2 defines singular symplectic curves and their equivalences, provides background on complex curve singularities and their resolutions, defines the contact manifold associated to a singular curve, and reviews previous topological obstructions to rational cuspidal curves. Section 3 develops the tools utilizing pseudoholomorphic curves and birational geometry that we constantly use throughout the paper. In particular, §§ 3.1 and 3.4 contain the crucial definitions of symplectic proper transform, and of birational derivation and birational equivalence of configurations of curves, respectively. Section 4 proves Theorem 1.4. Section 5 studies symplectic isotopy problems for reducible configurations, which will be used to study the isotopy problem for rational cuspidal curves. Sections 6 and 7 give the proofs of Theorems 1.2 and 1.3, respectively. In § 8 we give an example, communicated to us by Stepan Orevkov, of a symplectic rational (non-cuspidal) curve that is not isotopic to any complex curve. The appendix contains a result about rational homology ball symplectic fillings of lens spaces that we use in the proof of Theorem 1.2.
## 2. Symplectic singular curves

In this section, we start by reviewing some basic facts about singular complex curves. A good reference for the material we cover here is [Wal04]. Then we give our definitions of the symplectic analogues.

Let $C$ in $\mathbb{C}^2$ be the zero set of an analytic function $F(x, y)$, such that $F(0, 0) = 0$; we say that $C$ is singular at the origin if $(\partial F/\partial x)(0, 0) = (\partial F/\partial y)(0, 0) = 0$. We suppose that $F$ is locally irreducible, i.e. irreducible in the ring $\mathbb{C}[[x, y]]$ of power series in two variables; in this case, we say that the singularity is a *cusp*. (Note that certain authors call cusps only the singularities of type (2, 3); we refer to the latter as *simple cusps*.)

Up to diffeomorphism, $C$ can be parametrized (in a neighborhood of the origin) as the image of $\phi: \mathbb{C} \to \mathbb{C}^2$ defined by $\phi(t) = (t^m, a_1 t^{b_1} + \cdots + a_k t^{b_k})$ for some positive integers $m < b_1 < \cdots < b_k$ and some $a_1, \ldots, a_k \in \mathbb{C}^*$. Moreover, if we require that the sequence $e_i$ defined by $e_1 = \gcd(m, b_1)$, $e_i = \gcd(e_{i-1}, b_i)$ is strictly decreasing, the exponents $m, b_i$ are uniquely determined. In this case, $\phi$ is the Puiseux parametrization of $C$, and $(m; b_1, \ldots, b_k)$ are the *Puiseux exponents* of $C$; $m$ is called the *multiplicity* of the singularity.

**Remark 2.1.** Note that with the above coordinates from the Puiseux parametrization, the cusp curve $C$ has a unique tangent line $\{y = 0\}$. This is the limit of the tangent lines of the nearby smooth points as $dx/dt = mt^{m-1}$, $dy/dt = b_1a_1t^{b_1-1} + \cdots$, so $dy/dx \to 0$ as $t \to 0$ because $b_1 > m$. For a general complex curve singularity, there are a finitely many locally irreducible branches, each with a unique complex tangent line. The multiplicity of intersection of the tangent line at a cusp point is $b_1$ (the solutions to setting $y = 0$), whereas a line which is not tangent to the cusp will intersect the cusp with multiplicity $m$ (the solution to setting $y = ax = at^m$ for $a \neq 0$). See [Wal04, Section 2.3] for more details.

Recall that the *link* of the singularity is the diffeomorphism type of $(S_\varepsilon \cap C, S_\varepsilon)$, where $S_\varepsilon = \partial B_\varepsilon$ is the boundary of a ball of radius $\varepsilon \ll 1$. The fact that the singularity of $C$ at the origin is irreducible translates into the condition that the link is a knot (i.e. has one component).

The Puiseux exponents determine the topology of the singularity: two singularities have the same Puiseux exponents if and only if their links are diffeomorphic [Wal04, Proposition 5.3.1].

In this paper, we are almost exclusively concerned with singularities whose Puiseux expansion has $k = 1$, i.e. it is of the form $(m; b_1)$; we say that the *singularity is of type (m, b₁)*. Note that $m < b_1$; however, the link of a singularity of type $(m, b_1)$, which is the torus knot $T(m, b_1)$, is isotopic to the torus knot $T(b_1, m)$.

### 2.1 Resolution of singularities

Recall from [Wal04] that every curve singularity can be resolved by blowing up (sufficiently many times), and the diffeomorphism type of the link determines the topology of the *resolution*. If $\pi: \tilde{X} \to X$ is a (multiple) blow-up at a point $x \in X$, and $C \subset X$ is a curve, we call $\pi^{-1}(C \setminus \{x\})$ the *proper transform* of $C$ under $\pi$, and we denote it by $\tilde{C}$; we also call $\pi^{-1}(C)$ the *total transform* of $C$ under $\pi$, and we denote it by $\bar{C}$.

For every singular curve $C \subset X$, there exists a composition of blow-ups $\pi: \tilde{X} \to X$ such that $\tilde{C}$ is smooth; we call any such $\pi$ a *resolution of C*.

Given a resolution $\pi$ of $C$, consider the restriction $n := \pi|_{\tilde{C}}$ of $\pi$ to the proper transform $\tilde{C}$ of $C$. We call $n: \tilde{C} \to C$ (and, by abuse of notation, $\tilde{C}$) the *normalization* of $C$. Note that this is essentially independent of the choice of the resolution: if $\pi'$ is another resolution of $C$ and we construct $n'$ accordingly, then there is an isomorphism $\phi$ between the sources of $n$ and $n'$ such
## 2. Symplectic singular curves

In this section, we start by reviewing some basic facts about singular complex curves. A good reference for the material we cover here is [Wal04]. Then we give our definitions of the symplectic analogues.

Let $C$ in $\mathbb{C}^2$ be the zero set of an analytic function $F(x, y)$, such that $F(0, 0) = 0$; we say that $C$ is singular at the origin if $(\partial F/\partial x)(0, 0) = (\partial F/\partial y)(0, 0) = 0$. We suppose that $F$ is locally irreducible, i.e. irreducible in the ring $\mathbb{C}[[x, y]]$ of power series in two variables; in this case, we say that the singularity is a *cusp*. (Note that certain authors call cusps only the singularities of type (2, 3); we refer to the latter as *simple cusps*.)

Up to diffeomorphism, $C$ can be parametrized (in a neighborhood of the origin) as the image of $\phi: \mathbb{C} \to \mathbb{C}^2$ defined by $\phi(t) = (t^m, a_1 t^{b_1} + \cdots + a_k t^{b_k})$ for some positive integers $m < b_1 < \cdots < b_k$ and some $a_1, \ldots, a_k \in \mathbb{C}^*$. Moreover, if we require that the sequence $e_i$ defined by $e_1 = \gcd(m, b_1)$, $e_i = \gcd(e_{i-1}, b_i)$ is strictly decreasing, the exponents $m, b_i$ are uniquely determined. In this case, $\phi$ is the Puiseux parametrization of $C$, and $(m; b_1, \ldots, b_k)$ are the *Puiseux exponents* of $C$; $m$ is called the *multiplicity* of the singularity.

**Remark 2.1.** Note that with the above coordinates from the Puiseux parametrization, the cusp curve $C$ has a unique tangent line $\{y = 0\}$. This is the limit of the tangent lines of the nearby smooth points as $dx/dt = mt^{m-1}$, $dy/dt = b_1a_1t^{b_1-1} + \cdots$, so $dy/dx \to 0$ as $t \to 0$ because $b_1 > m$. For a general complex curve singularity, there are a finitely many locally irreducible branches, each with a unique complex tangent line. The multiplicity of intersection of the tangent line at a cusp point is $b_1$ (the solutions to setting $y = 0$), whereas a line which is not tangent to the cusp will intersect the cusp with multiplicity $m$ (the solution to setting $y = ax = at^m$ for $a \neq 0$). See [Wal04, Section 2.3] for more details.

Recall that the *link* of the singularity is the diffeomorphism type of $(S_\varepsilon \cap C, S_\varepsilon)$, where $S_\varepsilon = \partial B_\varepsilon$ is the boundary of a ball of radius $\varepsilon \ll 1$. The fact that the singularity of $C$ at the origin is irreducible translates into the condition that the link is a knot (i.e. has one component).

The Puiseux exponents determine the topology of the singularity: two singularities have the same Puiseux exponents if and only if their links are diffeomorphic [Wal04, Proposition 5.3.1].

In this paper, we are almost exclusively concerned with singularities whose Puiseux expansion has $k = 1$, i.e. it is of the form $(m; b_1)$; we say that the *singularity is of type (m, b₁)*. Note that $m < b_1$; however, the link of a singularity of type $(m, b_1)$, which is the torus knot $T(m, b_1)$, is isotopic to the torus knot $T(b_1, m)$.

### 2.1 Resolution of singularities

Recall from [Wal04] that every curve singularity can be resolved by blowing up (sufficiently many times), and the diffeomorphism type of the link determines the topology of the *resolution*. If $\pi: \tilde{X} \to X$ is a (multiple) blow-up at a point $x \in X$, and $C \subset X$ is a curve, we call $\pi^{-1}(C \setminus \{x\})$ the *proper transform* of $C$ under $\pi$, and we denote it by $\tilde{C}$; we also call $\pi^{-1}(C)$ the *total transform* of $C$ under $\pi$, and we denote it by $\bar{C}$.

For every singular curve $C \subset X$, there exists a composition of blow-ups $\pi: \tilde{X} \to X$ such that $\tilde{C}$ is smooth; we call any such $\pi$ a *resolution of C*.

Given a resolution $\pi$ of $C$, consider the restriction $n := \pi|_{\tilde{C}}$ of $\pi$ to the proper transform $\tilde{C}$ of $C$. We call $n: \tilde{C} \to C$ (and, by abuse of notation, $\tilde{C}$) the *normalization* of $C$. Note that this is essentially independent of the choice of the resolution: if $\pi'$ is another resolution of $C$ and we construct $n'$ accordingly, then there is an isomorphism $\phi$ between the sources of $n$ and $n'$ such
<figure><img src="image_8.png" /><figcaption>FIGURE 1. A handle diagram interpretation of the blow-up of a singularity of type $(p, q)$. (Recall that $p < q$ in a singularity of type $(p, q)$.) There are $p$ strands on each side, and the rectangular box with the +1 denotes a full twist.</figcaption></figure>

that $n = n' \circ \phi$. We also observe that $C$ is cuspidal if and only if the normalization is one-to-one; in particular, in this case it is a homeomorphism onto its image.

There are two natural stopping points when resolving a singularity: the *minimal resolution* is the smallest resolution such that the proper transform $\tilde{C}$ of $C$ is smooth; the *normal crossing resolution* is the smallest resolution such that the total transform $\bar{C}$ of $C$ is a normal crossing divisor, i.e. all singularities are double points.<fnref n="1" />

Recall that the *multiplicity* $m_p$ of a singularity $p$ of $C \subset X$ is the minimal intersection of a germ of a divisor $D$ at $p$ with $C$; that is, $m_p = \min_D(C \cdot D)_p$; the multiplicity of a singularity has the following interpretation: blow up $X$ at $p$, obtaining $\tilde{X}$, which contains the corresponding exceptional divisor $E$; then $m_p$ is the intersection number of $E$ and $\tilde{C}$, i.e. $m_p = \tilde{C} \cdot E$. In terms of homology, there is an orthogonal decomposition $H_2(\tilde{X}) = H_2(X) \oplus \mathbb{Z}[E]$, and we have $[\tilde{C}] = [C] - m_p[E]$. In particular, $[\tilde{C}]^2 = [C]^2 - m_p^2$. See Figure 1 for a Kirby calculus interpretation of the blow-up at a cusp with multiplicity $p$.

We record the singularity types at each step in the minimal resolution into a sequence of integers, called the *multiplicity sequence*, as follows. Suppose that the minimal resolution of a singularity of $C$ is a sequence of $k$ blow-ups, and call $C_j$ the proper transform of $C$ after the first $j-1$ blow-ups, so that $C_1 = C$, and $C_{k+1}$ is the proper transform of $C$ in the minimal resolution. Then $C_j$ has a singularity of multiplicity $m_j$, and we write $[m_1, \dots, m_k]$ for the multiplicity sequence of the singularity of $C$. Note that $m_{j+1} \le m_j$ for each $j$, but that not every sequence of integers correspond to the multiplicity sequence of a singularity; for instance, $[3, 2, 2]$ is not the multiplicity sequence of any singularity. The multiplicity sequence of an irreducible singularity determines the singularity [Wal04, Theorem 3.5.6]; however, the multiplicity sequence is defined both for irreducible and reducible singularities.

**Remark 2.2.** Algebraic geometers often consider the multiplicity sequence associated with the normal crossing divisors resolution of the singularity, rather than that associated with the minimal resolution. In fact, one determines the other: if the latter ends in $m_k > 1$, the former coincides with it until the $k$th entry, and then ends with a sequence of 1s of length $m_k$. These last entries

<footnote n="1">¹ The terminology in the log algebraic geometry community (e.g. in [KP17, PP17, PP20, KP22]) seems to by slightly different: they call *minimal weak resolution* (or, in earlier papers, *minimal embedded resolution*) what we call minimal resolution, and *minimal log resolution* what we call normal crossing resolution.</footnote>
<figure><img src="image_8.png" /><figcaption>FIGURE 1. A handle diagram interpretation of the blow-up of a singularity of type $(p, q)$. (Recall that $p < q$ in a singularity of type $(p, q)$.) There are $p$ strands on each side, and the rectangular box with the +1 denotes a full twist.</figcaption></figure>

that $n = n' \circ \phi$. We also observe that $C$ is cuspidal if and only if the normalization is one-to-one; in particular, in this case it is a homeomorphism onto its image.

There are two natural stopping points when resolving a singularity: the *minimal resolution* is the smallest resolution such that the proper transform $\tilde{C}$ of $C$ is smooth; the *normal crossing resolution* is the smallest resolution such that the total transform $\bar{C}$ of $C$ is a normal crossing divisor, i.e. all singularities are double points.<fnref n="1" />

Recall that the *multiplicity* $m_p$ of a singularity $p$ of $C \subset X$ is the minimal intersection of a germ of a divisor $D$ at $p$ with $C$; that is, $m_p = \min_D(C \cdot D)_p$; the multiplicity of a singularity has the following interpretation: blow up $X$ at $p$, obtaining $\tilde{X}$, which contains the corresponding exceptional divisor $E$; then $m_p$ is the intersection number of $E$ and $\tilde{C}$, i.e. $m_p = \tilde{C} \cdot E$. In terms of homology, there is an orthogonal decomposition $H_2(\tilde{X}) = H_2(X) \oplus \mathbb{Z}[E]$, and we have $[\tilde{C}] = [C] - m_p[E]$. In particular, $[\tilde{C}]^2 = [C]^2 - m_p^2$. See Figure 1 for a Kirby calculus interpretation of the blow-up at a cusp with multiplicity $p$.

We record the singularity types at each step in the minimal resolution into a sequence of integers, called the *multiplicity sequence*, as follows. Suppose that the minimal resolution of a singularity of $C$ is a sequence of $k$ blow-ups, and call $C_j$ the proper transform of $C$ after the first $j-1$ blow-ups, so that $C_1 = C$, and $C_{k+1}$ is the proper transform of $C$ in the minimal resolution. Then $C_j$ has a singularity of multiplicity $m_j$, and we write $[m_1, \dots, m_k]$ for the multiplicity sequence of the singularity of $C$. Note that $m_{j+1} \le m_j$ for each $j$, but that not every sequence of integers correspond to the multiplicity sequence of a singularity; for instance, $[3, 2, 2]$ is not the multiplicity sequence of any singularity. The multiplicity sequence of an irreducible singularity determines the singularity [Wal04, Theorem 3.5.6]; however, the multiplicity sequence is defined both for irreducible and reducible singularities.

**Remark 2.2.** Algebraic geometers often consider the multiplicity sequence associated with the normal crossing divisors resolution of the singularity, rather than that associated with the minimal resolution. In fact, one determines the other: if the latter ends in $m_k > 1$, the former coincides with it until the $k$th entry, and then ends with a sequence of 1s of length $m_k$. These last entries

<footnote n="1">¹ The terminology in the log algebraic geometry community (e.g. in [KP17, PP17, PP20, KP22]) seems to by slightly different: they call *minimal weak resolution* (or, in earlier papers, *minimal embedded resolution*) what we call minimal resolution, and *minimal log resolution* what we call normal crossing resolution.</footnote>
corresponds to the blow-ups needed to make the last exceptional divisor in the minimal smooth resolution disjoint from the proper transform of the curve.

Given a cuspidal curve C, we say that the multiplicity multi-sequence of C is the union of all multiplicity sequences of singularities of C. This is only well-defined up to choosing an order of the cusps. Instead of making the choice, we just sort all the entries decreasingly. We use double-bracket notation to denote the multiplicity multi-sequence: [[m₁, ..., mₙ]].

**Example 2.3.** The multiplicity of a singularity whose link is the torus knot T(p, q) is min{p, q}. For a singularity of type (p, p + 1), one blow-up already yields a smooth curve, and so the multiplicity sequence is [p]. In general, after blowing up once at a singularity of type (p, q), the resulting curve has a singularity whose link is T(p, q - p) (and, therefore, is of type (p, q - p) or (q - p, p), depending on whether q is larger or smaller than 2p). Iterating this process, one eventually reaches a smooth curve.

The following are the singularities that will appear in our examples.

(i) For a singularity of type (p, kp + 1), the multiplicity is p, and singularity of the proper transform in the first blow-up is of type (p, (k - 1)p + 1). It follows inductively that the multiplicity sequence is the string of length k, [p, ..., p]. In particular, a curve with a singularity of type (p, 2p + 1) and a curve with two singularities of type (p, p + 1) have the same multiplicity multi-sequence, [[p, p]].

(ii) A singularity of type (p + 1, 2p + 1) has multiplicity sequence [p + 1, p]. In particular the singularity of type (3, 5) has multiplicity sequence [3, 2].

(iii) A singularity of type (p, 4p - 1) has multiplicity sequence [p, p, p, p - 1] (p ≥ 3).

Here we describe the normal crossing resolution of singularities of type (p, q). This is best described in terms of continued fraction expansions; given a rational number r, we write

$$r = [a_1, \dots, a_k]^- = a_1 - \frac{1}{a_2 - \frac{1}{\dots - \frac{1}{a_k}}}$$

for its (negative or Hirzebruch-Jung) continued fraction expansion, where $a_i \ge 2$ for each $i \ge 2$.

In what follows, we adopt the notation $a^{[\ell]}$ to denote a string of $\ell$ entries, all equal to a; for instance, $[2^{[\ell]}]$ will be the multiplicity sequence of the singularity of type (2, 2$\ell$ + 1).

It consists of a plumbing of spheres along a three-legged star-shaped graph decorated by Euler classes. The Euler class on the central vertex is -1; that is, the central vertex is the exceptional divisor corresponding to the last blow-up in the resolution. Two of the legs are linear chains whose decorations give the continued fraction expansions of

$$\frac{p}{p - q^*} \quad \text{and} \quad \frac{q}{q - p^*},$$

where $qq^* \equiv 1 \mod p$ and $0 < q^* < p$, and $pp^* \equiv 1 \mod q$ and $0 < p^* < q$.

More precisely, the resolution graph of the singularity of type (p, q) looks like the following:

<figure><img src="image_9.png" /></figure>
corresponds to the blow-ups needed to make the last exceptional divisor in the minimal smooth resolution disjoint from the proper transform of the curve.

Given a cuspidal curve $C$, we say that the multiplicity multi-sequence of $C$ is the union of all multiplicity sequences of singularities of $C$. This is only well-defined up to choosing an order of the cusps. Instead of making the choice, we just sort all the entries decreasingly. We use double-bracket notation to denote the multiplicity multi-sequence: [[$m_1, \dots, m_n$]].

**Example 2.3.** The multiplicity of a singularity whose link is the torus knot $T(p, q)$ is $\min\{p, q\}$. For a singularity of type $(p, p+1)$, one blow-up already yields a smooth curve, and so the multiplicity sequence is $[p]$. In general, after blowing up once at a singularity of type $(p, q)$, the resulting curve has a singularity whose link is $T(p, q-p)$ (and, therefore, is of type $(p, q-p)$ or $(q-p, p)$, depending on whether $q$ is larger or smaller than $2p$). Iterating this process, one eventually reaches a smooth curve.

The following are the singularities that will appear in our examples.

(i) For a singularity of type $(p, kp+1)$, the multiplicity is $p$, and singularity of the proper transform in the first blow-up is of type $(p, (k-1)p+1)$. It follows inductively that the multiplicity sequence is the string of length $k$, $[p, \dots, p]$. In particular, a curve with a singularity of type $(p, 2p+1)$ and a curve with two singularities of type $(p, p+1)$ have the same multiplicity multi-sequence, $[[p, p]]$.

