tight contact structure $\xi_0$, namely the boundary of the plumbing of $2m - 1$ copies of $T^*S^2$, whose corresponding contact structure has deg $c(\xi_0) = (-2m + 1)/4$.

To conclude the proof, we need to show that $\xi'$ is universally tight. As mentioned previously, however, $\xi'$ is tight (because it is filled by $W'$), and $L' = L(m, m-1)$ admits a unique tight contact structure (by [Hon00]). As on each lens space there is always a universally tight contact structure (obtained by taking a quotient of $S^3$ with its standard contact structure), $\xi'$ is universally tight. $\square$

Alternatively, to exclude the case $\gcd(k, m) = 2$ in the proof, one can argue that the unique contact structure on $L(2m, 2m-1)$ is also universally tight, and then so is $\xi$. Then Lisca's classification of fillings [Lis08, Corollary 1.2] rules this possibility out.

The second result concerns the uniqueness of the rational homology ball filling. It is very likely that the result is known to experts (either in the context of Milnor fibers of cyclic quotient singularities, or in the context of symplectic fillings of contact structures), but we were unable to locate the precise statement.

**PROPOSITION A.2.** The standard contact structure on the lens space $L(m^2, mk-1)$ has a unique rational homology ball symplectic filling up to symplectic deformation.

The proof is, in fact, somewhat implicit in Lisca's classification paper [Lis08]. We refer to the paper for notation and context. We start by recalling the following two facts about continued fractions. Both facts follow from Riemenschneier's point rule [Rie74] (see [Lis08, Lemma 2.6]).

If we have two continued fraction expansions $[\mathbf{a}]^-$ and $[\mathbf{b}]^-$ such that

$$1/[a_1, \dots, a_m]^- + 1/[b_1, \dots, b_\ell]^- = 1$$

(or, equivalently, the two associated fractions are of the form $p/q$ and $p/(p-q)$ for some $q < p$ coprime positive integers), we say that $[\mathbf{a}]^-$ and $[\mathbf{b}]^-$ are dual to each other.

If $[\mathbf{a}]^-$ and $[\mathbf{b}]^-$ are dual to each other, then $\sum (a_i - 1) = \sum (b_j - 1)$. We also have that

$$[n_1, \dots, n_{j-1}, 1, n_{j+1}, \dots, n_\ell]^- = 0$$

if and only if $[n_{j-1}, \dots, n_2, n_1]^-$ and $[n_{j+1}, \dots, n_\ell]$ are dual to each other.

*Proof.* The (negative) continued fraction expansion of $m^2/(mk-1)$ is of the form $[n_1, \dots, n_{j-1}, 2, n_{j+1}, \dots, n_\ell]^{-}$, such that $[n_1, \dots, n_{j-1}, 1, n_{j+1}, \dots, n_\ell]^- = 0$ (see the proof of [Lis08, Theorem 1.2(c)]).

Recall from the proof of [Lis08, Theorem 1.1] that symplectic deformation classes of symplectic fillings of $L(p, q)$ with $p/q = [n_1, \dots, n_\ell]^{-}$ correspond to strings $\mathbf{m} = (m_1, \dots, m_\ell)$ such that $1 \le m_i \le n_i$ for each $i$ and $[m_1, \dots, m_\ell]^- = 0$.

Moreover, the Euler characteristics of the filling corresponding to $\mathbf{m}$ is $\sum (n_i - m_i)$, so that the filling is a rational homology ball if and only if $m_i = n_i$ for all indices except for one, for which $m_i = n_i - 1$. We claim that for such any $p, q$ there is at most one index $j$ such that the corresponding sequence $\mathbf{m} = (n_1, \dots, n_{j-1}, n_j - 1, n_{j+1}, n_\ell)$ has $[\mathbf{m}]^- = 0$, and in that case $n_j = 2$.