(ii) A singularity of type $(p+1, 2p+1)$ has multiplicity sequence $[p+1, p]$. In particular the singularity of type $(3, 5)$ has multiplicity sequence $[3, 2]$.

(iii) A singularity of type $(p, 4p-1)$ has multiplicity sequence $[p, p, p, p-1]$ ($p \ge 3$).

Here we describe the normal crossing resolution of singularities of type $(p, q)$. This is best described in terms of continued fraction expansions; given a rational number $r$, we write

$$r = [a_1, \dots, a_k]^- = a_1 - \frac{1}{a_2 - \frac{1}{\dots - \frac{1}{a_k}}}$$

for its (negative or Hirzebruch-Jung) continued fraction expansion, where $a_i \ge 2$ for each $i \ge 2$.

In what follows, we adopt the notation $a^{[\ell]}$ to denote a string of $\ell$ entries, all equal to $a$; for instance, $[2^{[\ell]}]$ will be the multiplicity sequence of the singularity of type $(2, 2\ell+1)$.

It consists of a plumbing of spheres along a three-legged star-shaped graph decorated by Euler classes. The Euler class on the central vertex is $-1$; that is, the central vertex is the exceptional divisor corresponding to the last blow-up in the resolution. Two of the legs are linear chains whose decorations give the continued fraction expansions of

$$\frac{p}{p-q^*} \quad \text{and} \quad \frac{q}{q-p^*},$$

where $qq^* \equiv 1 \mod p$ and $0 < q^* < p$, and $pp^* \equiv 1 \mod q$ and $0 < p^* < q$.

More precisely, the resolution graph of the singularity of type $(p, q)$ looks like the following:

<figure><img src="image_9.png" /></figure>
where $p/(p-q^*) = [a_1, \dots, a_m]^{-}$ and $q/(q-p^*) = [b_1, \dots, b_n]^{-}$, and the hollow dot represents the proper transform of $C$. (It might be helpful to recall that $p/(p-q) = [a_m, \dots, a_1]^{-}$ and $q/(q-r(p,q)) = [b_n, \dots, b_1]^{-}$, where $r(p,q)$ is the remainder of the division of $p$ by $q$.)

The following lemma is well-known among algebraic geometers; we include a proof both for completeness and to give an example of the techniques for low-dimensional topologists.

**LEMMA 2.4.** Suppose $C$ is a singular curve in $X^4$ with a singularity of type $(p, q)$ at $x_0$. Consider the normal crossing divisor resolution of the singularity of $C$ at $x_0$ in $\tilde{X}$. Then the proper transform $\tilde{C}$ of $C$ in $\tilde{X}$ satisfies $\tilde{C} \cdot \tilde{C} = C \cdot C - pq$.

*Proof.* We prove the result by induction on the length $\ell$ of the multiplicity sequence of the singularity.

If $\ell = 1$, the singularity is of type $(p, p+1)$ for some $p$, and the first blow-up already gives the minimal smooth resolution. The total transform of $C$ in the minimal resolution consists of the proper transform of $C$ and the exceptional divisor, and the two have a tangency of order $p$; as observed previously, the proper transform has self-intersection $C \cdot C - p^2$. To get to the normal crossing divisor resolution, we blow up $p$ times at the tangency; because the proper transform of $C$ was already smooth, each of these blow-ups decrease the self-intersection by one, hence $\tilde{C} \cdot \tilde{C} = C \cdot C - p^2 - p \cdot 1 = C \cdot C - p(p+1)$, thus proving the base case.

If $\ell > 1$, blow up once at $x_0$; the singularity of the proper transform of $C$ will be of type $(p, q-p)$ (or, possibly, $(q-p, p)$), and the length of its multiplicity sequence decreases by 1 by definition; therefore $\tilde{C} \cdot \tilde{C} = C \cdot C - p^2 - p(q-p) = C \cdot C - pq$, as desired. $\square$

If $C$ is a curve of self-intersection $s$ with a unique singularity of type $(p, q)$, then the total transform of $C$ is the full plumbing graph, where the hollow dot is replaced by a vertex with Euler class $s - pq$. In our applications, in the case of a unique singularity, if $s - pq > 1$, we will blow up further times along the edge between the central vertex and this short leg to change the coefficient on the central vertex to $-2$ and add a chain of $s - pq - 2$ more $(-2)$-vertices, a single $(-1)$-vertex, and a single $+1$-vertex.

In general, we can resolve each singularity of $C$ independently of the others; furthermore, if all singularities are of type $(p_i, q_i)$, the total transform of $C$ will look like a plumbing tree, obtained by fusing the resolution graphs of the individual singularities along the hollow vertex, whose weight is $s - \sum_i p_i q_i$.

For instance, when $C$ is a rational curve with self-intersection 16, with a singularity of type (2,3) and one of type (2,5), the total transform $\bar{C}$ of $C$ in the normal crossing resolution of $C$ is described by the following plumbing diagram:

<figure><img src="image_10.png" /></figure>

Here the vertex labeled with $0 = 16 - 2 \cdot 3 - 2 \cdot 5$ is the proper transform $\tilde{C}$ of $C$, and is obtained by fusing the corresponding hollow vertices in the resolution diagrams for the two singularities of $C$.
where $p/(p-q^*) = [a_1, \dots, a_m]^{-}$ and $q/(q-p^*) = [b_1, \dots, b_n]^{-}$, and the hollow dot represents the proper transform of $C$. (It might be helpful to recall that $p/(p-q) = [a_m, \dots, a_1]^{-}$ and $q/(q-r(p,q)) = [b_n, \dots, b_1]^{-}$, where $r(p,q)$ is the remainder of the division of $p$ by $q$.)

The following lemma is well-known among algebraic geometers; we include a proof both for completeness and to give an example of the techniques for low-dimensional topologists.

**LEMMA 2.4.** Suppose $C$ is a singular curve in $X^4$ with a singularity of type $(p, q)$ at $x_0$. Consider the normal crossing divisor resolution of the singularity of $C$ at $x_0$ in $\tilde{X}$. Then the proper transform $\tilde{C}$ of $C$ in $\tilde{X}$ satisfies $\tilde{C} \cdot \tilde{C} = C \cdot C - pq$.

*Proof.* We prove the result by induction on the length $\ell$ of the multiplicity sequence of the singularity.

If $\ell = 1$, the singularity is of type $(p, p+1)$ for some $p$, and the first blow-up already gives the minimal smooth resolution. The total transform of $C$ in the minimal resolution consists of the proper transform of $C$ and the exceptional divisor, and the two have a tangency of order $p$; as observed previously, the proper transform has self-intersection $C \cdot C - p^2$. To get to the normal crossing divisor resolution, we blow up $p$ times at the tangency; because the proper transform of $C$ was already smooth, each of these blow-ups decrease the self-intersection by one, hence $\tilde{C} \cdot \tilde{C} = C \cdot C - p^2 - p \cdot 1 = C \cdot C - p(p+1)$, thus proving the base case.

If $\ell > 1$, blow up once at $x_0$; the singularity of the proper transform of $C$ will be of type $(p, q-p)$ (or, possibly, $(q-p, p)$), and the length of its multiplicity sequence decreases by 1 by definition; therefore $\tilde{C} \cdot \tilde{C} = C \cdot C - p^2 - p(q-p) = C \cdot C - pq$, as desired. $\square$

If $C$ is a curve of self-intersection $s$ with a unique singularity of type $(p, q)$, then the total transform of $C$ is the full plumbing graph, where the hollow dot is replaced by a vertex with Euler class $s - pq$. In our applications, in the case of a unique singularity, if $s - pq > 1$, we will blow up further times along the edge between the central vertex and this short leg to change the coefficient on the central vertex to $-2$ and add a chain of $s - pq - 2$ more $(-2)$-vertices, a single $(-1)$-vertex, and a single $+1$-vertex.

In general, we can resolve each singularity of $C$ independently of the others; furthermore, if all singularities are of type $(p_i, q_i)$, the total transform of $C$ will look like a plumbing tree, obtained by fusing the resolution graphs of the individual singularities along the hollow vertex, whose weight is $s - \sum_i p_i q_i$.

For instance, when $C$ is a rational curve with self-intersection 16, with a singularity of type (2,3) and one of type (2,5), the total transform $\bar{C}$ of $C$ in the normal crossing resolution of $C$ is described by the following plumbing diagram:

<figure><img src="image_10.png" /></figure>

Here the vertex labeled with $0 = 16 - 2 \cdot 3 - 2 \cdot 5$ is the proper transform $\tilde{C}$ of $C$, and is obtained by fusing the corresponding hollow vertices in the resolution diagrams for the two singularities of $C$.
## 2.2 Singular symplectic curves

We are now ready to define the main objects of interest in the paper.

**DEFINITION 2.5.** A singular symplectic curve $C$ in a symplectic 4-manifold $(X, \omega)$ is a subset $C \subset X$ such that:

(i) there exists a finite subset $\text{Sing}(C) \in C$ such that $C \setminus \text{Sing}(C)$ is a smooth symplectic submanifold of $X$;

(ii) every $q \in \text{Sing}(C)$ has an open neighborhood $U \subset X$, such that there exists a symplectomorphism $(U, C \cap U) \to (V, D \cap V)$, where $V \subset \mathbb{C}^2$, and $D$ is a complex algebraic curve.

We call $\text{Sing}(C)$ the set of singularities of $C$, $C \setminus \text{Sing}(C)$ the smooth part of $C$. We say that $C$ is cuspidal if all its singularities are cusps (i.e. locally irreducible).

As the singularities are locally modeled on complex curves, we have the resolutions of § 2.1. In particular, $C$ can be parameterized as a locally injective smooth image of a parameterization $f: \Sigma \to X$ which is an embedding away from the singular points ($\Sigma$ is the normalization). When this normalization $\Sigma$ is connected, the curve $C$ is *irreducible*. When $\Sigma$ is disconnected, $C$ is *reducible* and we consider it as a curve with *labeled components*.

When we want to refer to the abstract topological type of the singular curve, namely the labeled components and their homology classes, and the topological types of the singularities (encoded by the links of the singularities up to isotopy), we call it a configuration and denote it with script font. A specific *symplectic realization* $C$ of a configuration $C$ will be a singular symplectic curve with the homology and singularity data specified by $C$.

We restrict to singularities modeled by complex curves because these are the singularity types that can occur in pseudoholomorphic curves by two results of McDuff, that we summarize in the following theorem.

**THEOREM 2.6** (McDuff [McD92b]; see also Micallef and White [MW95]). Let $C$ be a $J$-holomorphic curve in an almost complex 4-manifold $(X, J)$.

(1) If $C$ is not multiply covered, there is a neighborhood $U$ of each of its singular points such that the pair $(U, U \cap C)$ is homeomorphic to the cone over $(S^3, K_x)$ where $K_x$ is an algebraic link in $S^3$ that depends only on the germ of $C$ at $x$.

(2) There are an almost complex structure $J_0$ and a $J_0$-holomorphic curve $C_0$, such that $J_0$ is integrable near each of the singular points of $C_0$ and $(X, C)$ is homeomorphic to $(X, C_0)$; moreover, $J_0$ can be chosen to be arbitrarily close to $J$ in the $C^0$-topology and $C_0$ can be chosen to be arbitrarily close to $C$ in the $C^1$-topology.

As the symplectic condition is $C^1$-open, it is preserved by $C^1$-small isotopies. It follows that any $J$-holomorphic singular curve for any $\omega$-compatible $J$ is $C^1$-close to a curve $C_0$ whose singular points have complex models. Since $C^1$-small isotopies do not affect questions of existence or uniqueness, we model singular symplectic curves with complex Darboux charts as given in Definition 2.5.

There are two different notions of genus for a singular curve $C$. Recall that every singularity can be perturbed; that is, it can be replaced with a Milnor fiber, locally defined by a perturbation of the equation for the singularity. The genus of the Milnor fiber is also the 3-genus of its link, and it is a topological invariant of the singularity. If $q$ is a singular point of $C$, with $r$ branches, we define $\mu(q)$, the *Milnor number* of $q$, as the first Betti number of the Milnor fiber of the singularity of $C$ at $q$; we define $\delta(q)$ by $2\delta(q) = \mu(q) + r - 1$. If $q$ is a cusp point of $C$, then $\delta(q)$ is just the genus of the Milnor fiber of the singularity.
## 2.2 Singular symplectic curves

We are now ready to define the main objects of interest in the paper.

**DEFINITION 2.5.** A singular symplectic curve $C$ in a symplectic 4-manifold $(X, \omega)$ is a subset $C \subset X$ such that:

(i) there exists a finite subset $\text{Sing}(C) \in C$ such that $C \setminus \text{Sing}(C)$ is a smooth symplectic submanifold of $X$;

(ii) every $q \in \text{Sing}(C)$ has an open neighborhood $U \subset X$, such that there exists a symplectomorphism $(U, C \cap U) \to (V, D \cap V)$, where $V \subset \mathbb{C}^2$, and $D$ is a complex algebraic curve.

We call $\text{Sing}(C)$ the set of singularities of $C$, $C \setminus \text{Sing}(C)$ the smooth part of $C$. We say that $C$ is cuspidal if all its singularities are cusps (i.e. locally irreducible).

As the singularities are locally modeled on complex curves, we have the resolutions of § 2.1. In particular, $C$ can be parameterized as a locally injective smooth image of a parameterization $f: \Sigma \to X$ which is an embedding away from the singular points ($\Sigma$ is the normalization). When this normalization $\Sigma$ is connected, the curve $C$ is *irreducible*. When $\Sigma$ is disconnected, $C$ is *reducible* and we consider it as a curve with *labeled components*.

When we want to refer to the abstract topological type of the singular curve, namely the labeled components and their homology classes, and the topological types of the singularities (encoded by the links of the singularities up to isotopy), we call it a configuration and denote it with script font. A specific *symplectic realization* $C$ of a configuration $C$ will be a singular symplectic curve with the homology and singularity data specified by $C$.

We restrict to singularities modeled by complex curves because these are the singularity types that can occur in pseudoholomorphic curves by two results of McDuff, that we summarize in the following theorem.

**THEOREM 2.6 (McDuff [McD92b]; see also Micallef and White [MW95]).** Let $C$ be a $J$-holomorphic curve in an almost complex 4-manifold $(X, J)$.

(1) If $C$ is not multiply covered, there is a neighborhood $U$ of each of its singular points such that the pair $(U, U \cap C)$ is homeomorphic to the cone over $(S^3, K_x)$ where $K_x$ is an algebraic link in $S^3$ that depends only on the germ of $C$ at $x$.

(2) There are an almost complex structure $J_0$ and a $J_0$-holomorphic curve $C_0$, such that $J_0$ is integrable near each of the singular points of $C_0$ and $(X, C)$ is homeomorphic to $(X, C_0)$; moreover, $J_0$ can be chosen to be arbitrarily close to $J$ in the $C^0$-topology and $C_0$ can be chosen to be arbitrarily close to $C$ in the $C^1$-topology.

As the symplectic condition is $C^1$-open, it is preserved by $C^1$-small isotopies. It follows that any $J$-holomorphic singular curve for any $\omega$-compatible $J$ is $C^1$-close to a curve $C_0$ whose singular points have complex models. Since $C^1$-small isotopies do not affect questions of existence or uniqueness, we model singular symplectic curves with complex Darboux charts as given in Definition 2.5.

There are two different notions of genus for a singular curve $C$. Recall that every singularity can be perturbed; that is, it can be replaced with a Milnor fiber, locally defined by a perturbation of the equation for the singularity. The genus of the Milnor fiber is also the 3-genus of its link, and it is a topological invariant of the singularity. If $q$ is a singular point of $C$, with $r$ branches, we define $\mu(q)$, the *Milnor number* of $q$, as the first Betti number of the Milnor fiber of the singularity of $C$ at $q$; we define $\delta(q)$ by $2\delta(q) = \mu(q) + r - 1$. If $q$ is a cusp point of $C$, then $\delta(q)$ is just the genus of the Milnor fiber of the singularity.
For the following definition, we borrow the terminology from algebraic geometry.

**DEFINITION 2.7.** Let $C$ be a singular symplectic curve. We call $p_g(C) := g(\Sigma)$ where $\Sigma$ is the normalization of $C$, the geometric genus of $C$, and we will say that $C$ is rational if its geometric genus is zero.

We call

$$p_a(C) := p_g(C) + \sum_{p \in \text{Sing}(C)} \delta(p) \qquad (2.1)$$

the arithmetic genus of $C$.

As $p_a(C)$ is the genus of a symplectic smoothing of $C$, $C$ satisfies the adjunction formula

$$K_X \cdot C + C \cdot C + 2 - 2p_a(C) = 0. \qquad (2.2)$$

It follows from the adjunction formula that for a singularity with multiplicity sequence $[m_1, \dots, m_n]$

$$\delta(p) = \frac{1}{2} \sum m_j(m_j - 1). \qquad (2.3)$$

Our focus will be on singular symplectic curves in $\mathbb{CP}^2$. In this case, because $H_2(\mathbb{CP}^2; \mathbb{Z}) \cong \mathbb{Z}$, the homology class of a curve is determined by an integer. We use the convention that $[\mathbb{CP}^1]$ with the complex orientation is the positive generator which we often denote by $h$. Note that $\langle [\omega], [\mathbb{CP}^1] \rangle = \int_{\mathbb{CP}^1} \omega > 0$. As all symplectic surfaces have positive symplectic area, their integral homology classes will correspond to positive integers. Extending the terminology from the algebraic case, we have the following definition.

**DEFINITION 2.8.** The degree of a (singular) symplectic curve $C$ in $\mathbb{CP}^2$ is the positive integer $d$ such that $[C] = d[\mathbb{CP}^1] \in H_2(\mathbb{CP}^2; \mathbb{Z})$.

We focus on classifying pairs $(X, C)$ which are relatively minimal, meaning there are no exceptional spheres in $X$ that are disjoint from $C$. The following definition will be convenient.

**DEFINITION 2.9.** A singular symplectic curve $C$ is said to be minimally embedded in a symplectic 4-manifold $(X, \omega)$ if $X \setminus C$ contains no exceptional symplectic $(-1)$-spheres.

There is a natural equivalence relation between different singular curves.

**DEFINITION 2.10.** A symplectic equisingular isotopy of singular symplectic curves is a one-parameter family of singular symplectic curves $C_t \subset (X, \omega)$ such that for each $t, t' \in [0, 1]$ the curves $C_t$ and $C_{t'}$ have topologically equivalent singularities. If $C_t$ is reducible, we label the components of $C_t$ and require the discrete labeling to vary continuously in $t$ (i.e. the labeling is preserved by the family).

We may sometimes drop the term equisingular for brevity and simply say that $C_0$ and $C_1$ are symplectically isotopic, with the understanding that all of the isotopies we consider in this article are equisingular.

Note that each singularity is locally uniquely determined up to equisingular isotopy by the smooth isotopy class of its link. Our definition of a symplectic singular curve required the existence of a local symplectomorphism to a complex curve singularity. Here we verify that complex curve singularities with the same topological type have the same symplectomorphism type. Thus, we can take one representative complex model for each topological type of curve singularity.

**LEMMA 2.11.** Suppose $C_0$ and $C_1$ are two complex curves in $(\mathbb{C}^2, \omega_{\text{std}})$ where $(0, 0)$ is a singular point for $C_i$ such that the links of the singularities at $(0, 0)$ in $C_i$ for $i = 0, 1$ are smoothly isotopic.
For the following definition, we borrow the terminology from algebraic geometry.

**DEFINITION 2.7.** Let $C$ be a singular symplectic curve. We call $p_g(C) := g(\Sigma)$ where $\Sigma$ is the normalization of $C$, the geometric genus of $C$, and we will say that $C$ is rational if its geometric genus is zero.

We call

$$p_a(C) := p_g(C) + \sum_{p \in \mathrm{Sing}(C)} \delta(p) \qquad (2.1)$$

the arithmetic genus of $C$.

As $p_a(C)$ is the genus of a symplectic smoothing of $C$, $C$ satisfies the adjunction formula

$$K_X \cdot C + C \cdot C + 2 - 2p_a(C) = 0. \qquad (2.2)$$

It follows from the adjunction formula that for a singularity with multiplicity sequence $[m_1, \dots, m_n]$

$$\delta(p) = \frac{1}{2} \sum m_j(m_j - 1). \qquad (2.3)$$

Our focus will be on singular symplectic curves in $\mathbb{CP}^2$. In this case, because $H_2(\mathbb{CP}^2; \mathbb{Z}) \cong \mathbb{Z}$, the homology class of a curve is determined by an integer. We use the convention that $[\mathbb{CP}^1]$ with the complex orientation is the positive generator which we often denote by $h$. Note that $\langle [\omega], [\mathbb{CP}^1] \rangle = \int_{\mathbb{CP}^1} \omega > 0$. As all symplectic surfaces have positive symplectic area, their integral homology classes will correspond to positive integers. Extending the terminology from the algebraic case, we have the following definition.

**DEFINITION 2.8.** The degree of a (singular) symplectic curve $C$ in $\mathbb{CP}^2$ is the positive integer $d$ such that $[C] = d[\mathbb{CP}^1] \in H_2(\mathbb{CP}^2; \mathbb{Z})$.

We focus on classifying pairs $(X, C)$ which are relatively minimal, meaning there are no exceptional spheres in $X$ that are disjoint from $C$. The following definition will be convenient.

**DEFINITION 2.9.** A singular symplectic curve $C$ is said to be minimally embedded in a symplectic 4-manifold $(X, \omega)$ if $X \setminus C$ contains no exceptional symplectic $(-1)$-spheres.

There is a natural equivalence relation between different singular curves.

**DEFINITION 2.10.** A symplectic equisingular isotopy of singular symplectic curves is a one-parameter family of singular symplectic curves $C_t \subset (X, \omega)$ such that for each $t, t' \in [0, 1]$ the curves $C_t$ and $C_{t'}$ have topologically equivalent singularities. If $C_t$ is reducible, we label the components of $C_t$ and require the discrete labeling to vary continuously in $t$ (i.e. the labeling is preserved by the family).

We may sometimes drop the term equisingular for brevity and simply say that $C_0$ and $C_1$ are symplectically isotopic, with the understanding that all of the isotopies we consider in this article are equisingular.

Note that each singularity is locally uniquely determined up to equisingular isotopy by the smooth isotopy class of its link. Our definition of a symplectic singular curve required the existence of a local symplectomorphism to a complex curve singularity. Here we verify that complex curve singularities with the same topological type have the same symplectomorphism type. Thus, we can take one representative complex model for each topological type of curve singularity.

**LEMMA 2.11.** Suppose $C_0$ and $C_1$ are two complex curves in $(\mathbb{C}^2, \omega_{\mathrm{std}})$ where $(0, 0)$ is a singular point for $C_i$ such that the links of the singularities at $(0, 0)$ in $C_i$ for $i = 0, 1$ are smoothly isotopic.
Then there exist an equisingular family of curves $(C'_t)_{t \in [0,1]}$ such that $C'_t$ has a singularity at the origin for each $t$, and there are neighborhoods $U_0$ and $U_1$ of $(0,0)$ such that $C_i \cap U_i = C'_i \cap U_i$ for $i = 0, 1$.

*Proof.* We can choose two complex charts around $(0,0)$ such that, for $i = 0, 1$, $C_i$ has a Puiseux expansions

$$y = c_{b_1}^i x^{b_1/m} + c_{b_1+1}^i x^{(b_1+1)/m} + \dots$$

Two singularities are topologically equivalent if and only if they have the same Puiseux exponents $(m; b_1, \dots, b_k)$. In terms of the Puiseux expansion, this means that $c_{b_j}^i \neq 0$ and that $c_\ell = 0$ for every $b_j < \ell < b_{j+1}$ such that $e_j \nmid \ell$. (Recall from the beginning of the section that the sequence $\{e_i\}$ is defined recursively as $e_1 = \gcd(m, b_1)$ and $e_i = \gcd(e_{i-1}, b_i)$ for $i > 1$.) As the space of sequences $\{c_\ell\}$ satisfying these constraints and the space of local charts are both connected, we can always isotope one singularity into the other without changing the singularity type. $\square$

*Remark 2.12.* For smooth symplectic curves in any symplectic 4-manifold $(X, \omega)$, any 1-parameter family of curves $C_t$ is induced by an ambient symplectic isotopy. Namely there exist symplectomorphisms $\Psi_t : (X, \omega) \to (X, \omega)$ with $\Psi_0 = \mathrm{Id}_X$ and $\Psi_t(C_0) = C_t$.

On the other hand, singular curves carry local symplectomorphism invariants beyond their topological type. Therefore, there can be a 1-parameter equisingular family of symplectic (or even complex) curves which may not be related by an ambient symplectic isotopy.

The differences stem from the preservation by symplectomorphisms of ‘angles’ between collections of symplectic planes intersecting at the origin. For a cusp singularity, the angles are not as easily visible, but after blowing up to a normal crossing resolution, the position of the intersection points of the exceptional divisors can vary. After blowing down one exceptional divisor, different relative intersection positions of other curves with that exceptional divisor will change the symplectic angles between the resulting transversally intersecting curves. These angles will have an effect on the geometry of the cusp when you blow down completely to undo the resolution.

This is the analogue of the situation in the complex category, where, for instance, there are families of ordinary quadruple points in $(\mathbb{C}^2, 0)$ such that different elements are related by deformations but are not isomorphic. These are distinguished by the cross-ratio of the four points the four branches determine in $\mathbb{P}(T_{(0,0)}\mathbb{C}^2) \cong \mathbb{CP}^1$.

In order to see the symplectic analogue of the complex statement, we reduce to the linear case, and argue by dimension-counting. Consider two ordinary $n$-tuple points in $(\mathbb{C}^2, (0,0))$ for some $n > 1$. If there is a global symplectomorphism $\psi : (U, (0,0)) \to (V, (0,0))$ between open sets containing them that sends one singularity to the other, then the differential $d\psi_{(0,0)}$ is a linear symplectomorphism of $(T_{(0,0)}\mathbb{C}^2, \omega)$ that identifies the $n$-tuple of tangent spaces to the branches of the two singularities. The space $\mathrm{Sp}(4)$ of linear symplectomorphisms of $(\mathbb{R}^4, \omega = \omega_{\mathrm{std}})$ has finite dimension $d_{\mathrm{ls}}$; the Grassmannian $\mathrm{Gr}_\omega(4,2)$ of symplectic planes in $(\mathbb{R}^4, \omega)$ also has finite dimension $d_{\mathrm{Gr}}$. As the diagonal action of $\mathrm{Sp}(4)$ on $\mathrm{Gr}_\omega(4,2)^{\times n}$ is smooth for each $n$, the dimension of the orbits is at most $d_{\mathrm{ls}}$. The space of $\omega$-planes intersecting positively and transversely is open and non-empty in $\mathrm{Gr}_\omega(4,2)^{\times n}$. This proves that, as soon as $d_{\mathrm{ls}} < nd_{\mathrm{Gr}}$, the action cannot be transitive on any connected component and, in particular, that there are isotopic ordinary $n$-tuple points that are isotopic but not ambient-symplectomorphic. In fact, one can compute $d_{\mathrm{ls}} = 10$ and $d_{\mathrm{Gr}} = 4$, because the symplectic condition is open and, therefore, $d_{\mathrm{Gr}}$ is the same as the dimension of the Grassmannian of 2-planes in $\mathbb{R}^4$. This shows that $n = 3$ already suffices.
Then there exist an equisingular family of curves $(C'_t)_{t \in [0,1]}$ such that $C'_t$ has a singularity at the origin for each $t$, and there are neighborhoods $U_0$ and $U_1$ of $(0,0)$ such that $C_i \cap U_i = C'_i \cap U_i$ for $i = 0, 1$.

*Proof.* We can choose two complex charts around $(0,0)$ such that, for $i = 0, 1$, $C_i$ has a Puiseux expansions

$$y = c_{b_1}^i x^{b_1/m} + c_{b_1+1}^i x^{(b_1+1)/m} + \dots$$

Two singularities are topologically equivalent if and only if they have the same Puiseux exponents $(m; b_1, \dots, b_k)$. In terms of the Puiseux expansion, this means that $c_{b_j}^i \neq 0$ and that $c_\ell = 0$ for every $b_j < \ell < b_{j+1}$ such that $e_j \nmid \ell$. (Recall from the beginning of the section that the sequence $\{e_i\}$ is defined recursively as $e_1 = \gcd(m, b_1)$ and $e_i = \gcd(e_{i-1}, b_i)$ for $i > 1$.) As the space of sequences $\{c_\ell\}$ satisfying these constraints and the space of local charts are both connected, we can always isotope one singularity into the other without changing the singularity type. $\square$

*Remark 2.12.* For smooth symplectic curves in any symplectic 4-manifold $(X, \omega)$, any 1-parameter family of curves $C_t$ is induced by an ambient symplectic isotopy. Namely there exist symplectomorphisms $\Psi_t : (X, \omega) \to (X, \omega)$ with $\Psi_0 = \mathrm{Id}_X$ and $\Psi_t(C_0) = C_t$.

On the other hand, singular curves carry local symplectomorphism invariants beyond their topological type. Therefore, there can be a 1-parameter equisingular family of symplectic (or even complex) curves which may not be related by an ambient symplectic isotopy.

The differences stem from the preservation by symplectomorphisms of ‘angles’ between collections of symplectic planes intersecting at the origin. For a cusp singularity, the angles are not as easily visible, but after blowing up to a normal crossing resolution, the position of the intersection points of the exceptional divisors can vary. After blowing down one exceptional divisor, different relative intersection positions of other curves with that exceptional divisor will change the symplectic angles between the resulting transversally intersecting curves. These angles will have an effect on the geometry of the cusp when you blow down completely to undo the resolution.

This is the analogue of the situation in the complex category, where, for instance, there are families of ordinary quadruple points in $(\mathbb{C}^2, 0)$ such that different elements are related by deformations but are not isomorphic. These are distinguished by the cross-ratio of the four points the four branches determine in $\mathbb{P}(T_{(0,0)}\mathbb{C}^2) \cong \mathbb{CP}^1$.

In order to see the symplectic analogue of the complex statement, we reduce to the linear case, and argue by dimension-counting. Consider two ordinary $n$-tuple points in $(\mathbb{C}^2, (0,0))$ for some $n > 1$. If there is a global symplectomorphism $\psi : (U, (0,0)) \to (V, (0,0))$ between open sets containing them that sends one singularity to the other, then the differential $d\psi_{(0,0)}$ is a linear symplectomorphism of $(T_{(0,0)}\mathbb{C}^2, \omega)$ that identifies the $n$-tuple of tangent spaces to the branches of the two singularities. The space $\mathrm{Sp}(4)$ of linear symplectomorphisms of $(\mathbb{R}^4, \omega = \omega_{\mathrm{std}})$ has finite dimension $d_{\mathrm{ls}}$; the Grassmannian $\mathrm{Gr}_\omega(4, 2)$ of symplectic planes in $(\mathbb{R}^4, \omega)$ also has finite dimension $d_{\mathrm{Gr}}$. As the diagonal action of $\mathrm{Sp}(4)$ on $\mathrm{Gr}_\omega(4, 2)^{\times n}$ is smooth for each $n$, the dimension of the orbits is at most $d_{\mathrm{ls}}$. The space of $\omega$-planes intersecting positively and transversely is open and non-empty in $\mathrm{Gr}_\omega(4, 2)^{\times n}$. This proves that, as soon as $d_{\mathrm{ls}} < nd_{\mathrm{Gr}}$, the action cannot be transitive on any connected component and, in particular, that there are isotopic ordinary $n$-tuple points that are isotopic but not ambient-symplectomorphic. In fact, one can compute $d_{\mathrm{ls}} = 10$ and $d_{\mathrm{Gr}} = 4$, because the symplectic condition is open and, therefore, $d_{\mathrm{Gr}}$ is the same as the dimension of the Grassmannian of 2-planes in $\mathbb{R}^4$. This shows that $n = 3$ already suffices.
## 2.3 The cuspidal contact structure and its caps

**THEOREM 2.13.** Given a singular symplectic curve $C$, with specified singularity types and normal Euler number $s$. Suppose $s = C \cdot C > 0$. Then there exists a symplectic manifold $(N, \omega)$ with concave boundary such that $N$ is a regular neighborhood of $C$, such that $[C]^2 = s$ in $N$. Moreover, every symplectic embedding of $C$ into a symplectic manifold $(X, \omega)$ has a concave neighborhood inducing the same contact boundary.

*Proof.* Let $\bar{C}$ denote the normal crossing resolution of $C$ which is a collection of transversally intersecting curves with specified genus, intersections, and normal Euler numbers. The intersections are generic double points.

A regular neighborhood $U$ of $\bar{C}$ can be constructed as a plumbing of surfaces. Moreover, by undoing the normal crossing resolution, we see that $U$ is the blow-up of a regular neighborhood $N$ of $C$. In particular, the intersection form of $U$ is $\langle s \rangle \oplus \langle -1 \rangle^{\oplus n}$ for some $n \ge 0$.

Because $s = C \cdot C > 0$, the intersection form of the plumbing is not negative definite. We claim that the inclusion $\partial U \to U$ induces the trivial map on the second cohomology group: $H_{\mathrm{DR}}^2(U) \to H_{\mathrm{DR}}^2(\partial U)$. This follows from the long exact sequence of the pair $(U, \partial U)$: in fact, the map $H^2(U, \partial U; \mathbb{Z}) \to H^2(U; \mathbb{Z})$ is presented by the intersection form of $U$; because the latter is non-degenerate, the map $H_{\mathrm{DR}}^2(U, \partial U) \to H_{\mathrm{DR}}^2(U)$ is an isomorphism. It follows that $\omega_U$ is exact on $\partial U$. (The exactness assumption is automatically satisfied if $C$ is rational and cuspidal, because in that case $\partial U$ is a rational homology sphere and, therefore, $H_{\mathrm{DR}}^2(\partial U) = 0$.)

Construct a concave symplectic structure on $U$, such that $\bar{C}$ is a symplectic submanifold using the construction of [GS09] adapted to the concave case [LM19, Theorem 1.3]. Blow down to obtain the symplectic structure on $N$.

The existence of an equivalent concave neighborhood in any symplectic embedding follows similarly from blowing up to the normal crossing resolution and using the analogous result of [GS09, LM19]. □

**DEFINITION 2.14.** Given a singular symplectic curve $C$, defined by the collection of its singularities and its self-intersection, we define $(Y_C, \xi_C)$ by $Y_C = -\partial N$, and $\xi_C$ as the contact structure induced by the concave symplectic structure.

Note that the contact structure depends only on the topological types of the singularities of $C$, its geometric genus, and its self-intersection. In addition, observe that, in order to define $(Y_C, \xi_C)$, we are using that there exists a normal crossing resolution.

This is a generalization of the contact structures introduced by Chen (see [Che18, Section 4]) for rational curves. The focus in [Che18] was on curves in $\mathbb{CP}^2$ (or in symplectic manifolds with the same algebraic topology). Chen described the contact structure using transverse symplectic handle attachment, instead of using the normal crossing resolution. The two contact structures are actually the same because they are both the canonical contact structure on the boundary of a small concave neighborhood of the curve, though we do not use this fact here.

*Remark 2.15.* We recall that, if $C$ is an algebraic curve in $\mathbb{CP}^2$ of degree $d$, then the contact structure $\xi_C$ is defined algebraically. It is a hyperplane section of the Veronese embedding of degree $d$ and, hence, its complement inherits a Stein structure from $C^M = \mathbb{P}^M \setminus H$.

*Remark 2.16.* Note that in our setup, $C$ need not be realized as a symplectic curve embedded in a closed Kähler surface. Instead its regular neighborhood can be defined using its self-intersection number, genus, and singularities. We exhibit examples of such curves which do not embed in a closed symplectic manifold (see § 7).
## 2.3 The cuspidal contact structure and its caps

**THEOREM 2.13.** Given a singular symplectic curve $C$, with specified singularity types and normal Euler number $s$. Suppose $s = C \cdot C > 0$. Then there exists a symplectic manifold $(N, \omega)$ with concave boundary such that $N$ is a regular neighborhood of $C$, such that $[C]^2 = s$ in $N$. Moreover, every symplectic embedding of $C$ into a symplectic manifold $(X, \omega)$ has a concave neighborhood inducing the same contact boundary.

*Proof.* Let $\bar{C}$ denote the normal crossing resolution of $C$ which is a collection of transversally intersecting curves with specified genus, intersections, and normal Euler numbers. The intersections are generic double points.

A regular neighborhood $U$ of $\bar{C}$ can be constructed as a plumbing of surfaces. Moreover, by undoing the normal crossing resolution, we see that $U$ is the blow-up of a regular neighborhood $N$ of $C$. In particular, the intersection form of $U$ is $\langle s \rangle \oplus \langle -1 \rangle^{\oplus n}$ for some $n \ge 0$.

Because $s = C \cdot C > 0$, the intersection form of the plumbing is not negative definite. We claim that the inclusion $\partial U \to U$ induces the trivial map on the second cohomology group: $H_{\mathrm{DR}}^2(U) \to H_{\mathrm{DR}}^2(\partial U)$. This follows from the long exact sequence of the pair $(U, \partial U)$: in fact, the map $H^2(U, \partial U; \mathbb{Z}) \to H^2(U; \mathbb{Z})$ is presented by the intersection form of $U$; because the latter is non-degenerate, the map $H_{\mathrm{DR}}^2(U, \partial U) \to H_{\mathrm{DR}}^2(U)$ is an isomorphism. It follows that $\omega_U$ is exact on $\partial U$. (The exactness assumption is automatically satisfied if $C$ is rational and cuspidal, because in that case $\partial U$ is a rational homology sphere and, therefore, $H_{\mathrm{DR}}^2(\partial U) = 0$.)

Construct a concave symplectic structure on $U$, such that $\bar{C}$ is a symplectic submanifold using the construction of [GS09] adapted to the concave case [LM19, Theorem 1.3]. Blow down to obtain the symplectic structure on $N$.

The existence of an equivalent concave neighborhood in any symplectic embedding follows similarly from blowing up to the normal crossing resolution and using the analogous result of [GS09, LM19]. □

**DEFINITION 2.14.** Given a singular symplectic curve $C$, defined by the collection of its singularities and its self-intersection, we define $(Y_C, \xi_C)$ by $Y_C = -\partial N$, and $\xi_C$ as the contact structure induced by the concave symplectic structure.

Note that the contact structure depends only on the topological types of the singularities of $C$, its geometric genus, and its self-intersection. In addition, observe that, in order to define $(Y_C, \xi_C)$, we are using that there exists a normal crossing resolution.

This is a generalization of the contact structures introduced by Chen (see [Che18, Section 4]) for rational curves. The focus in [Che18] was on curves in $\mathbb{CP}^2$ (or in symplectic manifolds with the same algebraic topology). Chen described the contact structure using transverse symplectic handle attachment, instead of using the normal crossing resolution. The two contact structures are actually the same because they are both the canonical contact structure on the boundary of a small concave neighborhood of the curve, though we do not use this fact here.

*Remark 2.15.* We recall that, if $C$ is an algebraic curve in $\mathbb{CP}^2$ of degree $d$, then the contact structure $\xi_C$ is defined algebraically. It is a hyperplane section of the Veronese embedding of degree $d$ and, hence, its complement inherits a Stein structure from $C^M = \mathbb{P}^M \setminus H$.

*Remark 2.16.* Note that in our setup, $C$ need not be realized as a symplectic curve embedded in a closed Kähler surface. Instead its regular neighborhood can be defined using its self-intersection number, genus, and singularities. We exhibit examples of such curves which do not embed in a closed symplectic manifold (see § 7).
## 2.4 Topological obstructions

We mention here two techniques used previously to obstruct rational cuspidal curves in $CP^2$ which have topological interpretations and can thus obstruct symplectic curves (not just complex algebraic curves).

The semigroup condition was first discovered by Borodzik and Livingston [BL14]. Recall that to each curve singularity $x$ there is an associated semigroup $\Gamma_x$, that records the local multiplicities of intersections of germs of curves with the singularity; for instance, for a singularity of type $(p, q)$, the semigroup is generated by $p$ and $q$. By convention, we declare that $0$ belongs to $\Gamma_x$, so $0$ is the first element of the semigroup, and the multiplicity is the second. We define the counting function $R_x$ of $\Gamma_x$ as $R_x(n) = \#(\Gamma_x \cap (-\infty, n-1))$, and the minimum convolution $F \diamond G$ of two functions as $(F \diamond G)(n) = \min_{k \in \mathbb{Z}} F(k) + G(n-k)$; given a rational cuspidal curve $C$ of degree $d$, with cusps $x_1, \dots, x_k$, and associated $R$-functions $R_1, \dots, R_k$, let $R = R_1 \diamond \dots \diamond R_k$. Then, [BL14, Theorem 6.5] asserts that, for each $j = -1, 0, \dots, d-2$,

$$R(jd + 1) = \frac{(j + 1)(j + 2)}{2}. \qquad (2.4)$$

Note that this is a smooth obstruction: it obstructs the existence of a rational homology 4-ball which glues by a diffeomorphism of the boundaries to a regular neighborhood of $C$.

**Remark 2.17.** In fact, the function $R$ only depends on the multiplicity multi-sequence, not the way it is split into multiplicity sequences for individual cusps [BN16, Theorem 1.3.12]. Moreover, one can see that the semigroup condition is a generalization of Bézout’s theorem: see, for instance, [FLMN06, Proposition 3.2.1].