If $n_j > 2$, then $[\mathbf{m}]^-$ is the continued fraction expansion of a rational number $p'/q' > 1$. Suppose now $n_j = 2$. If $[\mathbf{m}]^- = 0$, then $[n_{j-1}, \dots, n_1]^-$ and $[n_{j+1}, \dots, n_\ell]^-$ are dual to each other; however, as $j$ varies between 1 and $\ell$, the difference

$$\sum_{h<j} (n_h - 1) - \sum_{h>J} (n_h - 1)$$
M. GOLLA AND L. STARKSTON

is strictly decreasing, hence there is at most one value of $j$ such that it vanishes. As we recalled previously, the difference must vanish if $[n_{j-1}, \dots, n_1]^{-}$ and $[n_{j+1}, \dots, n_l]^{-}$ are dual to each other.

The proposition now follows.

□

REFERENCES

ACC03 E. Artal Bartolo, J. Carmona Ruber and J. I. Cogolludo Agustín, *Braid monodromy and topology of plane curves*, Duke Math. J. **118** (2003), 261–278; MR 1980995.

ACCM05 E. Artal Bartolo, J. Carmona Ruber, J. I. Cogolludo-Agustín and M. Marco Buzunáriz, *Topology and combinatorics of real line arrangements*, Compos. Math. **141** (2005), 1578–1588; MR 2188450.

AG17 P. Aceto and M. Golla, *Dehn surgeries and rational homology balls*, Algebr. Geom. Topol. **17** (2017), 487–527; MR 3604383.

AGLL20 P. Aceto, M. Golla, K. Larson and A. G. Lecuona, *Surgeries on torus knots, rational balls, and cabling*, Preprint (2020), arXiv:2008.06760.

ASST21 M. Amram, R. Shwartz, U. Sinichkin and S.-L. Tan, *Zariski pairs of conic-line arrangements of degree 7 via fundamental groups*, Preprint (2021), arXiv:2106.03507.

Aub76 T. Aubin, *Équations du type Monge-Ampère sur les variétés kähleriennes compactes*, C. R. Acad. Sci. Paris Sér. A-B **283** (1976), A119–A121; MR 0433520.

Aur00 D. Auroux, *Symplectic 4-manifolds as branched coverings of $\mathbb{CP}^2$*, Invent. Math. **139** (2000), 551–602; MR 1738061.

Aur06 D. Auroux, *Symplectic 4-manifolds, singular plane curves, and isotopy problems*, in *Floer homology, gauge theory, and low-dimensional topology: Clay Math. Proc.*, vol. 5 (American Mathematical Society, Providence, RI, 2006), 263–276; MR 2249258.

Bar99 J.-F. Barraud, *Nodal symplectic spheres in $\mathbb{CP}^2$ with positive self-intersection*, Int. Math. Res. Not. IMRN **9** (1999), 495–508; MR 1692591.

BCG16 J. Bodnár, D. Celoria, and M. Golla, *Cuspidal curves and Heegaard Floer homology*, Proc. Lond. Math. Soc. **112** (2016), 512–548.

BHS18 M. Borodzik, J. Hom and A. Schinzel, *Involutive Heegaard Floer homology and rational cuspidal curves*, Proc. Lond. Math. Soc. **118** (2018), 441–472.

BL14 M. Borodzik and C. Livingston, *Heegaard Floer homology and rational cuspidal curves*, Forum Math. Sigma 2 (2014), e28, 23.

BN12 M. Borodzik and A. Némethi, *Spectrum of plane curves via knot theory*, J. Lond. Math. Soc. **86** (2012), 87–110.

BN16 J. Bodnár and A. Némethi, *Lattice cohomology and rational cuspidal curves*, Math. Res. Lett. **23** (2016), 339–375.

BO18 M. Bhupal and B. Özbağci, *Symplectic fillings of lens spaces as Lefschetz fibrations*, J. Eur. Math. Soc. (JEMS) **18** (2018), 1515–1535.

Che18 W. Chen, *Contact splitting of symplectic $\mathbb{Q}$-homology $\mathbb{CP}^2$, in Proceedings of the Gökova Geometry-Topology Conference 2017* (International Press, Somerville, MA, 2018), 53–72.