**Example 2.18.** In the case of quintics, the only multiplicity multi-sequence that is excluded by the semigroup criterion is $[3, 3]$, which corresponds to two rational cuspidal curves, one with a singularity of type $(3, 7)$ and one with two singularities of type $(3, 4)$.

**Remark 2.19.** There is a strengthening of [BL14] to curves of odd degree, using involutive Heegaard Floer homology, due to Borodzik, Hom, and Schinzel [BHS18]; in particular, by [BHS18, Section 5.1], the curves with cusps of types $\{[2, 2, 2], [2, 2, 2]\}$ and $\{[2, 2, 2, 2, 2], [2]\}$ are also obstructed.

We call spectrum semicontinuity an inequality on the Levine–Tristram signature and nullity functions of the links, $\sigma_\bullet, \eta_\bullet: S^1 \to \mathbb{Z}$. It is related to the algebro-geometric spectrum semicontinuity, formulated in terms of Hodge-theoretic data, and then recast in more topological terms by Borodzik and Némethi [BN12, Corollary 2.5.4]. From a curve $C$, by choosing a generic line $\ell$, we obtain a genus-$p_g(C)$ cobordism in $S^3 \times [0, 1]$ from the connected sum $K$ of links of all singularities of $C$, to the torus link $T(d, d)$; this cobordism is obtained from removing a neighborhood of a path connecting all singularities of $C$, and a neighborhood of the line $\ell$. Suppose that $C$ is rational and its singularities are $k$ cusps. Then $K$ is a knot, and for every $\zeta \in S_1^1 \subset S^1$ we have

$$|\sigma_{T(d,d)}(\zeta) - \sigma_K(\zeta)| + |\eta_{T(d,d)}(\zeta) - \eta_K(\zeta)| \le d - 1. \qquad (2.5)$$

Here $S_1^1$ is the unit circle with all Knotennullstellen (roots of integer polynomials such that $p(1) = 1$) removed; that is,

$$S_1^1 = S^1 \setminus \{\alpha \mid \exists p(t) \in \mathbb{Z}[t], p(1) = 1, p(\alpha) = 0\}.$$

This inequality was essentially proved by Nagel and Powell [NP17], who also introduced the notation and terminology for $S_1^1$ (see also [Con21, Theorem 2.12]).
## 2.4 Topological obstructions

We mention here two techniques used previously to obstruct rational cuspidal curves in $CP^2$ which have topological interpretations and can thus obstruct symplectic curves (not just complex algebraic curves).

The semigroup condition was first discovered by Borodzik and Livingston [BL14]. Recall that to each curve singularity $x$ there is an associated semigroup $\Gamma_x$, that records the local multiplicities of intersections of germs of curves with the singularity; for instance, for a singularity of type $(p, q)$, the semigroup is generated by $p$ and $q$. By convention, we declare that $0$ belongs to $\Gamma_x$, so $0$ is the first element of the semigroup, and the multiplicity is the second. We define the counting function $R_x$ of $\Gamma_x$ as $R_x(n) = \#(\Gamma_x \cap (-\infty, n-1))$, and the minimum convolution $F \diamond G$ of two functions as $(F \diamond G)(n) = \min_{k \in \mathbb{Z}} F(k) + G(n-k)$; given a rational cuspidal curve $C$ of degree $d$, with cusps $x_1, \dots, x_k$, and associated $R$-functions $R_1, \dots, R_k$, let $R = R_1 \diamond \dots \diamond R_k$. Then, [BL14, Theorem 6.5] asserts that, for each $j = -1, 0, \dots, d-2$,

$$R(jd + 1) = \frac{(j + 1)(j + 2)}{2}. \qquad (2.4)$$

Note that this is a smooth obstruction: it obstructs the existence of a rational homology 4-ball which glues by a diffeomorphism of the boundaries to a regular neighborhood of $C$.

*Remark 2.17.* In fact, the function $R$ only depends on the multiplicity multi-sequence, not the way it is split into multiplicity sequences for individual cusps [BN16, Theorem 1.3.12]. Moreover, one can see that the semigroup condition is a generalization of Bézout’s theorem: see, for instance, [FLMN06, Proposition 3.2.1].

*Example 2.18.* In the case of quintics, the only multiplicity multi-sequence that is excluded by the semigroup criterion is $[3, 3]$, which corresponds to two rational cuspidal curves, one with a singularity of type $(3, 7)$ and one with two singularities of type $(3, 4)$.

*Remark 2.19.* There is a strengthening of [BL14] to curves of odd degree, using involutive Heegaard Floer homology, due to Borodzik, Hom, and Schinzel [BHS18]; in particular, by [BHS18, Section 5.1], the curves with cusps of types $\{[2, 2, 2], [2, 2, 2]\}$ and $\{[2, 2, 2, 2, 2], [2]\}$ are also obstructed.

We call spectrum semicontinuity an inequality on the Levine–Tristram signature and nullity functions of the links, $\sigma_\bullet, \eta_\bullet: S^1 \to \mathbb{Z}$. It is related to the algebro-geometric spectrum semicontinuity, formulated in terms of Hodge-theoretic data, and then recast in more topological terms by Borodzik and Némethi [BN12, Corollary 2.5.4]. From a curve $C$, by choosing a generic line $\ell$, we obtain a genus-$p_g(C)$ cobordism in $S^3 \times [0, 1]$ from the connected sum $K$ of links of all singularities of $C$, to the torus link $T(d, d)$; this cobordism is obtained from removing a neighborhood of a path connecting all singularities of $C$, and a neighborhood of the line $\ell$. Suppose that $C$ is rational and its singularities are $k$ cusps. Then $K$ is a knot, and for every $\zeta \in S_1^1 \subset S^1$ we have

$$|\sigma_{T(d,d)}(\zeta) - \sigma_K(\zeta)| + |\eta_{T(d,d)}(\zeta) - \eta_K(\zeta)| \le d - 1. \qquad (2.5)$$

Here $S_1^1$ is the unit circle with all Knotennullstellen (roots of integer polynomials such that $p(1) = 1$) removed; that is,

$$S_1^1 = S^1 \setminus \{\alpha \mid \exists p(t) \in \mathbb{Z}[t], p(1) = 1, p(\alpha) = 0\}.$$

This inequality was essentially proved by Nagel and Powell [NP17], who also introduced the notation and terminology for $S_1^1$ (see also [Con21, Theorem 2.12]).
To be concrete: all transcendental complex numbers of norm 1 belong to $S_1^1$, so the set of *Knotennullstellen* has measure 0. A root of unity belongs to $S_1^1$ if and only if its order is a prime power; for this it suffices to evaluate cyclotomic polynomials at $1: \Phi_{p^r}(1) = p$ for every prime $p$ and positive integer $r$, whereas $\Phi_n(1) = 1$ whenever $n$ has at least two distinct prime factors.

The obstruction (2.5) is topological, once we assume that there exists a locally flatly embedded sphere $\ell$ in the homology class $h$ that intersects $C$ transversely and positively in $d$ points. This assumption is satisfied when $C$ is a symplectic curve, because we can choose an almost complex structure such that $C$ is $J$-holomorphic (by Lemma 3.4) and then choose a generic $J$-holomorphic line. Then each $J$-holomorphic line intersects $C$ positively, and the space of $J$-holomorphic lines has (real) dimension 4. The $J$-holomorphic lines which intersect $C$ at a singular point or tangentially has positive codimension (real codimension 2) in the space of all $J$-holomorphic lines. Therefore there is a $J$-holomorphic line which intersects $C$ positively and transversally in $d$ points.

*Example 2.20.* The spectrum semicontinuity obstructs the existence of a quintic with two cusps of type (2,7). As mentioned in Remark 2.19, this was also obstructed by using Floer-theoretic techniques. The spectrum semicontinuity is a stronger obstruction, because it holds in the topologically locally flat category, rather just the smooth category.

# 3. Pseudoholomorphic techniques

## 3.1 Proper transforms in symplectic blow-ups

In the symplectic category, blowing up and blowing down take on a slightly different character than they do in algebraic geometry [McD91]. To perform a symplectic blow-up at a point $p$ in a symplectic manifold $(X, \omega)$, we first choose a Darboux chart centered at $p$. Remove a ball inside this chart of radius $\lambda$, and collapse the boundary by the Hopf fibration to obtain the exceptional divisor. Alternatively, remove a ball of radius $\lambda + \varepsilon$ centered at $p$ and replace it with an $\varepsilon$-neighborhood of a copy of $\mathbb{CP}^1$ of symplectic area $\lambda$ in $\mathcal{O}(-1)$. The symplectic form on the ring of the ball between radius $\lambda$ and $\lambda + \varepsilon$ agrees with the symplectic form on $\mathcal{O}(-1)$ when the zero section is a symplectic submanifold of area $\lambda$. A symplectic blow-down reverses the operation, replacing an $\varepsilon$-neighborhood of an exceptional divisor of symplectic area $\lambda$ with a ball of radius $\lambda + \varepsilon$ (or, equivalently, deleting the exceptional divisor of symplectic area $\lambda$ and replacing it by a closed ball of radius $\lambda$).

*Remark 3.1.* In the algebraic geometric (or smooth) category, the blow-up has a well-defined effect on both the variety and its subvarieties. The effect on the subvarieties gives rise to the notions of the total and proper transforms discussed in § 2.1. The proper transform turns first-order data of the subvariety at the point $p$ into zeroth-order information (and second-order information into first-order information, etc.). Because the symplectic blow-up deletes an entire ball instead of just a point, we need to define the total and proper transforms of (singular) symplectic curves in symplectic 4-manifolds with a bit more care.

When we perform a symplectic blow-up at a point, we always choose the radius $\lambda + \varepsilon$ of the symplectic ball to be sufficiently small such that every sphere $S_r(p)$ of radius $0 < r \le \lambda + \varepsilon$ intersects the symplectic curves transversally. Note that using the radial vector field in the Darboux chart as a Liouville vector field, the spheres $S_r(p)$ are contact-type hypersurfaces.

Suppose now that we have a symplectic curve $C$ in $X$, and that we blow $X$ up at a (possibly singular) point $p$ on $C$. The transverse intersections of $C$ with the contact spheres are transverse links $T_r$ in the spheres $S_r(p)$. Note that if we identify spheres of different radii by a rescaling
To be concrete: all transcendental complex numbers of norm 1 belong to $S_1^1$, so the set of *Knotennullstellen* has measure 0. A root of unity belongs to $S_1^1$ if and only if its order is a prime power; for this it suffices to evaluate cyclotomic polynomials at 1: $\Phi_{p^r}(1) = p$ for every prime $p$ and positive integer $r$, whereas $\Phi_n(1) = 1$ whenever $n$ has at least two distinct prime factors.

The obstruction (2.5) is topological, once we assume that there exists a locally flatly embedded sphere $\ell$ in the homology class $h$ that intersects $C$ transversely and positively in $d$ points. This assumption is satisfied when $C$ is a symplectic curve, because we can choose an almost complex structure such that $C$ is $J$-holomorphic (by Lemma 3.4) and then choose a generic $J$-holomorphic line. Then each $J$-holomorphic line intersects $C$ positively, and the space of $J$-holomorphic lines has (real) dimension 4. The $J$-holomorphic lines which intersect $C$ at a singular point or tangentially has positive codimension (real codimension 2) in the space of all $J$-holomorphic lines. Therefore there is a $J$-holomorphic line which intersects $C$ positively and transversally in $d$ points.

*Example 2.20.* The spectrum semicontinuity obstructs the existence of a quintic with two cusps of type (2,7). As mentioned in Remark 2.19, this was also obstructed by using Floer-theoretic techniques. The spectrum semicontinuity is a stronger obstruction, because it holds in the topologically locally flat category, rather just the smooth category.

# 3. Pseudoholomorphic techniques

## 3.1 Proper transforms in symplectic blow-ups

In the symplectic category, blowing up and blowing down take on a slightly different character than they do in algebraic geometry [McD91]. To perform a symplectic blow-up at a point $p$ in a symplectic manifold $(X, \omega)$, we first choose a Darboux chart centered at $p$. Remove a ball inside this chart of radius $\lambda$, and collapse the boundary by the Hopf fibration to obtain the exceptional divisor. Alternatively, remove a ball of radius $\lambda + \varepsilon$ centered at $p$ and replace it with an $\varepsilon$-neighborhood of a copy of $\mathbb{CP}^1$ of symplectic area $\lambda$ in $\mathcal{O}(-1)$. The symplectic form on the ring of the ball between radius $\lambda$ and $\lambda + \varepsilon$ agrees with the symplectic form on $\mathcal{O}(-1)$ when the zero section is a symplectic submanifold of area $\lambda$. A symplectic blow-down reverses the operation, replacing an $\varepsilon$-neighborhood of an exceptional divisor of symplectic area $\lambda$ with a ball of radius $\lambda + \varepsilon$ (or, equivalently, deleting the exceptional divisor of symplectic area $\lambda$ and replacing it by a closed ball of radius $\lambda$).

*Remark 3.1.* In the algebraic geometric (or smooth) category, the blow-up has a well-defined effect on both the variety and its subvarieties. The effect on the subvarieties gives rise to the notions of the total and proper transforms discussed in § 2.1. The proper transform turns first-order data of the subvariety at the point $p$ into zeroth-order information (and second-order information into first-order information, etc.). Because the symplectic blow-up deletes an entire ball instead of just a point, we need to define the total and proper transforms of (singular) symplectic curves in symplectic 4-manifolds with a bit more care.

When we perform a symplectic blow-up at a point, we always choose the radius $\lambda + \varepsilon$ of the symplectic ball to be sufficiently small such that every sphere $S_r(p)$ of radius $0 < r \le \lambda + \varepsilon$ intersects the symplectic curves transversally. Note that using the radial vector field in the Darboux chart as a Liouville vector field, the spheres $S_r(p)$ are contact-type hypersurfaces.

Suppose now that we have a symplectic curve $C$ in $X$, and that we blow $X$ up at a (possibly singular) point $p$ on $C$. The transverse intersections of $C$ with the contact spheres are transverse links $T_r$ in the spheres $S_r(p)$. Note that if we identify spheres of different radii by a rescaling
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

contactomorphism, the transverse links $T_r$ in the spheres of different radii $0 < r < \lambda + \varepsilon$ are all transversally isotopic to each other. If $p$ is a smooth point, the transverse link $T_r$ will be the standard unknot of maximal self-linking number. If $p$ is a singularity, $T_r$ will be transversally isotopic to an algebraic link.

For the algebraic proper transform in the blow-up, the exceptional divisor $E$ replaces the point $p$, and the proper transform $\tilde{C}$ intersects $E$ according to its tangent derivative information at that point. Thus, the proper transform is a finite-point compactification in the algebraic blow-up of the family of transverse links $T_r \subset S_r(p)$. We would like to have a similar proper transform in the symplectic blow-up, but in this case we delete a ball of radius $\lambda$ instead of only a point. Therefore, the links $T_r$ for $0 < r \le \lambda$ would be cut out. We cannot guarantee that the transverse links $T_r$ for $\lambda < r < \lambda + \varepsilon$ will finite-point compactify as $r \to \lambda$ with the same diffeomorphism type as the family $T_r$ for $0 < r < \lambda + \varepsilon$ as $r \to 0$.

To overcome this issue, we squeeze the entire transverse isotopy $T_r$ for $0 < r < \lambda + \varepsilon$ into the collar where $\lambda < r < \lambda + \varepsilon$ by reparametrizing the radial coordinate by a smooth, strictly increasing function $\rho$ such that $\rho(\tilde{r}) = \tilde{r} - \lambda$ near $\tilde{r} = \lambda$ and $\rho(\tilde{r}) = \tilde{r}$ near $\lambda + \varepsilon$. In other words, in the symplectic blow-up, for $\lambda < \tilde{r} < \lambda + \varepsilon$, the proper transform intersects the sphere $S_{\tilde{r}}$ in a link $\tilde{T}_{\tilde{r}}$ defined by

$$ \tilde{T}_{\tilde{r}} := T_{\rho(\tilde{r})}. $$

As $\tilde{r} \to \lambda$, $\rho(\tilde{r}) \to 0$. Therefore, this family extends via a finite-point compactification in the exceptional divisor in the same manner as the algebraic proper transform. Note that we do not change the symplectic structure on the ambient symplectic 4-manifold, we only modify the symplectic curve.

LEMMA 3.2. The surface $\bigcup_{\lambda < \tilde{r} < \lambda + \varepsilon} \tilde{T}_{\tilde{r}}$ is a symplectic curve.

Proof. We want to prove that the tangent spaces to the surface are still symplectic subspaces; the idea is that this is true because the bases for the tangent spaces only differ by a positive scaling factor $\rho'(r)$. Namely, if $\phi(r, t)$ denotes a polar parametrization of the original surface such that $\phi(r, t) = (T_r(t), r)$, then $\tilde{\phi}(r, t) = (T_{\rho(r)}(t), \rho(r))$ is a parametrization for the new surface and

$$ \frac{\partial \tilde{\phi}}{\partial r}(r, t) = \rho'(r) \frac{\partial \phi}{\partial r}(\rho(r), t), \quad \frac{\partial \tilde{\phi}}{\partial t}(r, t) = \frac{\partial \phi}{\partial t}(\rho(r), t), $$

so the value of $\omega$ must be positive on an oriented basis for the curve parametrized by $\tilde{\phi}$ because it is positive on an oriented basis for the curve parametrized by $\phi$. $\square$

A symplectic blow-down reverses this procedure. Deleting an $\varepsilon$-neighborhood of an exceptional sphere of weight $\lambda$ (i.e. the symplectic area is $\pi\lambda^2$), we replace it with a symplectic ball of radius $\lambda + \varepsilon$. To define the image of a symplectic curve which intersects the exceptional divisor over $B_{\lambda+\varepsilon}(p)$ we reverse the squeezing procedure above to stretch the transverse link family back out. Namely we take the surface defined by the union of the center $p$ of the ball together with the transverse links $T_r \subset S_r(p)$ where

$$ T_r := \tilde{T}_{\rho^{-1}(r)}. $$

Note that the definition of the proper transform depends on the choice of the Darboux chart, of $\varepsilon$ and of $\rho$. However, the *symplectic isotopy class* is *independent* of all these choices.

PROPOSITION 3.3. The symplectic isotopy class of the proper transform of a singular symplectic curve $C \subset (X, \omega)$ is independent of the choice of the Darboux ball, of $\varepsilon$ and $\rho$, and is related to the
contactomorphism, the transverse links $T_r$ in the spheres of different radii $0 < r < \lambda + \varepsilon$ are all transversally isotopic to each other. If $p$ is a smooth point, the transverse link $T_r$ will be the standard unknot of maximal self-linking number. If $p$ is a singularity, $T_r$ will be transversally isotopic to an algebraic link.

For the algebraic proper transform in the blow-up, the exceptional divisor $E$ replaces the point $p$, and the proper transform $\tilde{C}$ intersects $E$ according to its tangent derivative information at that point. Thus, the proper transform is a finite-point compactification in the algebraic blow-up of the family of transverse links $T_r \subset S_r(p)$. We would like to have a similar proper transform in the symplectic blow-up, but in this case we delete a ball of radius $\lambda$ instead of only a point. Therefore, the links $T_r$ for $0 < r \le \lambda$ would be cut out. We cannot guarantee that the transverse links $T_r$ for $\lambda < r < \lambda + \varepsilon$ will finite-point compactify as $r \to \lambda$ with the same diffeomorphism type as the family $T_r$ for $0 < r < \lambda + \varepsilon$ as $r \to 0$.

To overcome this issue, we squeeze the entire transverse isotopy $T_r$ for $0 < r < \lambda + \varepsilon$ into the collar where $\lambda < r < \lambda + \varepsilon$ by reparametrizing the radial coordinate by a smooth, strictly increasing function $\rho$ such that $\rho(\tilde{r}) = \tilde{r} - \lambda$ near $\tilde{r} = \lambda$ and $\rho(\tilde{r}) = \tilde{r}$ near $\lambda + \varepsilon$. In other words, in the symplectic blow-up, for $\lambda < \tilde{r} < \lambda + \varepsilon$, the proper transform intersects the sphere $S_{\tilde{r}}$ in a link $\tilde{T}_{\tilde{r}}$ defined by

$$ \tilde{T}_{\tilde{r}} := T_{\rho(\tilde{r})}. $$

As $\tilde{r} \to \lambda$, $\rho(\tilde{r}) \to 0$. Therefore, this family extends via a finite-point compactification in the exceptional divisor in the same manner as the algebraic proper transform. Note that we do not change the symplectic structure on the ambient symplectic 4-manifold, we only modify the symplectic curve.

**LEMMA 3.2.** *The surface $\bigcup_{\lambda<\tilde{r}<\lambda+\varepsilon} \tilde{T}_{\tilde{r}}$ is a symplectic curve.*

*Proof.* We want to prove that the tangent spaces to the surface are still symplectic subspaces; the idea is that this is true because the bases for the tangent spaces only differ by a positive scaling factor $\rho'(r)$. Namely, if $\phi(r, t)$ denotes a polar parametrization of the original surface such that $\phi(r, t) = (T_r(t), r)$, then $\tilde{\phi}(r, t) = (T_{\rho(r)}(t), \rho(r))$ is a parametrization for the new surface and

$$ \frac{\partial \tilde{\phi}}{\partial r}(r, t) = \rho'(r) \frac{\partial \phi}{\partial r}(\rho(r), t), \quad \frac{\partial \tilde{\phi}}{\partial t}(r, t) = \frac{\partial \phi}{\partial t}(\rho(r), t), $$

so the value of $\omega$ must be positive on an oriented basis for the curve parametrized by $\tilde{\phi}$ because it is positive on an oriented basis for the curve parametrized by $\phi$. $\square$

A symplectic blow-down reverses this procedure. Deleting an $\varepsilon$-neighborhood of an exceptional sphere of weight $\lambda$ (i.e. the symplectic area is $\pi\lambda^2$), we replace it with a symplectic ball of radius $\lambda + \varepsilon$. To define the image of a symplectic curve which intersects the exceptional divisor over $B_{\lambda+\varepsilon}(p)$ we reverse the squeezing procedure above to stretch the transverse link family back out. Namely we take the surface defined by the union of the center $p$ of the ball together with the transverse links $T_r \subset S_r(p)$ where

$$ T_r := \tilde{T}_{\rho^{-1}(r)}. $$

Note that the definition of the proper transform depends on the choice of the Darboux chart, of $\varepsilon$ and of $\rho$. However, the *symplectic isotopy class* is *independent* of all these choices.

**PROPOSITION 3.3.** *The symplectic isotopy class of the proper transform of a singular symplectic curve $C \subset (X, \omega)$ is independent of the choice of the Darboux ball, of $\varepsilon$ and $\rho$, and is related to the*
algebraic proper transform by a diffeomorphism supported in a neighborhood of the exceptional divisor.

*Proof.* We start by showing independence of $\epsilon$. Without loss of generality assume that $\epsilon_1 > \epsilon_0$. If one blow-up/down is performed using $\epsilon_0$ with function $\rho_0$ and the other using $\epsilon_1$ with function $\rho_1$, we may extend $\rho$ to the interval $[\lambda, \lambda + \epsilon_1]$ by extending by the identity on the interval $[\lambda + \epsilon_0, \lambda + \epsilon_1]$. Therefore, without loss of generality we may assume $\epsilon_0 = \epsilon_1$ and simply show independence of $\rho_i$.

To find the symplectic isotopy connecting the proper transforms for different choices of $\rho_i$, define a 1-parameter family of proper transforms using $\rho_s(r) = (1-s)\rho_0(r) + s\rho_1(r)$. The required characteristics of $\rho$ are convex conditions so $\rho_s(r)$ defines a symplectic proper transform for $s \in [0, 1]$ interpolating between the two choices of proper transform.

We now note that the space of embeddings of Darboux balls (with varying radius) in any connected symplectic manifold is itself connected. Thus, in order to prove independence (up to isotopy) of the Darboux ball, we can suppose that we have a 1-parameter family $D_t$ of Darboux balls centered at $p$, parametrized by symplectomorphisms $(\phi_t: D_r^4 \to X)_{t \in [0,1]}$ connecting two Darboux balls $D_0$ and $D_1$. (We can assume that the balls have the same volume by passing to a subfamily.) Then, by choosing $\rho + \epsilon < r$, and applying the recipe above, we obtain a 1-parameter family $\tilde{C}_t$ of symplectic proper transforms that varies smoothly with $t$, i.e. a symplectic isotopy between $\tilde{C}_0$ and $\tilde{C}_1$.

To see that the symplectic proper transform is related to the algebraic proper transform by a diffeomorphism, simply re-stretch out the ring between radius $\lambda$ and $\lambda + \epsilon$ in the blow-up by reparametrizing the radial coordinate by $\rho^{-1}$ (and shrink by reparametrizing by $\rho$ in the blow-down). Note, this diffeomorphism will not be a symplectomorphism. $\square$

## 3.2 Pseudoholomorphic curves

Here we prove two technical lemmas that we use throughout.

**LEMMA 3.4.** Let $(M, \omega)$ be a symplectic 4-manifold and $C \subset M$ be a singular symplectic surface. The space $\mathcal{J}^\omega(C)$ of almost complex structures $J$ which are compatible with the symplectic structure $\omega$ such that $C$ is $J$-holomorphic is non-empty and contractible.

*Proof.* The proof is a mild upgrade of a standard proof that the space of almost complex structures compatible with a given symplectic structure is contractible. There are multiple ways to prove this classical fact, and here we follow one given in [MS17, Proposition 2.63].

The first observation to make is that the condition that $J \in \mathcal{J}^\omega(C)$ is a pointwise condition. Namely, at each point $x \in M$, $J_x$ must be an $\omega_x$-compatible almost complex structure (at the vector space level), and whenever $x \in C$, we also require that $J$ preserves all tangent spaces to $C$ at $x$. Note that at a singular point in $C$, there may be more than one branch defining finitely many distinct tangent spaces, however each branch does have a well-defined tangent space (Remark 2.1). Note that at a critical point of a $J$-holomorphic map $u: \Sigma \to M$, $du = 0$ so the $J$-holomorphic condition $du \circ j = J \circ du$ is actually vacuous at that point. However, at nearby smooth points, the $J$-holomorphic condition does require that $J$ preserves the tangent space to $C$, so if $J$ varies continuously over points in $M$, because the tangent space at a singular point is the limit of nearby tangent spaces, the requirement that $J$ preserve the tangent spaces at singular points will be automatically satisfied.

We want to choose $J$ as a continuous section of the bundle $\pi: E \to M$ whose fiber $E_x$ over $x$ is the space of $\omega_x$ compatible almost complex structures on $T_x M$, such that the section is constrained over points $x \in C$ to the subset $E_x(C) \subset E_x$ of compatible almost complex structures
algebraic proper transform by a diffeomorphism supported in a neighborhood of the exceptional divisor.

*Proof.* We start by showing independence of $\epsilon$. Without loss of generality assume that $\epsilon_1 > \epsilon_0$. If one blow-up/down is performed using $\epsilon_0$ with function $\rho_0$ and the other using $\epsilon_1$ with function $\rho_1$, we may extend $\rho$ to the interval $[\lambda, \lambda + \epsilon_1]$ by extending by the identity on the interval $[\lambda + \epsilon_0, \lambda + \epsilon_1]$. Therefore, without loss of generality we may assume $\epsilon_0 = \epsilon_1$ and simply show independence of $\rho_i$.

To find the symplectic isotopy connecting the proper transforms for different choices of $\rho_i$, define a 1-parameter family of proper transforms using $\rho_s(r) = (1-s)\rho_0(r) + s\rho_1(r)$. The required characteristics of $\rho$ are convex conditions so $\rho_s(r)$ defines a symplectic proper transform for $s \in [0, 1]$ interpolating between the two choices of proper transform.

We now note that the space of embeddings of Darboux balls (with varying radius) in any connected symplectic manifold is itself connected. Thus, in order to prove independence (up to isotopy) of the Darboux ball, we can suppose that we have a 1-parameter family $D_t$ of Darboux balls centered at $p$, parametrized by symplectomorphisms $(\phi_t: D_r^4 \to X)_{t \in [0,1]}$ connecting two Darboux balls $D_0$ and $D_1$. (We can assume that the balls have the same volume by passing to a subfamily.) Then, by choosing $\rho + \epsilon < r$, and applying the recipe above, we obtain a 1-parameter family $\tilde{C}_t$ of symplectic proper transforms that varies smoothly with $t$, i.e. a symplectic isotopy between $\tilde{C}_0$ and $\tilde{C}_1$.

To see that the symplectic proper transform is related to the algebraic proper transform by a diffeomorphism, simply re-stretch out the ring between radius $\lambda$ and $\lambda + \epsilon$ in the blow-up by reparametrizing the radial coordinate by $\rho^{-1}$ (and shrink by reparametrizing by $\rho$ in the blow-down). Note, this diffeomorphism will not be a symplectomorphism. $\square$

## 3.2 Pseudoholomorphic curves

Here we prove two technical lemmas that we use throughout.

**LEMMA 3.4.** Let $(M, \omega)$ be a symplectic 4-manifold and $C \subset M$ be a singular symplectic surface. The space $\mathcal{J}^\omega(C)$ of almost complex structures $J$ which are compatible with the symplectic structure $\omega$ such that $C$ is $J$-holomorphic is non-empty and contractible.

*Proof.* The proof is a mild upgrade of a standard proof that the space of almost complex structures compatible with a given symplectic structure is contractible. There are multiple ways to prove this classical fact, and here we follow one given in [MS17, Proposition 2.63].

The first observation to make is that the condition that $J \in \mathcal{J}^\omega(C)$ is a pointwise condition. Namely, at each point $x \in M$, $J_x$ must be an $\omega_x$-compatible almost complex structure (at the vector space level), and whenever $x \in C$, we also require that $J$ preserves all tangent spaces to $C$ at $x$. Note that at a singular point in $C$, there may be more than one branch defining finitely many distinct tangent spaces, however each branch does have a well-defined tangent space (Remark 2.1). Note that at a critical point of a $J$-holomorphic map $u: \Sigma \to M$, $du = 0$ so the $J$-holomorphic condition $du \circ j = J \circ du$ is actually vacuous at that point. However, at nearby smooth points, the $J$-holomorphic condition does require that $J$ preserves the tangent space to $C$, so if $J$ varies continuously over points in $M$, because the tangent space at a singular point is the limit of nearby tangent spaces, the requirement that $J$ preserve the tangent spaces at singular points will be automatically satisfied.

We want to choose $J$ as a continuous section of the bundle $\pi: E \to M$ whose fiber $E_x$ over $x$ is the space of $\omega_x$ compatible almost complex structures on $T_x M$, such that the section is constrained over points $x \in C$ to the subset $E_x(C) \subset E_x$ of compatible almost complex structures
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

on $T_x M$ which preserve the tangent space(s) to $C$. We show that the contraction of $E_x$ to a
point from [MS17, Proposition 2.50] preserves the subset $E_x(C)$. Therefore, the space $J^\omega(C)$ is
contractible.

One way to show that $E_x$ is contractible is by showing that $E_x$ is homeomorphic to the space of symmetric positive definite symplectic matrices [MS17, Proposition 2.50(i), (iii)]. Fixing a standard symplectic basis on $T_x M$, let $J_0$ be the matrix for the standard complex structure. Then we can identify any almost complex structure $J$ on $T_x M$ with a symmetric, positive-definite, symplectic matrix $P$ by $P = -J_0J$ (and $J = J_0P$). Here, we choose the basis on $T_x M$ with some care along points of $C$. Near singular points of $C$, there is, by definition, a symplectic identification with a subset of $\mathbb{C}^2$ such that $C$ is identified with a complex curve (with respect to the standard complex structure). We choose the basis of $T_x M$ compatibly with the standard coordinates on $\mathbb{C}^2$ under this identification so that at singular points, $J_0$ preserves all tangent spaces to $C$ in $T_x M$. Along smooth points of $C$, the basis for $T_x M$ should extend a symplectic basis for $T_x C$. Therefore, $J_0$ preserves the tangent space(s) of $C$ at every point. Away from $C$ Darboux coordinates can be extended arbitrarily.

A deformation contraction from the space of symmetric, positive-definite, symplectic matrices to a point is given by sending $P$ to $P^{1-t}$ for $t \in [0, 1]$. It is proven in [MS17, Lemma 2.21] that $P^{1-t}$ is a (symmetric, positive-definite) symplectic matrix whenever $P$ is. The contraction deforming $J = J_0P$ to $J_0P^{1-t}$ gives the contraction of $E_x$. Therefore, it suffices to check that this contraction preserves the subset $E_x(C)$ of almost complex structures $J$ on $T_x M$ which preserve the tangent spaces of $C$.

As $J_0$ preserves the tangent spaces of $C$ by assumption on the choice of frame for $TM$
along $C$, $J$ preserves the tangent spaces of $C$ if and only if the corresponding matrix $P$ does.
As $P$ is symmetric, it is diagonalizable. A subspace of $T_x M$ is an invariant subspace of $P$
if and only if it is spanned by eigenvectors of $P$. Similarly a subspace is invariant under
$P^{1-t}$ if and only if it is spanned by eigenvectors of $P^{1-t}$. As the eigenvectors of $P$ are the
same as the eigenvectors of $P^{1-t}$, their invariant subspaces are the same. Therefore, if $P$ pres-
serves the tangent spaces of $C$, $P^{1-t}$ does as well. Thus, the contraction preserves the subset
$E_x(C)$. $\quad \square$

The strength of using $J$-holomorphic curves is that we have much greater control over their
geometric intersections. Two general symplectic surfaces may intersect with a canceling pair of
positive and negative intersections, but in this case they could not be realized as $J$-holomorphic
simultaneously for the same almost complex structure $J$. When $(M, \omega)$ is a 4-manifold with
a compatible almost complex structure $J$, the orientation induced by $\omega$ and $J$ agree. Two
transversally intersecting $J$-holomorphic curves can be easily seen to have positive intersections
because the almost complex structure induces complex orientations on each of the curves at
an intersection point, which adds up to the positive orientation on the 4-manifold induced by
$J$. More generally, any (not necessarily transverse) intersection between simple $J$-holomorphic
curves (possibly with singularities) contributes positively (see [MS12, Section 2.6,
Appendix E.2]).

LEMMA 3.5 [McD90]. Let $h, e_1, \dots, e_N$ be the standard basis for $H_2(\mathbb{CP}^2\#N\overline{\mathbb{CP}}^2)$ with $h^2 = 1$ and $e_i^2 = -1$. Suppose $C$ is a configuration of positively intersecting symplectic surfaces in $\mathbb{CP}^2\#N\overline{\mathbb{CP}}^2$. Let $e_{i_1}, \dots, e_{i_\ell}$ be exceptional classes which have non-negative algebraic intersections with each of the symplectic surfaces in the configuration $C$. Then there exist disjoint exceptional spheres $E_{i_1}, \dots, E_{i_\ell}$ representing the classes $e_{i_1}, \dots, e_{i_\ell}$, respectively, such that any geometric intersections of $E$ with $C$ are positive.
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

on $T_x M$ which preserve the tangent space(s) to $C$. We show that the contraction of $E_x$ to a
point from [MS17, Proposition 2.50] preserves the subset $E_x(C)$. Therefore, the space $J^\omega(C)$ is
contractible.

One way to show that $E_x$ is contractible is by showing that $E_x$ is homeomorphic to the space of symmetric positive definite symplectic matrices [MS17, Proposition 2.50(i), (iii)]. Fixing a standard symplectic basis on $T_x M$, let $J_0$ be the matrix for the standard complex structure. Then we can identify any almost complex structure $J$ on $T_x M$ with a symmetric, positive-definite, symplectic matrix $P$ by $P = -J_0J$ (and $J = J_0P$). Here, we choose the basis on $T_x M$ with some care along points of $C$. Near singular points of $C$, there is, by definition, a symplectic identification with a subset of $\mathbb{C}^2$ such that $C$ is identified with a complex curve (with respect to the standard complex structure). We choose the basis of $T_x M$ compatibly with the standard coordinates on $\mathbb{C}^2$ under this identification so that at singular points, $J_0$ preserves all tangent spaces to $C$ in $T_x M$. Along smooth points of $C$, the basis for $T_x M$ should extend a symplectic basis for $T_x C$. Therefore, $J_0$ preserves the tangent space(s) of $C$ at every point. Away from $C$ Darboux coordinates can be extended arbitrarily.

A deformation contraction from the space of symmetric, positive-definite, symplectic matrices to a point is given by sending $P$ to $P^{1-t}$ for $t \in [0, 1]$. It is proven in [MS17, Lemma 2.21] that $P^{1-t}$ is a (symmetric, positive-definite) symplectic matrix whenever $P$ is. The contraction deforming $J = J_0P$ to $J_0P^{1-t}$ gives the contraction of $E_x$. Therefore, it suffices to check that this contraction preserves the subset $E_x(C)$ of almost complex structures $J$ on $T_x M$ which preserve the tangent spaces of $C$.

As $J_0$ preserves the tangent spaces of $C$ by assumption on the choice of frame for $TM$
along $C$, $J$ preserves the tangent spaces of $C$ if and only if the corresponding matrix $P$ does.
As $P$ is symmetric, it is diagonalizable. A subspace of $T_x M$ is an invariant subspace of $P$
if and only if it is spanned by eigenvectors of $P$. Similarly a subspace is invariant under
$P^{1-t}$ if and only if it is spanned by eigenvectors of $P^{1-t}$. As the eigenvectors of $P$ are the
same as the eigenvectors of $P^{1-t}$, their invariant subspaces are the same. Therefore, if $P$ pre-
serves the tangent spaces of $C$, $P^{1-t}$ does as well. Thus, the contraction preserves the subset
$E_x(C)$. $\square$

The strength of using $J$-holomorphic curves is that we have much greater control over their
geometric intersections. Two general symplectic surfaces may intersect with a canceling pair of
positive and negative intersections, but in this case they could not be realized as $J$-holomorphic
simultaneously for the same almost complex structure $J$. When $(M, \omega)$ is a 4-manifold with
a compatible almost complex structure $J$, the orientation induced by $\omega$ and $J$ agree. Two
transversally intersecting $J$-holomorphic curves can be easily seen to have positive intersections
because the almost complex structure induces complex orientations on each of the curves at
an intersection point, which adds up to the positive orientation on the 4-manifold induced by
$J$. More generally, any (not necessarily transverse) intersection between simple $J$-holomorphic
curves (possibly with singularities) contributes positively (see [MS12, Section 2.6,
Appendix E.2]).

LEMMA 3.5 [McD90]. Let $h, e_1, \dots, e_N$ be the standard basis for $H_2(\mathbb{CP}^2\#N\overline{\mathbb{CP}}^2)$ with $h^2 = 1$ and $e_i^2 = -1$. Suppose $C$ is a configuration of positively intersecting symplectic surfaces in $\mathbb{CP}^2\#N\overline{\mathbb{CP}}^2$. Let $e_{i_1}, \dots, e_{i_\ell}$ be exceptional classes which have non-negative algebraic intersections with each of the symplectic surfaces in the configuration $C$. Then there exist disjoint exceptional spheres $E_{i_1}, \dots, E_{i_\ell}$ representing the classes $e_{i_1}, \dots, e_{i_\ell}$, respectively, such that any geometric intersections of $E$ with $C$ are positive.
Blowing down exceptional spheres using this lemma, together with exceptional spheres
appearing in a configuration, eventually we blow down spheres in all $e_i$ classes and reach a
configuration in $\mathbb{CP}^2$.

Proof. Fix an almost complex structure $J \in \mathcal{J}^\omega(C)$ which is $C^\infty$. Let $\Lambda = \langle e_{i_1}, \dots, e_{i_\ell} \rangle$. As in [McD90, Theorem 3.4], one can find a maximal collection of disjoint exceptional $J$-holomorphic curves generating $\Lambda$. (These necessarily represent $e_i$ classes, because these are the only classes of square $-1$ which are positively oriented by $J$.) They will intersect $C$ positively because both are $J$-holomorphic. $\square$

The order of the $e_i$ which we choose to $J$-holomorphically blow down does depend to some extent on the configuration $C$ because if one of the surfaces in $C$ represents the class $e_{i_0} - e_{i_1} - \cdots - e_{i_k}$, then we cannot blow down a $J$-holomorphic exceptional sphere in the class $e_{i_0}$ until we have blown down exceptional spheres in the classes $e_{i_1}, \dots, e_{i_k}$ first. This is because $e_{i_0}$ has negative algebraic intersection number with $e_{i_0} - e_{i_1} - \cdots - e_{i_k}$. If $k=0$, then the surface in $C$ represents the class $e_{i_0}$ so it can itself act as the $J$-holomorphic exceptional sphere. However, this is the only restriction on the ordering because the positively intersecting exceptional classes can be geometrically realized disjointly.

## 3.3 Embeddings of plumbings into $\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$

Suppose $P$ is a plumbing of symplectic spheres, such that one of the spheres has self-intersection $+1$. In our context, this will typically be a neighborhood of the normal crossing resolution of a rational cuspidal curve (or possibly a further blow-up). A theorem of McDuff strongly restricts the closed symplectic manifolds in which $P$ can symplectically embed.

**THEOREM 3.6** [McD90]. If $(X, \omega)$ is a closed symplectic 4-manifold and $C_0 \subset X$ is a smooth symplectic sphere of self-intersection number $+1$, then there is a symplectomorphism of $(X, \omega)$ to a symplectic blow up of $(\mathbb{CP}^2, \lambda\omega_{\mathrm{FS}})$ for some $\lambda > 0$, such that $C_0$ is identified with $\mathbb{CP}^1$.