Con21 A. Conway, *The Levine-Tristram signature: a survey*, in *2019–2020 MATRIX annals*, MATRIX Book Series, vol. 4 (Springer, 2021), 31–56.

CS10 D. I. Cartwright and T. Steger, *Enumeration of the 50 fake projective planes*, C. R. Math. Acad. Sci. Paris **348** (2010), 11–13; MR 2586735.

Eli90 Y. Eliashberg, *Filling by holomorphic discs and its applications*, in *Geometry of low-dimensional manifolds, Vol. 2 (Durham, 1989)*, London Mathematical Society Lecture Note Series, vol. 151 (Cambridge University Press, 1990), 45–67.
M. GOLLA AND L. STARKSTON

is strictly decreasing, hence there is at most one value of $j$ such that it vanishes. As we recalled previously, the difference must vanish if $[n_{j-1}, \dots, n_1]^{-}$ and $[n_{j+1}, \dots, n_l]^{-}$ are dual to each other.

The proposition now follows.

□

REFERENCES

ACC03 E. Artal Bartolo, J. Carmona Ruber and J. I. Cogolludo Agustín, *Braid monodromy and topology of plane curves*, Duke Math. J. **118** (2003), 261–278; MR 1980995.

ACCM05 E. Artal Bartolo, J. Carmona Ruber, J. I. Cogolludo-Agustín and M. Marco Buzunáriz, *Topology and combinatorics of real line arrangements*, Compos. Math. **141** (2005), 1578–1588; MR 2188450.

AG17 P. Aceto and M. Golla, *Dehn surgeries and rational homology balls*, Algebr. Geom. Topol. **17** (2017), 487–527; MR 3604383.

AGLL20 P. Aceto, M. Golla, K. Larson and A. G. Lecuona, *Surgeries on torus knots, rational balls, and cabling*, Preprint (2020), arXiv:2008.06760.

ASST21 M. Amram, R. Shwartz, U. Sinichkin and S.-L. Tan, *Zariski pairs of conic-line arrangements of degree 7 via fundamental groups*, Preprint (2021), arXiv:2106.03507.

Aub76 T. Aubin, *Équations du type Monge-Ampère sur les variétés kähleriennes compactes*, C. R. Acad. Sci. Paris Sér. A-B **283** (1976), A119–A121; MR 0433520.

Aur00 D. Auroux, *Symplectic 4-manifolds as branched coverings of $\mathbb{CP}^2$*, Invent. Math. **139** (2000), 551–602; MR 1738061.

Aur06 D. Auroux, *Symplectic 4-manifolds, singular plane curves, and isotopy problems*, in *Floer homology, gauge theory, and low-dimensional topology: Clay Math. Proc.*, vol. 5 (American Mathematical Society, Providence, RI, 2006), 263–276; MR 2249258.

Bar99 J.-F. Barraud, *Nodal symplectic spheres in $\mathbb{CP}^2$ with positive self-intersection*, Int. Math. Res. Not. IMRN **9** (1999), 495–508; MR 1692591.

BCG16 J. Bodnár, D. Celoria, and M. Golla, *Cuspidal curves and Heegaard Floer homology*, Proc. Lond. Math. Soc. **112** (2016), 512–548.

BHS18 M. Borodzik, J. Hom and A. Schinzel, *Involutive Heegaard Floer homology and rational cuspidal curves*, Proc. Lond. Math. Soc. **118** (2018), 441–472.

BL14 M. Borodzik and C. Livingston, *Heegaard Floer homology and rational cuspidal curves*, Forum Math. Sigma 2 (2014), e28, 23.

BN12 M. Borodzik and A. Némethi, *Spectrum of plane curves via knot theory*, J. Lond. Math. Soc. **86** (2012), 87–110.

BN16 J. Bodnár and A. Némethi, *Lattice cohomology and rational cuspidal curves*, Math. Res. Lett. **23** (2016), 339–375.

BO18 M. Bhupal and B. Özbağci, *Symplectic fillings of lens spaces as Lefschetz fibrations*, J. Eur. Math. Soc. (JEMS) **18** (2018), 1515–1535.