More generally, McDuff's result shows that if the plumbing contains a sphere of square $p \neq 4$, $p > 0$, then the symplectic manifold is a blow-up of the $p$th Hirzebruch surface and the sphere is identified with the associated 0-section. A sphere of square 4 might be the conic in $\mathbb{CP}^2$ or the sphere representing $2[S^2 \times \{*\}] + [\{*\} \times S^2]$ in $S^2 \times S^2$. A sphere of square 0 is a fiber in a ruled surface. We typically work in cases where we can guarantee that our symplectic 4-manifold is a blow-up of $\mathbb{CP}^2$ (or occasionally in $S^2 \times S^2$). (Note that $S^2 \times S^2 \# \overline{\mathbb{CP}}^2 \cong \mathbb{CP}^2 \# 2\overline{\mathbb{CP}}^2$.)

We now discuss how to classify all symplectic embeddings of $P$ into $\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$. A symplectic embedding of a rational cuspidal curve is equivalent (by a sequence of blow-ups supported in a neighborhood of the rational cuspidal curve) to a symplectic embedding of its normal crossing resolution plumbing $(U, \omega_U)$.

We classify embeddings of $P$ into $\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$ in two steps (similar arguments appear in many classification of fillings results, starting with Lisca [Lis08]). First, we determine the possibilities for the map on second homology induced by the embedding. As the core spheres of the plumbing form a basis for $H_2(P)$, we just need to classify the possible classes in $H_2(\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2)$ that these symplectic spheres can represent. The restrictions here are given by the adjunction formula and the intersection form on $P$, as well as the constraint that the $+1$-sphere in $P$ is identified with $\mathbb{CP}^1 \subset \mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$. Next, for each possible adjunctive embedding $H_2(P) \to H_2(\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2)$, we classify the geometric realizations of embeddings $P \to \mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$ up to symplectic isotopy. Typically, we show that such a geometric embedding is unique or that it cannot exist. This will
Blowing down exceptional spheres using this lemma, together with exceptional spheres
appearing in a configuration, eventually we blow down spheres in all $e_i$ classes and reach a
configuration in $\mathbb{CP}^2$.

Proof. Fix an almost complex structure $J \in \mathcal{J}^\omega(C)$ which is $C^\infty$. Let $\Lambda = \langle e_{i_1}, \dots, e_{i_\ell} \rangle$. As in [McD90, Theorem 3.4], one can find a maximal collection of disjoint exceptional $J$-holomorphic curves generating $\Lambda$. (These necessarily represent $e_i$ classes, because these are the only classes of square $-1$ which are positively oriented by $J$.) They will intersect $C$ positively because both are $J$-holomorphic. $\square$

The order of the $e_i$ which we choose to $J$-holomorphically blow down does depend to some extent on the configuration $C$ because if one of the surfaces in $C$ represents the class $e_{i_0} - e_{i_1} - \cdots - e_{i_k}$, then we cannot blow down a $J$-holomorphic exceptional sphere in the class $e_{i_0}$ until we have blown down exceptional spheres in the classes $e_{i_1}, \dots, e_{i_k}$ first. This is because $e_{i_0}$ has negative algebraic intersection number with $e_{i_0} - e_{i_1} - \cdots - e_{i_k}$. If $k=0$, then the surface in $C$ represents the class $e_{i_0}$ so it can itself act as the $J$-holomorphic exceptional sphere. However, this is the only restriction on the ordering because the positively intersecting exceptional classes can be geometrically realized disjointly.

## 3.3 Embeddings of plumbings into $\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$

Suppose $P$ is a plumbing of symplectic spheres, such that one of the spheres has self-intersection $+1$. In our context, this will typically be a neighborhood of the normal crossing resolution of a rational cuspidal curve (or possibly a further blow-up). A theorem of McDuff strongly restricts the closed symplectic manifolds in which $P$ can symplectically embed.

**THEOREM 3.6** [McD90]. If $(X, \omega)$ is a closed symplectic 4-manifold and $C_0 \subset X$ is a smooth symplectic sphere of self-intersection number $+1$, then there is a symplectomorphism of $(X, \omega)$ to a symplectic blow up of $(\mathbb{CP}^2, \lambda\omega_{\mathrm{FS}})$ for some $\lambda > 0$, such that $C_0$ is identified with $\mathbb{CP}^1$.

More generally, McDuff's result shows that if the plumbing contains a sphere of square $p \neq 4$, $p > 0$, then the symplectic manifold is a blow-up of the $p$th Hirzebruch surface and the sphere is identified with the associated 0-section. A sphere of square 4 might be the conic in $\mathbb{CP}^2$ or the sphere representing $2[S^2 \times \{*\}] + [\{*\} \times S^2]$ in $S^2 \times S^2$. A sphere of square 0 is a fiber in a ruled surface. We typically work in cases where we can guarantee that our symplectic 4-manifold is a blow-up of $\mathbb{CP}^2$ (or occasionally in $S^2 \times S^2$). (Note that $S^2 \times S^2 \# \overline{\mathbb{CP}}^2 \cong \mathbb{CP}^2 \# 2\overline{\mathbb{CP}}^2$.)

We now discuss how to classify all symplectic embeddings of $P$ into $\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$. A symplectic embedding of a rational cuspidal curve is equivalent (by a sequence of blow-ups supported in a neighborhood of the rational cuspidal curve) to a symplectic embedding of its normal crossing resolution plumbing $(U, \omega_U)$.

We classify embeddings of $P$ into $\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$ in two steps (similar arguments appear in many classification of fillings results, starting with Lisca [Lis08]). First, we determine the possibilities for the map on second homology induced by the embedding. As the core spheres of the plumbing form a basis for $H_2(P)$, we just need to classify the possible classes in $H_2(\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2)$ that these symplectic spheres can represent. The restrictions here are given by the adjunction formula and the intersection form on $P$, as well as the constraint that the $+1$-sphere in $P$ is identified with $\mathbb{CP}^1 \subset \mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$. Next, for each possible adjunctive embedding $H_2(P) \to H_2(\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2)$, we classify the geometric realizations of embeddings $P \to \mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$ up to symplectic isotopy. Typically, we show that such a geometric embedding is unique or that it cannot exist. This will
be done by supposing that we have such a geometric embedding into $CP^2#N\overline{CP}^2$, keeping track of the configuration formed by the core symplectic spheres of the plumbing, and then blowing down using Lemma 3.5. We then look at the possible images of this configuration after blowing down to $CP^2$. This will generally be a reducible configuration, which we will then try to classify up to symplectic isotopy. This classification of reducible symplectic configurations in $CP^2$ is the subject of § 5.

Now we analyze the possible homology classes represented by the spheres in P. A symplectic surface $\Sigma$ in a symplectic 4-manifold $(X, \omega)$ satisfies the adjunction formula (2.2). When $X = CP^2#N\overline{CP}^2$, $K_X = -c_1(X) = -3H + E_1 + \cdots + E_N$. Here our convention is that $h$ is the class of $CP^1$, with dual $H$ and $e_i$ is represented by the $i$th exceptional sphere with dual $E_i$. The following lemma is a generalization of [Lis08, Propositions 4.4]

LEMMA 3.7. Suppose $\Sigma$ is a smooth symplectic sphere in $CP^2#N\overline{CP}^2$ intersecting $CP^1$ non-negatively. Then writing $[\Sigma] = a_0h + a_1e_1 + \cdots + a_Ne_N$ (so $a_0 \ge 0$), we have:

$$ (i) \sum (a_i^2 + a_i) = 2 + a_0^2 - 3a_0; $$

$$ (ii) \text{ if } a_0 = 0, \text{ there is one } i_0 \text{ such that } a_{i_0} = 1 \text{ and all other } a_i \in \{0, -1\}; $$

$$ (iii) \text{ if } a_0 \neq 0, \text{ then for all } i \geq 1, a_i \leq 0. $$

Some particular cases which we use often are:

$$ (iv) \text{ if } a_0 = 1 \text{ or } a_0 = 2, a_i \in \{0, -1\} \text{ for all } 1 \leq i \leq N; $$

$$ (v) \text{ if } a_0 = 3, \text{ then there exists a unique } i_0 \text{ such that } a_{i_0} = -2, \text{ and } a_i \in \{0, -1\} \text{ for all other } i. $$

The self-intersection number of $\Sigma$ can be used to compute how many $a_i$ have coefficient 0 versus -1.

Proof. As $\Sigma$ is a sphere ($g=0$), the adjunction formula gives

$$ 3a_0 + a_1 + \cdots + a_N = 2 + a_0^2 - a_1^2 - \cdots - a_N^2, $$

which can be rearranged to item (i). Note that $a_i^2 + a_i \geq 0$ because $a_i \in \mathbb{Z}$, so the coefficient $a_0$ determines a bound on the possible coefficients $a_i$.

To prove items (ii) and (iii), we use positivity of intersections. Fix an index $0 < i_0 \leq N$ and suppose first that $a_{i_0} > 0$. As $\Sigma$ and $CP^1$ are symplectic with non-negative intersections, there is an almost complex structure on $CP^2#N\overline{CP}^2$ such that $\Sigma$ and $CP^1$ are $J$-holomorphic by Lemma 3.4. By Lemma 3.5, we can blow-down a $J$-holomorphic exceptional sphere in the class $e_i$ for some $i$. If we blow down a $J$-holomorphic sphere corresponding to $e_i$ for $i \neq i_0$, there is an induced $J'$ on the blow-down such that $\Sigma$ is a singular $J'$-holomorphic curve. Therefore, eventually we have a situation where both $\Sigma$ and $e_{i_0}$ are represented by $J$-holomorphic spheres, so if $a_{i_0} > 0$, then $[\Sigma] \cdot e_{i_0} < 0$. As $J$-holomorphic spheres must intersect non-negatively, this can only occur if $\Sigma$ is exactly equal to the $J$-holomorphic exceptional sphere representing $e_{i_0}$. In particular, if $a_{i_0} > 0$, then $a_0 = 0$, thus proving item (iii). If $a_0 = 0$, it implies $a_{i_0} = 1$ and $\Sigma$ is a blow-up of the exceptional class representing $e_{i_0}$. In this case, for $i_j \neq i_0$ the same argument shows that $a_{i_j}$ cannot be positive so we get item (ii).

The particular cases follow from combining items (i) and (iii) and observing which integers give low values of $n^2 + n$. $\square$

In the following lemmas, $C_i$ and $C_j$ are smooth symplectic spheres in a positive plumbing in $CP^2#N\overline{CP}^2$ such that $[C_i] \cdot h = [C_j] \cdot h = 0$.
be done by supposing that we have such a geometric embedding into $CP^2#N\overline{CP}^2$, keeping track of the configuration formed by the core symplectic spheres of the plumbing, and then blowing down using Lemma 3.5. We then look at the possible images of this configuration after blowing down to $CP^2$. This will generally be a reducible configuration, which we will then try to classify up to symplectic isotopy. This classification of reducible symplectic configurations in $CP^2$ is the subject of § 5.

Now we analyze the possible homology classes represented by the spheres in P. A symplectic surface $\Sigma$ in a symplectic 4-manifold $(X, \omega)$ satisfies the adjunction formula (2.2). When $X = CP^2#N\overline{CP}^2$, $K_X = -c_1(X) = -3H + E_1 + \cdots + E_N$. Here our convention is that $h$ is the class of $CP^1$, with dual $H$ and $e_i$ is represented by the $i$th exceptional sphere with dual $E_i$. The following lemma is a generalization of [Lis08, Propositions 4.4]

LEMMA 3.7. Suppose $\Sigma$ is a smooth symplectic sphere in $CP^2#N\overline{CP}^2$ intersecting $CP^1$ non-negatively. Then writing $[\Sigma] = a_0h + a_1e_1 + \cdots + a_Ne_N$ (so $a_0 \ge 0$), we have:

(i) $\sum (a_i^2 + a_i) = 2 + a_0^2 - 3a_0$;

(ii) if $a_0 = 0$, there is one $i_0$ such that $a_{i_0} = 1$ and all other $a_i \in \{0, -1\}$;

(iii) if $a_0 \neq 0$, then for all $i \ge 1$, $a_i \le 0$.

Some particular cases which we use often are:

(iv) if $a_0 = 1$ or $a_0 = 2$, $a_i \in \{0, -1\}$ for all $1 \le i \le N$;

(v) if $a_0 = 3$, then there exists a unique $i_0$ such that $a_{i_0} = -2$, and $a_i \in \{0, -1\}$ for all other $i$.

The self-intersection number of $\Sigma$ can be used to compute how many $a_i$ have coefficient 0 versus $-1$.

Proof. As $\Sigma$ is a sphere ($g=0$), the adjunction formula gives

$$3a_0 + a_1 + \cdots + a_N = 2 + a_0^2 - a_1^2 - \cdots - a_N^2,$$

which can be rearranged to item (i). Note that $a_i^2 + a_i \ge 0$ because $a_i \in \mathbb{Z}$, so the coefficient $a_0$ determines a bound on the possible coefficients $a_i$.

To prove items (ii) and (iii), we use positivity of intersections. Fix an index $0 < i_0 \le N$ and suppose first that $a_{i_0} > 0$. As $\Sigma$ and $CP^1$ are symplectic with non-negative intersections, there is an almost complex structure on $CP^2#N\overline{CP}^2$ such that $\Sigma$ and $CP^1$ are $J$-holomorphic by Lemma 3.4. By Lemma 3.5, we can blow-down a $J$-holomorphic exceptional sphere in the class $e_i$ for some $i$. If we blow down a $J$-holomorphic sphere corresponding to $e_i$ for $i \neq i_0$, there is an induced $J'$ on the blow-down such that $\Sigma$ is a singular $J'$-holomorphic curve. Therefore, eventually we have a situation where both $\Sigma$ and $e_{i_0}$ are represented by $J$-holomorphic spheres, so if $a_{i_0} > 0$, then $[\Sigma] \cdot e_{i_0} < 0$. As $J$-holomorphic spheres must intersect non-negatively, this can only occur if $\Sigma$ is exactly equal to the $J$-holomorphic exceptional sphere representing $e_{i_0}$. In particular, if $a_{i_0} > 0$, then $a_0 = 0$, thus proving item (iii). If $a_0 = 0$, it implies $a_{i_0} = 1$ and $\Sigma$ is a blow-up of the exceptional class representing $e_{i_0}$. In this case, for $i_j \neq i_0$ the same argument shows that $a_{i_j}$ cannot be positive so we get item (ii).

The particular cases follow from combining items (i) and (iii) and observing which integers give low values of $n^2 + n$. $\square$

In the following lemmas, $C_i$ and $C_j$ are smooth symplectic spheres in a positive plumbing in $CP^2#N\overline{CP}^2$ such that $[C_i] \cdot h = [C_j] \cdot h = 0$.
LEMMA 3.8. If $[C_i] \cdot [C_j] = 1$ (and $[C_i] \cdot h = [C_j] \cdot h = 0$), there is exactly one exceptional class $e_i$ which appears with non-zero coefficient in both $[C_i]$ and $[C_j]$. The coefficient of $e_i$ is +1 in one of $[C_i]$, $[C_j]$ and $-1$ in the other.

Proof. It follows from Lemma 3.7(ii), that $[C_i]$ and $[C_j]$ have the form $[C_i] = e_{n_0} - e_{n_1} - \cdots - e_{n_k}$, $[C_j] = e_{m_0} - e_{m_1} - \cdots - e_{m_l}$. As the algebraic intersection is +1, either $e_{n_0} = e_{m_q}$ or $e_{m_0} = e_{n_p}$ for some $q, p$. To rule out the possibility that both of these occur, we consider the symplectic areas. As $C_i$ and $C_j$ are both symplectic spheres, $\langle \omega, [C_i] \rangle, \langle \omega, [C_j] \rangle > 0$. Each of the exceptional classes also has positive symplectic area. Let $a := \langle \omega, e_{n_0} \rangle > 0$ and $b := \langle \omega, e_{m_0} \rangle > 0$. Then

$$0 < \langle \omega, [C_i] \rangle < a - b \quad \text{and} \quad 0 < \langle \omega, [C_j] \rangle < b - a,$$

which is a contradiction. □

LEMMA 3.9. If $e_m$ appears with coefficient +1 in $[C_i]$, then it does not appear with coefficient +1 in the homology class of any other sphere in the plumbing.

Proof. This follows from positivity of intersection and Lemma 3.7. □

LEMMA 3.10. If $[C_i] \cdot [C_j] = 0$, then either there is no exceptional class which appears with non-zero coefficients in both, or there are exactly two exceptional classes $e_m$ and $e_n$ appearing with non-zero coefficients in both. One of these classes $e_m$ has coefficient $-1$ in both $[C_i]$ and $[C_j]$ and the other $e_n$ appears with coefficient +1 in one of $[C_i]$ or $[C_j]$ and coefficient $-1$ in the other.

Proof. This follows from a similar argument as in Lemma 3.8, but with an additional exceptional class appearing with coefficient $-1$ in both $[C_i]$ and $[C_j]$ to cancel out the positive intersection. □

When there is a linear chain of such symplectic spheres with self-intersection $-2$ (consecutive spheres intersect once), there are few options for the homology classes of the spheres in that chain.

LEMMA 3.11. Suppose $\Sigma_1, \dots, \Sigma_k$ are a chain of symplectic spheres of self-intersection $-2$ disjoint from $\mathbb{CP}^1$ in $\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$. Then the homology classes are given by one of the following two options up to re-indexing the exceptional classes:

(A) $[\Sigma_i] = e_i - e_{i+1}$ for $i = 1, \dots, k$;

(B) $[\Sigma_i] = e_{i+1} - e_i$ for $i = 1, \dots, k$.

The homology class of any surface disjoint from the chain has the same coefficient for $e_1, \dots, e_{k+1}$.

Proof. By Lemma 3.7(ii), each $[\Sigma_i] = e_{k_1^i} - e_{k_2^i}$. As consecutive spheres intersect once positively, by Lemma 3.10 either:

(i) $k_2^i = k_1^{i+1}$; or

(ii) $k_1^i = k_2^{i+1}$.

Each exceptional class can appear with positive coefficient at most once by Lemma 3.9 so all $k_1^i$ are distinct. Therefore, if for any $i_0$, $k_1^{i_0} = k_2^{i_0+1}$, we must have that $k_1^i = k_2^{i+1}$ for all $i_0 \le i \le k$. We can never switch from choosing option (ii) to choosing option (i) as we go down the chain.

If $k_2^{i_0} = k_1^{i_0+1}$ and $k_1^{i_0+1} = k_2^{i_0+2}$ (i.e. if we switch from option (i) to option (ii)), then we have $[\Sigma_{i_0}] = e_a - e_b$, $[\Sigma_{i_0+1}] = e_b - e_c$, and $[\Sigma_{i_0+2}] = e_d - e_b$. As $[\Sigma_{i_0}] \cdot [\Sigma_{i_0+2}] = 0$ we must have $(e_a - e_b) \cdot (e_d - e_b) = 0$ but $(e_a - e_b) \cdot (e_d - e_b) = e_a \cdot e_d - 1 < 0$, so we cannot switch from option (i) to option (ii) either.
LEMMA 3.8. If $[C_i] \cdot [C_j] = 1$ (and $[C_i] \cdot h = [C_j] \cdot h = 0$), there is exactly one exceptional class $e_i$ which appears with non-zero coefficient in both $[C_i]$ and $[C_j]$. The coefficient of $e_i$ is +1 in one of $[C_i]$, $[C_j]$ and $-1$ in the other.

*Proof.* It follows from Lemma 3.7(ii), that $[C_i]$ and $[C_j]$ have the form $[C_i] = e_{n_0} - e_{n_1} - \cdots - e_{n_k}$, $[C_j] = e_{m_0} - e_{m_1} - \cdots - e_{m_l}$. As the algebraic intersection is +1, either $e_{n_0} = e_{m_q}$ or $e_{m_0} = e_{n_p}$ for some $q, p$. To rule out the possibility that both of these occur, we consider the symplectic areas. As $C_i$ and $C_j$ are both symplectic spheres, $\langle \omega, [C_i] \rangle, \langle \omega, [C_j] \rangle > 0$. Each of the exceptional classes also has positive symplectic area. Let $a := \langle \omega, e_{n_0} \rangle > 0$ and $b := \langle \omega, e_{m_0} \rangle > 0$. Then

$$0 < \langle \omega, [C_i] \rangle < a - b \quad \text{and} \quad 0 < \langle \omega, [C_j] \rangle < b - a,$$

which is a contradiction.

LEMMA 3.9. If $e_m$ appears with coefficient +1 in $[C_i]$, then it does not appear with coefficient +1 in the homology class of any other sphere in the plumbing.

*Proof.* This follows from positivity of intersection and Lemma 3.7.

LEMMA 3.10. If $[C_i] \cdot [C_j] = 0$, then either there is no exceptional class which appears with non-zero coefficients in both, or there are exactly two exceptional classes $e_m$ and $e_n$ appearing with non-zero coefficients in both. One of these classes $e_m$ has coefficient $-1$ in both $[C_i]$ and $[C_j]$ and the other $e_n$ appears with coefficient +1 in one of $[C_i]$ or $[C_j]$ and coefficient $-1$ in the other.

*Proof.* This follows from a similar argument as in Lemma 3.8, but with an additional exceptional class appearing with coefficient $-1$ in both $[C_i]$ and $[C_j]$ to cancel out the positive intersection.

When there is a linear chain of such symplectic spheres with self-intersection $-2$ (consecutive spheres intersect once), there are few options for the homology classes of the spheres in that chain.

LEMMA 3.11. Suppose $\Sigma_1, \dots, \Sigma_k$ are a chain of symplectic spheres of self-intersection $-2$ disjoint from $\mathbb{CP}^1$ in $\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$. Then the homology classes are given by one of the following two options up to re-indexing the exceptional classes:

(A) $[\Sigma_i] = e_i - e_{i+1}$ for $i = 1, \dots, k$;

(B) $[\Sigma_i] = e_{i+1} - e_i$ for $i = 1, \dots, k$.

The homology class of any surface disjoint from the chain has the same coefficient for $e_1, \dots, e_{k+1}$.

*Proof.* By Lemma 3.7(ii), each $[\Sigma_i] = e_{k_1^i} - e_{k_2^i}$. As consecutive spheres intersect once positively, by Lemma 3.10 either:

(i) $k_2^i = k_1^{i+1}$; or

(ii) $k_1^i = k_2^{i+1}$.

Each exceptional class can appear with positive coefficient at most once by Lemma 3.9 so all $k_1^i$ are distinct. Therefore, if for any $i_0$, $k_1^{i_0} = k_2^{i_0+1}$, we must have that $k_1^i = k_2^{i+1}$ for all $i_0 \le i \le k$. We can never switch from choosing option (ii) to choosing option (i) as we go down the chain.

If $k_2^{i_0} = k_1^{i_0+1}$ and $k_1^{i_0+1} = k_2^{i_0+2}$ (i.e. if we switch from option (i) to option (ii)), then we have $[\Sigma_{i_0}] = e_a - e_b$, $[\Sigma_{i_0+1}] = e_b - e_c$, and $[\Sigma_{i_0+2}] = e_d - e_b$. As $[\Sigma_{i_0}] \cdot [\Sigma_{i_0+2}] = 0$ we must have $(e_a - e_b) \cdot (e_d - e_b) = 0$ but $(e_a - e_b) \cdot (e_d - e_b) = e_a \cdot e_d - 1 < 0$, so we cannot switch from option (i) to option (ii) either.
Therefore, if we start with option (i), we get choice (A), and if we start with option (ii) we get choice (B). For last statement, let $a_i$ denote the coefficient of $e_i$ in $[\Sigma]$. Then the statement follows from $0 = [\Sigma] \cdot [\Sigma_i] = \mp(a_i - a_{i+1})$. $\square$

Often, only one of these options can occur. The following particular case will appear frequently in our homological classifications.

LEMMA 3.12. If $\Sigma_1, \dots, \Sigma_k$ form a chain of $(-2)$-spheres as in Lemma 3.11, such that the chain is attached to another symplectic sphere $\Sigma_0$ which does intersect $\mathbb{CP}^1$, option (B) can only occur if $e_2, \dots, e_{k+1}$ all appear with coefficient $-1$ in $[\Sigma_0]$. In particular if $[\Sigma_0] \cdot h = 1$, option (B) can only occur if $[\Sigma_0]^2 \le 1 - k$.

*Proof.* By Lemma 3.7, all coefficients of exceptional classes in $[\Sigma_0]$ are negative. Let $[\Sigma_0] = a_0h + \sum_i a_i e_i$. As $[\Sigma_0] \cdot [\Sigma_1] = 1$ because they are joined in the chain, the exceptional class which appears with coefficient $+1$ in $[\Sigma_1]$ must appear with coefficient $-1$ in $[\Sigma_0]$. If the chain $\Sigma_1, \dots, \Sigma_k$ has homology classes as in option (B), this means $a_2 = -1$. As $[\Sigma_0] \cdot [\Sigma_i] = 0$ for $i = 2, \dots, k$, we find that $-a_{i+1} + a_i = 0$ for $i = 2, \dots, k$ so $a_2 = \dots = a_{k+1} = -1$.

When $a_0 = 1$, $a_i \in \{0, -1\}$ for all $i \ge 1$ by Lemma 3.7 so $[\Sigma_0]^2 = 1 - \#\{a_i \ne 0\}$. If we have the chain of $-2$ spheres with homology classes as in option (B), we must have at least $k$ non-zero coefficients so $[\Sigma_0]^2 \le 1 - k$. $\square$

These lemmas, together with some arithmetic considerations, generally suffice to allow us to classify all possibilities for the homology classes of the spheres in a normal crossing resolution. Given these homology classes we can apply Lemma 3.5 to blow down to a configuration in $\mathbb{CP}^2$. The way that the exceptional classes appeared in the homology classes of the plumbing spheres affects the intersections between the proper transforms. We record the data of these intersections, including the degree of tangency between surfaces at each intersection and when intersections between different components coincide. There may also be singularities in a single component which we record as well. Fixing this data of the singularities and intersections, we then try to classify symplectic configurations of surfaces in $\mathbb{CP}^2$ up to equisingular symplectic isotopy. In the next section we solve this classification for families of singular reducible configurations of symplectic surfaces in $\mathbb{CP}^2$ that we need.

## 3.4 Birational transformations

In complex dimension two, a birational transformation is a sequence of blow-ups and blow-downs. As blow-ups and blow-downs can be done symplectically [McD90], these transformations from algebraic geometry can be imported into the symplectic context. When we have a singular surface in $\mathbb{CP}^2$, the birational transformation may change the singularities and self-intersection number of the surface. The blow-ups can begin to resolve singularities, and blow-downs may create new singularities.

There are two ways that we will relate singular symplectic surfaces in $\mathbb{CP}^2$ using birational transformations. The first notion is weaker, but for two surfaces related in this way, the existence of one type of singular surface will imply the existence of another type of singular surface.

DEFINITION 3.13. A configuration $C_2$ in $(M_2, \omega_2)$ is birationally derived from another configuration $C_1 \subset (M_1, \omega_1)$ if for every symplectic realization $\Sigma_1$ of $C_1$ in $(M_1, \omega_1)$, there is a sequence of blow-ups of the pair $(M_1, \Sigma_1)$ to the total transform $(M_1 \# N\overline{\mathbb{CP}}^2, \overline{\Sigma}_1)$, followed by a sequence of blow-downs of exceptional spheres $\pi : M_1 \# N\overline{\mathbb{CP}}^2 \to M_2$, such that $\Sigma_2 = \pi(\overline{\Sigma}_1)$ is a realization of $C_2$.
Therefore, if we start with option (i), we get choice (A), and if we start with option (ii) we get choice (B). For last statement, let $a_i$ denote the coefficient of $e_i$ in $[\Sigma]$. Then the statement follows from $0 = [\Sigma] \cdot [\Sigma_i] = \mp (a_i - a_{i+1})$. $\square$

Often, only one of these options can occur. The following particular case will appear frequently in our homological classifications.

**LEMMA 3.12.** If $\Sigma_1, \dots, \Sigma_k$ form a chain of $(-2)$-spheres as in Lemma 3.11, such that the chain is attached to another symplectic sphere $\Sigma_0$ which does intersect $\mathbb{CP}^1$, option (B) can only occur if $e_2, \dots, e_{k+1}$ all appear with coefficient $-1$ in $[\Sigma_0]$. In particular if $[\Sigma_0] \cdot h = 1$, option (B) can only occur if $[\Sigma_0]^2 \le 1 - k$.

*Proof.* By Lemma 3.7, all coefficients of exceptional classes in $[\Sigma_0]$ are negative. Let $[\Sigma_0] = a_0h + \sum_i a_i e_i$. As $[\Sigma_0] \cdot [\Sigma_1] = 1$ because they are joined in the chain, the exceptional class which appears with coefficient $+1$ in $[\Sigma_1]$ must appear with coefficient $-1$ in $[\Sigma_0]$. If the chain $\Sigma_1, \dots, \Sigma_k$ has homology classes as in option (B), this means $a_2 = -1$. As $[\Sigma_0] \cdot [\Sigma_i] = 0$ for $i = 2, \dots, k$, we find that $-a_{i+1} + a_i = 0$ for $i = 2, \dots, k$ so $a_2 = \dots = a_{k+1} = -1$.

When $a_0 = 1$, $a_i \in \{0, -1\}$ for all $i \ge 1$ by Lemma 3.7 so $[\Sigma_0]^2 = 1 - \#\{a_i \ne 0\}$. If we have the chain of $-2$ spheres with homology classes as in option (B), we must have at least $k$ non-zero coefficients so $[\Sigma_0]^2 \le 1 - k$. $\square$

These lemmas, together with some arithmetic considerations, generally suffice to allow us to classify all possibilities for the homology classes of the spheres in a normal crossing resolution. Given these homology classes we can apply Lemma 3.5 to blow down to a configuration in $\mathbb{CP}^2$. The way that the exceptional classes appeared in the homology classes of the plumbing spheres affects the intersections between the proper transforms. We record the data of these intersections, including the degree of tangency between surfaces at each intersection and when intersections between different components coincide. There may also be singularities in a single component which we record as well. Fixing this data of the singularities and intersections, we then try to classify symplectic configurations of surfaces in $\mathbb{CP}^2$ up to equisingular symplectic isotopy. In the next section we solve this classification for families of singular reducible configurations of symplectic surfaces in $\mathbb{CP}^2$ that we need.

## 3.4 Birational transformations

In complex dimension two, a birational transformation is a sequence of blow-ups and blow-downs. As blow-ups and blow-downs can be done symplectically [McD90], these transformations from algebraic geometry can be imported into the symplectic context. When we have a singular surface in $\mathbb{CP}^2$, the birational transformation may change the singularities and self-intersection number of the surface. The blow-ups can begin to resolve singularities, and blow-downs may create new singularities.

There are two ways that we will relate singular symplectic surfaces in $\mathbb{CP}^2$ using birational transformations. The first notion is weaker, but for two surfaces related in this way, the existence of one type of singular surface will imply the existence of another type of singular surface.

**DEFINITION 3.13.** A configuration $C_2$ in $(M_2, \omega_2)$ is birationally derived from another configuration $C_1 \subset (M_1, \omega_1)$ if for every symplectic realization $\Sigma_1$ of $C_1$ in $(M_1, \omega_1)$, there is a sequence of blow-ups of the pair $(M_1, \Sigma_1)$ to the total transform $(M_1 \# N\overline{\mathbb{CP}}^2, \overline{\Sigma}_1)$, followed by a sequence of blow-downs of exceptional spheres $\pi : M_1 \# N\overline{\mathbb{CP}}^2 \to M_2$, such that $\Sigma_2 = \pi(\overline{\Sigma}_1)$ is a realization of $C_2$.
From this definition, if we birationally invert the blow-down $\pi$ by blowing-up, we see that the total transform of $\Sigma_2$ in $M_1\#N\overline{CP}^2$, contains the total transform of $\Sigma_1$, i.e. $\bar{\Sigma}_1 \subseteq \bar{\Sigma}_2$. This follows from the fact that a set is contained in the preimage of its image, so $\bar{\Sigma}_2 = \pi^{-1}(\Sigma_2) = \pi^{-1}(\pi(\bar{\Sigma}_1)) \supseteq \bar{\Sigma}_1$. For this reason, we can also refer to the birational derivation relation as saying that $C_1$ is birationally contained in $C_2$.

Note that this relation is directional. If $C_2$ is birationally derived from $C_1$, it is not typically true that $C_1$ is birationally derived from $C_2$. To strengthen this notion, we impose a more restrictive condition that the exceptional spheres that are blown down by $\pi$ are actually contained in $\bar{\Sigma}_1$.

**DEFINITION 3.14.** We say that $C_1$ in $(M_1, \omega_1)$ is birationally equivalent to $C_2$ in $(M_2, \omega_2)$ if for every symplectic realization $\Sigma_1$ of $C_1$ there is a sequence of blow-ups of the pair $(M_1, \Sigma_1)$ to the total transform $(M_1\#N\overline{CP}^2, \bar{\Sigma}_1)$, followed by a sequence of blow-downs of exceptional spheres $\pi: M_1\#N\overline{CP}^2 \to M_2$ such that the exceptional locus of $\pi$ is contained in $\bar{\Sigma}_1$ and $\Sigma_2 = \pi(\bar{\Sigma}_1)$ is a symplectic realization of $C_2$.

Equivalently, a birational equivalence is a birational derivation where in the blow-up $\bar{\Sigma}_1 = \bar{\Sigma}_2$. Reversing the sequence of blow-ups and blow-downs, we see that any realization $\Sigma_2$ of $C_2$ in $(M_2, \omega_2)$ will have exceptional spheres contained in its total transform $\bar{\Sigma}_2$ which can be blown down to obtain a realization of $C_1$ in $(M_1, \omega_1)$. Therefore, birational equivalence is a symmetric relation. If $C_1$ and $C_2$ are birationally equivalent, they are each birationally derived from the other.

Without the extra condition for birational equivalence, a birational derivation is not an equivalence relation. More precisely, suppose that a configuration $C_2$ that is birationally derived from a configuration $C_1$. Then for any realization $\Sigma_1$ of $C_1$ it has a resolution such that $\bar{\Sigma}_1 \subseteq \bar{\Sigma}_2$ for some realization $\Sigma_2$ of $C_2$. However, $\bar{\Sigma}_2$ may contain some additional exceptional spheres that were not contained in $\bar{\Sigma}_1$. Therefore, when we blow down the exceptional spheres in $\bar{\Sigma}_1$, the image of $\bar{\Sigma}_2$ will only contain $\Sigma_1$ instead of being equal to $\Sigma_1$. A weaker equivalence relation could be obtained from the definition of birational derivation by replacing the condition $\Sigma_2 = \pi(\bar{\Sigma}_1)$ by $\Sigma_2 \subseteq \pi(\bar{\Sigma}_1)$, however this relation will not be particularly useful to us.

Note that the number of components of the configuration is preserved by a birational equivalence but not by a birational derivation.

**Example 3.15.** Let $C_1$ be a configuration in $(\overline{CP}^2, \omega_{FS})$ whose realizations consist of a single symplectic conic $C$ with three symplectic lines $L_1, L_2, L_3$ that intersect the conic tangentially at three distinct points $p_1, p_2, p_3$, respectively, and intersect each other transversally at three distinct points. Blow up at each of the three tangential intersection points, and denote the resulting proper transforms by $\tilde{C}$ and $\tilde{L}_i$. The resulting self-intersection numbers satisfy $[\tilde{C}]^2 = 1$ and $[\tilde{L}_i]^2 = 0$. The intersections between $\tilde{C}$ and $\tilde{L}_i$ are transverse and there are three new exceptional spheres $E_1, E_2, E_3$ which pass through those intersection points. See Figure 2. Because $[\tilde{C}]^2 = +1$ and $\tilde{C}$ is a smooth sphere, we can find a symplectomorphism of $\overline{CP}^2\#3\overline{CP}^2$ which identifies $\tilde{C}$ with $\overline{CP}^1$. We calculate the possible homology classes of the other surfaces in the picture in terms of a standard basis which identifies $[\tilde{C}] = h$ using Lemma 3.7 and intersection numbers. We find the only option is

$$[\tilde{C}] = h, \quad [\tilde{L}_1] = h - e_1, \quad [\tilde{L}_2] = h - e_2, \quad [\tilde{L}_3] = h - e_3, \\ [E_1] = h - e_2 - e_3, \quad [E_2] = h - e_1 - e_3, \quad [E_3] = h - e_1 - e_2.$$

Lemma 3.5 implies that there exist exceptional spheres in classes $e_1, e_2, e_3$ which intersect the labeled surfaces non-negatively. Blowing down such exceptional spheres results in a configuration
From this definition, if we birationally invert the blow-down $\pi$ by blowing-up, we see that the total transform of $\Sigma_2$ in $M_1\#N\overline{CP}^2$, contains the total transform of $\Sigma_1$, i.e. $\bar{\Sigma}_1 \subseteq \bar{\Sigma}_2$. This follows from the fact that a set is contained in the preimage of its image, so $\bar{\Sigma}_2 = \pi^{-1}(\Sigma_2) = \pi^{-1}(\pi(\bar{\Sigma}_1)) \supseteq \bar{\Sigma}_1$. For this reason, we can also refer to the birational derivation relation as saying that $C_1$ is birationally contained in $C_2$.

Note that this relation is directional. If $C_2$ is birationally derived from $C_1$, it is not typically true that $C_1$ is birationally derived from $C_2$. To strengthen this notion, we impose a more restrictive condition that the exceptional spheres that are blown down by $\pi$ are actually contained in $\bar{\Sigma}_1$.

**DEFINITION 3.14.** We say that $C_1$ in $(M_1, \omega_1)$ is birationally equivalent to $C_2$ in $(M_2, \omega_2)$ if for every symplectic realization $\Sigma_1$ of $C_1$ there is a sequence of blow-ups of the pair $(M_1, \Sigma_1)$ to the total transform $(M_1\#N\overline{CP}^2, \bar{\Sigma}_1)$, followed by a sequence of blow-downs of exceptional spheres $\pi: M_1\#N\overline{CP}^2 \to M_2$ such that the exceptional locus of $\pi$ is contained in $\bar{\Sigma}_1$ and $\Sigma_2 = \pi(\bar{\Sigma}_1)$ is a symplectic realization of $C_2$.

Equivalently, a birational equivalence is a birational derivation where in the blow-up $\bar{\Sigma}_1 = \bar{\Sigma}_2$. Reversing the sequence of blow-ups and blow-downs, we see that any realization $\Sigma_2$ of $C_2$ in $(M_2, \omega_2)$ will have exceptional spheres contained in its total transform $\bar{\Sigma}_2$ which can be blown down to obtain a realization of $C_1$ in $(M_1, \omega_1)$. Therefore, birational equivalence is a symmetric relation. If $C_1$ and $C_2$ are birationally equivalent, they are each birationally derived from the other.

Without the extra condition for birational equivalence, a birational derivation is not an equivalence relation. More precisely, suppose that a configuration $C_2$ that is birationally derived from a configuration $C_1$. Then for any realization $\Sigma_1$ of $C_1$ it has a resolution such that $\bar{\Sigma}_1 \subseteq \bar{\Sigma}_2$ for some realization $\Sigma_2$ of $C_2$. However, $\bar{\Sigma}_2$ may contain some additional exceptional spheres that were not contained in $\bar{\Sigma}_1$. Therefore, when we blow down the exceptional spheres in $\bar{\Sigma}_1$, the image of $\bar{\Sigma}_2$ will only contain $\Sigma_1$ instead of being equal to $\Sigma_1$. A weaker equivalence relation could be obtained from the definition of birational derivation by replacing the condition $\Sigma_2 = \pi(\bar{\Sigma}_1)$ by $\Sigma_2 \subseteq \pi(\bar{\Sigma}_1)$, however this relation will not be particularly useful to us.

Note that the number of components of the configuration is preserved by a birational equivalence but not by a birational derivation.

**Example 3.15.** Let $C_1$ be a configuration in $(\overline{CP}^2, \omega_{FS})$ whose realizations consist of a single symplectic conic $C$ with three symplectic lines $L_1, L_2, L_3$ that intersect the conic tangentially at three distinct points $p_1, p_2, p_3$, respectively, and intersect each other transversally at three distinct points. Blow up at each of the three tangential intersection points, and denote the resulting proper transforms by $\tilde{C}$ and $\tilde{L}_i$. The resulting self-intersection numbers satisfy $[\tilde{C}]^2 = 1$ and $[\tilde{L}_i]^2 = 0$. The intersections between $\tilde{C}$ and $\tilde{L}_i$ are transverse and there are three new exceptional spheres $E_1, E_2, E_3$ which pass through those intersection points. See Figure 2. Because $[\tilde{C}]^2 = +1$ and $\tilde{C}$ is a smooth sphere, we can find a symplectomorphism of $\overline{CP}^2\#3\overline{CP}^2$ which identifies $\tilde{C}$ with $\overline{CP}^1$. We calculate the possible homology classes of the other surfaces in the picture in terms of a standard basis which identifies $[\tilde{C}] = h$ using Lemma 3.7 and intersection numbers. We find the only option is

$$[\tilde{C}] = h, \quad [\tilde{L}_1] = h - e_1, \quad [\tilde{L}_2] = h - e_2, \quad [\tilde{L}_3] = h - e_3, \\ [E_1] = h - e_2 - e_3, \quad [E_2] = h - e_1 - e_3, \quad [E_3] = h - e_1 - e_2.$$

Lemma 3.5 implies that there exist exceptional spheres in classes $e_1, e_2, e_3$ which intersect the labeled surfaces non-negatively. Blowing down such exceptional spheres results in a configuration
<figure><img src="image_11.png" /><figcaption>FIGURE 2. Birational derivation. The lines forming the triangle on the left are $L_1, L_2, L_3$, and the circle represents the conic $C$. After blowing up as indicated in the center figure, we obtain three exceptional spheres $E_1, E_2, E_3$. There exist three other exceptional spheres which are not visible in the center diagram that can be blown down to result in the configuration shown on the right.</figcaption></figure>

of seven symplectic lines intersecting in six triple points (the three triple points that already existed between $C, L_i, E_i$ for $i = 1, 2, 3$, and the three triple points that are created when blowing down $e_1, e_2, e_3$). Therefore, the configuration of seven lines with six triple points can be birationally derived from the configuration of a conic with three tangent lines. Note, the exceptional spheres which are blown down in the last step of the transformation are not included in the configuration (because none of the surfaces $\tilde{C}, \tilde{L}_i$ or $E_i$ represented a class $e_i$). Therefore, this is not a birational equivalence.