Che18 W. Chen, *Contact splitting of symplectic $\mathbb{Q}$-homology $\mathbb{CP}^2$, in Proceedings of the Gökova Geometry-Topology Conference 2017* (International Press, Somerville, MA, 2018), 53–72.

Con21 A. Conway, *The Levine-Tristram signature: a survey*, in *2019–2020 MATRIX annals*, MATRIX Book Series, vol. 4 (Springer, 2021), 31–56.

CS10 D. I. Cartwright and T. Steger, *Enumeration of the 50 fake projective planes*, C. R. Math. Acad. Sci. Paris **348** (2010), 11–13; MR 2586735.

Eli90 Y. Eliashberg, *Filling by holomorphic discs and its applications*, in *Geometry of low-dimensional manifolds, Vol. 2 (Durham, 1989)*, London Mathematical Society Lecture Note Series, vol. 151 (Cambridge University Press, 1990), 45–67.
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL cuspidal CURVES

EG91 Y. Eliashberg and M. Gromov, *Convex symplectic manifolds*, in *Several complex variables and complex geometry, Part 2*, Proceedings of Symposia in Pure Mathematics (AMS, 1991), 135–162.

Fen99 T. Fenske, *Rational 1- and 2-cuspidal plane curves*, Beitr. Algebra Geom. **40** (1999), 309–329; MR 1720107.

FLMN06 J. Fernández de Bobadilla, I. Luengo, A. Melle Hernández and A. Némethi, *On rational cuspidal projective plane curves*, Proc. Lond. Math. Soc. **92** (2006), 99–138.

FLMN07 J. Fernández de Bobadilla, I. Luengo, A. Melle Hernández and A. Némethi, *Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair*, in *Real and complex singularities*, Trends in Mathematics (Birkhäuser, Basel, 2007), 31–45; MR 2280130.

Fos20 E. Fossati, *Topological constraints for Stein fillings of tight structures on lens spaces*, J. Symplectic Geom. **18** (2020), 1475–1514.

Fra05 S. Francisco, *Symplectic isotopy for cuspidal curves*, PhD thesis, Massachusetts Institute of Technology (ProQuest LLC, Ann Arbor, MI, 2005); MR 2717257.

GK22 M. Golla and F. Kütle, *Symplectic isotopy of rational cuspidal sextics and septics*, Int. Math. Res. Not. IMRN (2022), rnab364. doi:10.1093/imrn/rnab364.

GL16 M. Golla and P. Lisca, *On Stein fillings of contact torus bundles*, Bull. Lond. Math. Soc. **48** (2016), 19–37; MR 3455745.

Gro85 M. Gromov, *Pseudo holomorphic curves in symplectic manifolds*, Invent. Math. **82** (1985), 307–347; MR 809718.

GS09 D. T. Gay and A. I. Stipsicz, *Symplectic surgeries and normal surface singularities*, Algebr. Geom. Topol. **9** (2009), 2203–2223; MR 2558309.

GS21 M. Golla and L. Starkston, *Rational cuspidal curves and symplectic fillings*, Preprint (2021), arXiv:2111.09700.

Har86 J. Harris, *On the Severi problem*, Invent. Math. **84** (1986), 445–461; MR 837522.

Hir83 F. Hirzebruch, *Arrangements of lines and algebraic surfaces*, in *Arithmetic and geometry, Vol. II*, Progress in Mathematics, vol. 36 (Birkhäuser, Boston, MA, 1983), 113–140; MR 717609.

HLS97 H. Hofer, V. Lizan and J.-C. Sikorav, *On genericity for holomorphic curves in four-dimensional almost-complex manifolds*, J. Geom. Anal. **7** (1997), 149–159; MR 1630789.

HW16 J. Hom and Z. Wu, *Four-ball genus bounds and a refinement of the Ozváth-Szabó tau invariant*, J. Symplectic Geom. **14** (2016), 305–323; MR 3523259.

Hon00 K. Honda, *On the classification of tight contact structures*, I, Geom. Topol. **4** (2000), 309–368.