In order to get a birational equivalence, we would need to augment the original configuration by adding in components which will become the exceptional spheres that eventually will be blown down. We can reverse engineer to predict what must be added to our configuration to obtain a birational equivalence to the same final configuration. The exceptional sphere $\tilde{S}_1$ in $\mathbb{CP}^2\#3\overline{\mathbb{CP}}^2$ representing $e_1$ intersects $E_2$ and $E_3$. Reversing the birational transformation by blowing down $E_2$ and $E_3$, we find that the image $S_1$ in $\mathbb{CP}^2$ will be a sphere of self-intersection +1 which must pass transversally through the tangential intersections between $C$ with $L_2$ and $C$ with $L_3$. The other exceptional spheres descend similarly to +1-spheres passing through two of the tangential intersections. Replace our starting configuration by a configuration consisting of the original conic $C$ and lines $L_1, L_2, L_3$ with $L_i$ tangent to $C$ at $p_i$ together with three additional symplectic lines $S_1, S_2, S_3$ such that $S_i$ intersects $C$ transversally at the points $p_{i+1}$ and $p_{i+2}$ (where indices are taken mod 3). Now the same birational transformation becomes a birational equivalence between a configuration consisting of a conic with three tangent lines and three lines through the three pairs of tangent points with a configuration of seven lines with six triple points and three double points. See Figure 3.

*Remark 3.16.* In this article, we prove the existence of birational derivations and birational equivalences using pseudoholomorphic curves. As demonstrated in Example 3.15, we primarily infer the existence of a particular birational derivation from a singular curve using Theorem 3.6 and Lemma 3.5. Alternatively, we can look for birational equivalences. As seen in Example 3.15, to upgrade a birational derivation to a birational equivalence, we generally need to augment the original configuration $\Sigma_1$ by adding extra components. If these components are symplectic lines (degree one) in $\mathbb{CP}^2$ with sufficiently simple intersections with the other components of $\Sigma_1$,
<figure><img src="image_11.png" /><figcaption>FIGURE 2. Birational derivation. The lines forming the triangle on the left are $L_1, L_2, L_3$, and the circle represents the conic $C$. After blowing up as indicated in the center figure, we obtain three exceptional spheres $E_1, E_2, E_3$. There exist three other exceptional spheres which are not visible in the center diagram that can be blown down to result in the configuration shown on the right.</figcaption></figure>

of seven symplectic lines intersecting in six triple points (the three triple points that already existed between $C, L_i, E_i$ for $i = 1, 2, 3$, and the three triple points that are created when blowing down $e_1, e_2, e_3$). Therefore, the configuration of seven lines with six triple points can be birationally derived from the configuration of a conic with three tangent lines. Note, the exceptional spheres which are blown down in the last step of the transformation are not included in the configuration (because none of the surfaces $\tilde{C}, \tilde{L}_i$ or $E_i$ represented a class $e_i$). Therefore, this is not a birational equivalence.

In order to get a birational equivalence, we would need to augment the original configuration by adding in components which will become the exceptional spheres that eventually will be blown down. We can reverse engineer to predict what must be added to our configuration to obtain a birational equivalence to the same final configuration. The exceptional sphere $\tilde{S}_1$ in $\mathbb{CP}^2\#3\overline{\mathbb{CP}}^2$ representing $e_1$ intersects $E_2$ and $E_3$. Reversing the birational transformation by blowing down $E_2$ and $E_3$, we find that the image $S_1$ in $\mathbb{CP}^2$ will be a sphere of self-intersection +1 which must pass transversally through the tangential intersections between $C$ with $L_2$ and $C$ with $L_3$. The other exceptional spheres descend similarly to +1-spheres passing through two of the tangential intersections. Replace our starting configuration by a configuration consisting of the original conic $C$ and lines $L_1, L_2, L_3$ with $L_i$ tangent to $C$ at $p_i$ together with three additional symplectic lines $S_1, S_2, S_3$ such that $S_i$ intersects $C$ transversally at the points $p_{i+1}$ and $p_{i+2}$ (where indices are taken mod 3). Now the same birational transformation becomes a birational equivalence between a configuration consisting of a conic with three tangent lines and three lines through the three pairs of tangent points with a configuration of seven lines with six triple points and three double points. See Figure 3.

*Remark 3.16.* In this article, we prove the existence of birational derivations and birational equivalences using pseudoholomorphic curves. As demonstrated in Example 3.15, we primarily infer the existence of a particular birational derivation from a singular curve using Theorem 3.6 and Lemma 3.5. Alternatively, we can look for birational equivalences. As seen in Example 3.15, to upgrade a birational derivation to a birational equivalence, we generally need to augment the original configuration $\Sigma_1$ by adding extra components. If these components are symplectic lines (degree one) in $\mathbb{CP}^2$ with sufficiently simple intersections with the other components of $\Sigma_1$,
<figure><img src="image_12.png" /><figcaption>FIGURE 3. An example of a birational equivalence. The lines forming the outer triangle in the left figure represent $L_1, L_2, L_3$, the circle represents the conic $C$, and the lines forming the inscribed triangle represent $S_1, S_2, S_3$. After blowing up at the marked points to get the center figure, the proper transforms $\tilde{S}_1, \tilde{S}_2, \tilde{S}_3$ are exceptional spheres. The short line segments in the center figure represent the exceptional spheres $E_1, E_2, E_3$. The right figure is obtained by blowing down $\tilde{S}_1, \tilde{S}_2, \tilde{S}_3$. The images of $E_1, E_2, E_3$ after this blow-down are lines in the right figure.</figcaption></figure>

we will see that we can use pseudoholomorphic curves to infer the existence of such augmenting curves through Proposition 5.1. In practice, we typically use Theorem 3.6 and Lemma 3.5 along with the homological analysis from §3.3 to discover a birational derivation. If desired, we can then reverse engineer to find an augmented configuration yielding a birational equivalence. Then, in proving our results, we may justify the existence of the birational derivation or justify the existence of the augmentation using the pseudoholomorphic curve results mentioned previously. Note that once a configuration is augmented to produce a birational equivalence, no pseudoholomorphic curve result is necessary to justify the existence of the birational equivalence because all of the exceptional curves which one needs to blow-down are visibly included in the total transform of the configuration. Instead, pseudoholomorphic curves are used for augmenting the configuration to get the birational equivalence. By contrast, to state the existence of a birational derivation from one configuration to another can require pseudoholomorphic curve machinery to imply the existence of appropriate exceptional divisors which may not be visible.

Our use of birational derivations and equivalences arises from their implications to symplectic isotopy problems, which we state next. These implications are somewhat immediate from the definitions, so the mathematical power goes into proving such birational derivations exist as described in the previous remark.

Now we give the relations between symplectic isotopy problems and birational derivations and equivalences.

**PROPOSITION 3.17.** If $C_1$ is a configuration in $(M_1, \omega_1)$, and $C_2$ in $(M_2, \omega_2)$ is birationally derived from $C_1$, then any subconfiguration of $C_2$ symplectically embeds into $(M_2, \omega)$.

This statement is immediate from the definition, but the utility of the statement comes from its contrapositive. Namely, we show that certain configurations $C_1$ cannot be symplectically realized in a closed symplectic manifold $(M_1, \omega_1)$, using the non-existence of a subconfiguration of symplectic curves that can be birationally derived from $C_1$.
<figure><img src="image_12.png" /><figcaption>FIGURE 3. An example of a birational equivalence. The lines forming the outer triangle in the left figure represent $L_1, L_2, L_3$, the circle represents the conic $C$, and the lines forming the inscribed triangle represent $S_1, S_2, S_3$. After blowing up at the marked points to get the center figure, the proper transforms $\tilde{S}_1, \tilde{S}_2, \tilde{S}_3$ are exceptional spheres. The short line segments in the center figure represent the exceptional spheres $E_1, E_2, E_3$. The right figure is obtained by blowing down $\tilde{S}_1, \tilde{S}_2, \tilde{S}_3$. The images of $E_1, E_2, E_3$ after this blow-down are lines in the right figure.</figcaption></figure>

we will see that we can use pseudoholomorphic curves to infer the existence of such augmenting curves through Proposition 5.1. In practice, we typically use Theorem 3.6 and Lemma 3.5 along with the homological analysis from §3.3 to discover a birational derivation. If desired, we can then reverse engineer to find an augmented configuration yielding a birational equivalence. Then, in proving our results, we may justify the existence of the birational derivation or justify the existence of the augmentation using the pseudoholomorphic curve results mentioned previously. Note that once a configuration is augmented to produce a birational equivalence, no pseudoholomorphic curve result is necessary to justify the existence of the birational equivalence because all of the exceptional curves which one needs to blow-down are visibly included in the total transform of the configuration. Instead, pseudoholomorphic curves are used for augmenting the configuration to get the birational equivalence. By contrast, to state the existence of a birational derivation from one configuration to another can require pseudoholomorphic curve machinery to imply the existence of appropriate exceptional divisors which may not be visible.

Our use of birational derivations and equivalences arises from their implications to symplectic isotopy problems, which we state next. These implications are somewhat immediate from the definitions, so the mathematical power goes into proving such birational derivations exist as described in the previous remark.

Now we give the relations between symplectic isotopy problems and birational derivations and equivalences.

**PROPOSITION 3.17.** If $C_1$ is a configuration in $(M_1, \omega_1)$, and $C_2$ in $(M_2, \omega_2)$ is birationally derived from $C_1$, then any subconfiguration of $C_2$ symplectically embeds into $(M_2, \omega)$.

This statement is immediate from the definition, but the utility of the statement comes from its contrapositive. Namely, we show that certain configurations $C_1$ cannot be symplectically realized in a closed symplectic manifold $(M_1, \omega_1)$, using the non-existence of a subconfiguration of symplectic curves that can be birationally derived from $C_1$.
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

**PROPOSITION 3.18.** Suppose a configuration $C_2$ in $(M_2, \omega_2)$ is birationally derived from $C_1$ in $(M_1, \omega_1)$, and suppose $C_2$ has a unique (non-empty) equisingular symplectic isotopy class in $(M_2, \omega_2)$. Then $C_1$ also has a unique (non-empty) symplectic isotopy class in $(M_1, \omega_1)$. If $(M_1, \omega_1) = (M_2, \omega_2) = (\mathbb{CP}^2, \omega_{\mathrm{FS}})$ and $C_2$ can be realized by a complex curve, then $C_1$ can also be realized by a complex curve.

*Proof.* Suppose any two symplectic embeddings of $C_2$ into $(M_2, \omega_2)$ are symplectically isotopic. Let $Q_0$ and $Q_1$ be two symplectic embeddings of $C_1$ into $(M_1, \omega_1)$. By the definition of birational derivation, for each $i = 0, 1$ there is a sequence of blow-ups of $Q_i$ to $\bar{Q}_i$ and a sequence of blow-downs that contract $\bar{Q}_i$ to $R_i$, where $R_i$ is a symplectic embedding of $C_2$ into $(M_2, \omega_2)$. Then there exists a family $R_t$ of equisingular symplectic embeddings of $C_2$ into $(M_2, \omega_2)$ for $t \in [0, 1]$ which connects $R_0$ and $R_1$. For each $R_t$, perform the sequence of blow-ups along the appropriate smooth or singular points in $R_t$ to obtain $\bar{R}_t$. By the definition of birational derivation, there is a distinguished subset of the components $\bar{Q}_t \subseteq \bar{R}_t$ for $t \in [0, 1]$, agreeing $\bar{Q}_0$ and $\bar{Q}_1$ for $t = 0, 1$. There are exceptional spheres in $\bar{Q}_t$ for each $t \in [0, 1]$ which can be blown down to give equisingular symplectic embeddings $Q_t$ of $C_1$ into $(M_1, \omega_1)$. This gives a symplectic family connecting $Q_0$ and $Q_1$.

As complex curves are preserved under birational transformations, the last statement follows from the same proof. □

If $C_1$ and $C_2$ are birationally equivalent, they are each birationally derived from the other yielding the following corollary.

**COROLLARY 3.19.** Suppose $C_1$ in $(M_1, \omega_1)$ and $C_2$ in $(M_2, \omega_2)$ are birationally equivalent. There is a unique equisingular symplectic isotopy class for $C_1$ in $(M_1, \omega_1)$, if and only if there is a unique equisingular symplectic isotopy class for $C_2$ in $(M_2, \omega_2)$. If $(M_1, \omega_1) = (M_2, \omega_2) = (\mathbb{CP}^2, \omega_{\mathrm{FS}})$ and if the equisingular symplectic isotopy class contains complex representatives for one configuration, it contains complex representatives for the other.

**3.5 Riemann–Hurwitz**

The Riemann–Hurwitz obstruction uses symplectic information in a more global way. Fix an almost complex structure $J$ on $\mathbb{CP}^2$ such that $C$ is $J$-holomorphic. Fix a point $p \in \mathbb{CP}^2$, and consider the pencil of $J$-holomorphic lines through $p$, and the associated projection $\pi_p : \mathbb{CP}^2 \setminus \{p\} \to \mathbb{CP}^1$. Restricting this projection to $\pi : C \to \mathbb{CP}^1$, and pre-composing with the normalization map $n : \tilde{C} \to C$, gives a ramified covering map $\pi \circ n : \tilde{C} \to \mathbb{CP}^1$. The fact that the only singularities are ramification points follows from positivity of intersections between $J$-holomorphic curves. Ramification points arise from tangencies between lines in the pencil with $C$ and from singular points of $C$.

For a point $q \neq p$, the ramification index of $n^{-1}(q)$ is equal to the multiplicity of intersection of the $J$-line through $q$ with $C$. For a smooth point, this is just the order of tangency between the $J$-line and $C$ at $q$. At a singular point $q \neq p$ with multiplicity sequence starting with $[m_q, m_{q,2}]$ the ramification index of $x = n^{-1}(q)$ is $m_q$ if the $J$-line from $p$ to $q$ is transverse to $C$ at $q$. If the $J$-line from $p$ to $q$ is tangent to $C$ at $q$, then $x$ will have ramification index at least $m_q + m_{q,2}$.

A priori, if $p \in C$, the map is not well-defined at $p$, but, in fact, $\pi$ has a unique continuous extension defined by sending $p$ to the image of the $J$-holomorphic line through $p$ which is tangent to $C$ at $p$. In this case, the ramification index of $p$ is equal to the multiplicity of intersection of the $J$-line tangent to $C$ at $p$ minus the multiplicity of $C$ at $p$. Therefore, if $p$ is a singular point whose multiplicity sequence starts with $[m_p, m_{p,2}]$ (where we set $m_{p,2} = 1$ if the multiplicity sequence has length one), then the ramification index of $\pi \circ n$ at $p$ is at least $m_{p,2}$.
**PROPOSITION 3.18.** Suppose a configuration $C_2$ in $(M_2, \omega_2)$ is birationally derived from $C_1$ in $(M_1, \omega_1)$, and suppose $C_2$ has a unique (non-empty) equisingular symplectic isotopy class in $(M_2, \omega_2)$. Then $C_1$ also has a unique (non-empty) symplectic isotopy class in $(M_1, \omega_1)$. If $(M_1, \omega_1) = (M_2, \omega_2) = (\mathbb{CP}^2, \omega_{\mathrm{FS}})$ and $C_2$ can be realized by a complex curve, then $C_1$ can also be realized by a complex curve.

*Proof.* Suppose any two symplectic embeddings of $C_2$ into $(M_2, \omega_2)$ are symplectically isotopic. Let $Q_0$ and $Q_1$ be two symplectic embeddings of $C_1$ into $(M_1, \omega_1)$. By the definition of birational derivation, for each $i = 0, 1$ there is a sequence of blow-ups of $Q_i$ to $\bar{Q}_i$ and a sequence of blow-downs that contract $\bar{Q}_i$ to $R_i$, where $R_i$ is a symplectic embedding of $C_2$ into $(M_2, \omega_2)$. Then there exists a family $R_t$ of equisingular symplectic embeddings of $C_2$ into $(M_2, \omega_2)$ for $t \in [0, 1]$ which connects $R_0$ and $R_1$. For each $R_t$, perform the sequence of blow-ups along the appropriate smooth or singular points in $R_t$ to obtain $\bar{R}_t$. By the definition of birational derivation, there is a distinguished subset of the components $\bar{Q}_t \subseteq \bar{R}_t$ for $t \in [0, 1]$, agreeing $\bar{Q}_0$ and $\bar{Q}_1$ for $t = 0, 1$. There are exceptional spheres in $\bar{Q}_t$ for each $t \in [0, 1]$ which can be blown down to give equisingular symplectic embeddings $Q_t$ of $C_1$ into $(M_1, \omega_1)$. This gives a symplectic family connecting $Q_0$ and $Q_1$.

As complex curves are preserved under birational transformations, the last statement follows from the same proof. □

If $C_1$ and $C_2$ are birationally equivalent, they are each birationally derived from the other yielding the following corollary.

**COROLLARY 3.19.** Suppose $C_1$ in $(M_1, \omega_1)$ and $C_2$ in $(M_2, \omega_2)$ are birationally equivalent. There is a unique equisingular symplectic isotopy class for $C_1$ in $(M_1, \omega_1)$, if and only if there is a unique equisingular symplectic isotopy class for $C_2$ in $(M_2, \omega_2)$. If $(M_1, \omega_1) = (M_2, \omega_2) = (\mathbb{CP}^2, \omega_{\mathrm{FS}})$ and if the equisingular symplectic isotopy class contains complex representatives for one configuration, it contains complex representatives for the other.

## 3.5 Riemann–Hurwitz

The Riemann–Hurwitz obstruction uses symplectic information in a more global way. Fix an almost complex structure $J$ on $\mathbb{CP}^2$ such that $C$ is $J$-holomorphic. Fix a point $p \in \mathbb{CP}^2$, and consider the pencil of $J$-holomorphic lines through $p$, and the associated projection $\pi_p : \mathbb{CP}^2 \setminus \{p\} \to \mathbb{CP}^1$. Restricting this projection to $\pi : C \to \mathbb{CP}^1$, and pre-composing with the normalization map $n : \tilde{C} \to C$, gives a ramified covering map $\pi \circ n : \tilde{C} \to \mathbb{CP}^1$. The fact that the only singularities are ramification points follows from positivity of intersections between $J$-holomorphic curves. Ramification points arise from tangencies between lines in the pencil with $C$ and from singular points of $C$.

For a point $q \neq p$, the ramification index of $n^{-1}(q)$ is equal to the multiplicity of intersection of the $J$-line through $q$ with $C$. For a smooth point, this is just the order of tangency between the $J$-line and $C$ at $q$. At a singular point $q \neq p$ with multiplicity sequence starting with $[m_q, m_{q,2}]$ the ramification index of $x = n^{-1}(q)$ is $m_q$ if the $J$-line from $p$ to $q$ is transverse to $C$ at $q$. If the $J$-line from $p$ to $q$ is tangent to $C$ at $q$, then $x$ will have ramification index at least $m_q + m_{q,2}$.

A priori, if $p \in C$, the map is not well-defined at $p$, but, in fact, $\pi$ has a unique continuous extension defined by sending $p$ to the image of the $J$-holomorphic line through $p$ which is tangent to $C$ at $p$. In this case, the ramification index of $p$ is equal to the multiplicity of intersection of the $J$-line tangent to $C$ at $p$ minus the multiplicity of $C$ at $p$. Therefore, if $p$ is a singular point whose multiplicity sequence starts with $[m_p, m_{p,2}]$ (where we set $m_{p,2} = 1$ if the multiplicity sequence has length one), then the ramification index of $\pi \circ n$ at $p$ is at least $m_{p,2}$.