IS99 S. Ivashkovich and V. Shevchishin, *Structure of the moduli space in a neighborhood of a cusp-curve and meromorphic hulls*, Invent. Math. **136** (1999), 571–602; MR 1695206.

KP17 M. Koras and K. Palka, *The Coolidge-Nagata conjecture*, Duke Math. J. **166** (2017), 3085–3145; MR 3715805.

KP22 M. Koras and K. Palka, *Complex planar curves homeomorphic to a line have at most four singular points*, J. Math. Pures Appl. (9) **158** (2022), 144–182; MR 4360367.

Lec12 A. G. Lecuona, *On the slice-ribbon conjecture for Montesinos knots*, Trans. Amer. Math. Soc. **364** (2012), 233–285.

Lib82 A. Libgober, *Alexander polynomial of plane algebraic curves and cyclic multiple planes*, Duke Math. J. **49** (1982), 833–851; MR 683005.

Lib89 A. Libgober, *Invariants of plane algebraic curves via representations of the braid groups*, Invent. Math. **95** (1989), 25–30; MR 969412.

Lis07 P. Lisca, *Lens spaces, rational balls and the ribbon conjecture*, Geom. Topol. **11** (2007), 429–472.
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL cuspidal CURVES

EG91 Y. Eliashberg and M. Gromov, *Convex symplectic manifolds*, in *Several complex variables and complex geometry, Part 2*, Proceedings of Symposia in Pure Mathematics (AMS, 1991), 135–162.

Fen99 T. Fenske, *Rational 1- and 2-cuspidal plane curves*, Beitr. Algebra Geom. **40** (1999), 309–329; MR 1720107.

FLMN06 J. Fernández de Bobadilla, I. Luengo, A. Melle Hernández and A. Némethi, *On rational cuspidal projective plane curves*, Proc. Lond. Math. Soc. **92** (2006), 99–138.

FLMN07 J. Fernández de Bobadilla, I. Luengo, A. Melle Hernández and A. Némethi, *Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair*, in *Real and complex singularities*, Trends in Mathematics (Birkhäuser, Basel, 2007), 31–45; MR 2280130.

Fos20 E. Fossati, *Topological constraints for Stein fillings of tight structures on lens spaces*, J. Symplectic Geom. **18** (2020), 1475–1514.

Fra05 S. Francisco, *Symplectic isotopy for cuspidal curves*, PhD thesis, Massachusetts Institute of Technology (ProQuest LLC, Ann Arbor, MI, 2005); MR 2717257.

GK22 M. Golla and F. Kütle, *Symplectic isotopy of rational cuspidal sextics and septics*, Int. Math. Res. Not. IMRN (2022), rnab364. doi:10.1093/imrn/rnab364.

GL16 M. Golla and P. Lisca, *On Stein fillings of contact torus bundles*, Bull. Lond. Math. Soc. **48** (2016), 19–37; MR 3455745.

Gro85 M. Gromov, *Pseudo holomorphic curves in symplectic manifolds*, Invent. Math. **82** (1985), 307–347; MR 809718.

GS09 D. T. Gay and A. I. Stipsicz, *Symplectic surgeries and normal surface singularities*, Algebr. Geom. Topol. **9** (2009), 2203–2223; MR 2558309.

GS21 M. Golla and L. Starkston, *Rational cuspidal curves and symplectic fillings*, Preprint (2021), arXiv:2111.09700.

Har86 J. Harris, *On the Severi problem*, Invent. Math. **84** (1986), 445–461; MR 837522.

Hir83 F. Hirzebruch, *Arrangements of lines and algebraic surfaces*, in *Arithmetic and geometry, Vol. II*, Progress in Mathematics, vol. 36 (Birkhäuser, Boston, MA, 1983), 113–140; MR 717609.

HLS97 H. Hofer, V. Lizan and J.-C. Sikorav, *On genericity for holomorphic curves in four-dimensional almost-complex manifolds*, J. Geom. Anal. **7** (1997), 149–159; MR 1630789.

HW16 J. Hom and Z. Wu, *Four-ball genus bounds and a refinement of the Ozváth-Szabó tau invariant*, J. Symplectic Geom. **14** (2016), 305–323; MR 3523259.

Hon00 K. Honda, *On the classification of tight contact structures*, I, Geom. Topol. **4** (2000), 309–368.

IS99 S. Ivashkovich and V. Shevchishin, *Structure of the moduli space in a neighborhood of a cusp-curve and meromorphic hulls*, Invent. Math. **136** (1999), 571–602; MR 1695206.

KP17 M. Koras and K. Palka, *The Coolidge-Nagata conjecture*, Duke Math. J. **166** (2017), 3085–3145; MR 3715805.

KP22 M. Koras and K. Palka, *Complex planar curves homeomorphic to a line have at most four singular points*, J. Math. Pures Appl. (9) **158** (2022), 144–182; MR 4360367.

Lec12 A. G. Lecuona, *On the slice-ribbon conjecture for Montesinos knots*, Trans. Amer. Math. Soc. **364** (2012), 233–285.

Lib82 A. Libgober, *Alexander polynomial of plane algebraic curves and cyclic multiple planes*, Duke Math. J. **49** (1982), 833–851; MR 683005.

Lib89 A. Libgober, *Invariants of plane algebraic curves via representations of the braid groups*, Invent. Math. **95** (1989), 25–30; MR 969412.

Lis07 P. Lisca, *Lens spaces, rational balls and the ribbon conjecture*, Geom. Topol. **11** (2007), 429–472.
M. GOLLA AND L. STARKSTON

Lis08 P. Lisca, *On symplectic fillings of lens spaces*, Trans. Amer. Math. Soc. **360** (2008), 765-799;
MR 2346471.

Liu96 A.-K. Liu, *Some new applications of general wall crossing formula, Gompf's conjecture and its applications*, Math. Res. Lett. **3** (1996), 569-585.

Liu14 T. Liu, *On planar rational cuspidal curves*, PhD thesis, Massachusetts Institute of Technology
(2014).

LM19 T.-J. Li and C. Y. Mak, *Symplectic divisorial capping in dimension 4*, J. Symplectic Geom.
**17** (2019), 1835-1852.

McD90 D. McDuff, *The structure of rational and ruled symplectic 4-manifolds*, J. Amer. Math. Soc.
**3** (1990), 679-712; MR 1049697.

McD91 D. McDuff, *Blow ups and symplectic embeddings in dimension 4*, Topology **30** (1991), 409-421;
MR 1113685.

McD92a D. McDuff, *Immersed spheres in symplectic 4-manifolds*, Ann. Inst. Fourier (Grenoble)
**42** (1992), 369-392; MR 1162567.

McD92b D. McDuff, *Singularities of J-holomorphic curves in almost complex 4-manifolds*, J. Geom.
Anal. **2** (1992), 249-266; MR 1164604.

MS96 D. McDuff and D. Salamon, *A survey of symplectic 4-manifolds with $b^+ = 1$*, Turkish J. Math.
**20** (1996), 47-60.

MS12 D. McDuff and D. Salamon, *J-holomorphic curves and symplectic topology*, second edition,
Colloquium Publications, vol. 52 (American Mathematical Society, Providence, RI, 2012);
MR 2954391.

MS17 D. McDuff and D. Salamon, *Introduction to symplectic topology*, Oxford Graduate Texts in
Mathematics (Oxford University Press, 2017).

Miy84 Y. Miyaoka, *The maximal number of quotient singularities on surfaces with given numerical
invariants*, Math. Ann. **268** (1984), 159-171; MR 744605.

Miy01 M. Miyanishi, *Open algebraic surfaces*, CRM Monograph Series, vol. 12 (American Mathe-
matical Society, Providence, RI, 2001); MR 1800276.

Moe08 T. K. Moe, *Rational cuspidal curves*, Master's thesis, University of Oslo (2008).

Moi81 B. G. Moishezon, *Stable branch curves and braid monodromies*, in *Algebraic geometry*
*(Chicago, Ill., 1980)*, Lecture Notes in Mathematics, vol. 862 (Springer, Berlin-New York),
1981, 107-192; MR 644819.

Mos71 L. Moser, *Elementary surgery along a torus knot*, Pacific J. Math. **38** (1971), 737-745.

MW95 M. J. Micallef and B. White, *The structure of branch points in minimal surfaces and in
pseudoholomorphic curves*, Ann. of Math. **(2)** **141** (1995), 35-85; MR 1314031.

Mum79 D. Mumford, *An algebraic surface with $K$ ample, ($K^2$) = 9, $p_g$ = $q$ = 0*, Amer. J. Math.
**101** (1979), 233-244; MR 527834.

Nam84 M. Namba, *Geometry of projective algebraic curves*, Monographs and Textbooks in Pure and
Applied Mathematics, vol. 88 (Marcel Dekker, New York, 1984); MR 768929.

NP17 M. Nagel and M. Powell, *Concordance invariance of Levine-Tristram signatures of links*, Doc.
Math. **22** (2017), 25-43.

OO05 H. Ohta and K. Ono, *Symplectic 4-manifolds containing singular rational curves with (2,3)-
cusp*, Sémin. Congr. **10** (2005), 233-241.

OSz03 P. S. Ozsváth and Z. Szabó, *Absolutely graded Floer homologies and intersection forms for
four-manifolds with boundary*, Adv. Math. **173** (2003), 179-261; MR 1957829.

OSz04a P. S. Ozsváth and Z. Szabó, *Holomorphic disks and three-manifold invariants: properties and
applications*, Ann. of Math. **(2)** **159** (2004), 1159-1245.
M. GOLLA AND L. STARKSTON

Lis08 P. Lisca, *On symplectic fillings of lens spaces*, Trans. Amer. Math. Soc. **360** (2008), 765-799;
MR 2346471.

Liu96 A.-K. Liu, *Some new applications of general wall crossing formula, Gompf's conjecture and its applications*, Math. Res. Lett. **3** (1996), 569-585.

Liu14 T. Liu, *On planar rational cuspidal curves*, PhD thesis, Massachusetts Institute of Technology
(2014).

LM19 T.-J. Li and C. Y. Mak, *Symplectic divisorial capping in dimension 4*, J. Symplectic Geom.
**17** (2019), 1835-1852.

McD90 D. McDuff, *The structure of rational and ruled symplectic 4-manifolds*, J. Amer. Math. Soc.
**3** (1990), 679-712; MR 1049697.

McD91 D. McDuff, *Blow ups and symplectic embeddings in dimension 4*, Topology **30** (1991), 409-421;
MR 1113685.

McD92a D. McDuff, *Immersed spheres in symplectic 4-manifolds*, Ann. Inst. Fourier (Grenoble)
**42** (1992), 369-392; MR 1162567.

McD92b D. McDuff, *Singularities of J-holomorphic curves in almost complex 4-manifolds*, J. Geom.
Anal. **2** (1992), 249-266; MR 1164604.

MS96 D. McDuff and D. Salamon, *A survey of symplectic 4-manifolds with $b^+ = 1$*, Turkish J. Math.
**20** (1996), 47-60.

MS12 D. McDuff and D. Salamon, *J-holomorphic curves and symplectic topology*, second edition,
Colloquium Publications, vol. 52 (American Mathematical Society, Providence, RI, 2012);
MR 2954391.

MS17 D. McDuff and D. Salamon, *Introduction to symplectic topology*, Oxford Graduate Texts in
Mathematics (Oxford University Press, 2017).

Miy84 Y. Miyaoka, *The maximal number of quotient singularities on surfaces with given numerical
invariants*, Math. Ann. **268** (1984), 159-171; MR 744605.

Miy01 M. Miyanishi, *Open algebraic surfaces*, CRM Monograph Series, vol. 12 (American Mathe-
matical Society, Providence, RI, 2001); MR 1800276.

Moe08 T. K. Moe, *Rational cuspidal curves*, Master's thesis, University of Oslo (2008).

Moi81 B. G. Moishezon, *Stable branch curves and braid monodromies*, in *Algebraic geometry*
*(Chicago, Ill., 1980)*, Lecture Notes in Mathematics, vol. 862 (Springer, Berlin-New York),
1981, 107-192; MR 644819.

Mos71 L. Moser, *Elementary surgery along a torus knot*, Pacific J. Math. **38** (1971), 737-745.

MW95 M. J. Micallef and B. White, *The structure of branch points in minimal surfaces and in
pseudoholomorphic curves*, Ann. of Math. **(2)** **141** (1995), 35-85; MR 1314031.

Mum79 D. Mumford, *An algebraic surface with $K$ ample, ($K^2$) = 9, $p_g$ = $q$ = 0*, Amer. J. Math.
**101** (1979), 233-244; MR 527834.

Nam84 M. Namba, *Geometry of projective algebraic curves*, Monographs and Textbooks in Pure and
Applied Mathematics, vol. 88 (Marcel Dekker, New York, 1984); MR 768929.

NP17 M. Nagel and M. Powell, *Concordance invariance of Levine-Tristram signatures of links*, Doc.
Math. **22** (2017), 25-43.

OO05 H. Ohta and K. Ono, *Symplectic 4-manifolds containing singular rational curves with (2,3)-
cusp*, Sémin. Congr. **10** (2005), 233-241.

OSz03 P. S. Ozsváth and Z. Szabó, *Absolutely graded Floer homologies and intersection forms for
four-manifolds with boundary*, Adv. Math. **173** (2003), 179-261; MR 1957829.

OSz04a P. S. Ozsváth and Z. Szabó, *Holomorphic disks and three-manifold invariants: properties and
applications*, Ann. of Math. **(2)** **159** (2004), 1159-1245.
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

OSz04b P. S. Ozsváth and Z. Szabó, *Holomorphic disks and topological invariants for closed three-manifolds*, Ann. of Math. (2) **159** (2004), 1027–1158.

OSz05 P. S. Ozsváth and Z. Szabó, *Heegaard Floer homology and contact structures*, Duke Math. J. **129** (2005), 39–61.

Pal19 K. Palka, *Cuspidal curves, minimal models and Zaidenberg’s finiteness conjecture*, J. Reine Angew. Math. **747** (2019), 147–174; MR 3905132.

PP17 K. Palka and T. Pěłka, *Classification of planar rational cuspidal curves I. $C^{**}$-fibrations*, Proc. Lond. Math. Soc. (3) **115** (2017), 638–692; MR 3694295.

PP20 K. Palka and T. Pěłka, *Classification of planar rational cuspidal curves. II. Log del Pezzo models*, Proc. Lond. Math. Soc. (3) **120** (2020), 642–703.

Pio07 J. Piontkowski, *On the number of cusps of rational cuspidal plane curves*, Exp. Math. **16** (2007), 251–255; MR 2339280.

PV10 O. Plamenevskaya and J. Van Horn-Morris, *Planar open books, monodromy factorizations and symplectic fillings*, Geom. Topol. **14** (2010), 2077–2101; MR 2740642.

PY07 G. Prasad and S.-K. Yeung, *Fake projective planes*, Invent. Math. **168** (2007), 321–370; MR 2289867.

Rie74 O. Riemenschneier, *Deformationen von Quotientensingularitäten (nach zyklischen Gruppen)*, Math. Ann. **209** (1974), 211–248.

RS19 D. Ruberman and L. Starkston, *Topological realizations of line arrangements*, Int. Math. Res. Not. IMRN **2019** (2019), 2295–2331.

Ryb11 G. L. Rybnikov, *On the fundamental group of the complement of a complex hyperplane arrangement*, Funktsional. Anal. i Priložhen. **45** (2011), 71–85; MR 2848779.

Sch07 S. Schönenberger, *Determining symplectic fillings from planar open books*, J. Symplectic Geom. **5** (2007), 19–41.

She00 V. V. Shevchishin, *Pseudoholomorphic curves and the symplectic isotopy problem*, Preprint (2000), arXiv:math/0010262.

She04 V. V. Shevchishin, *On the local Severi problem*, Int. Math. Res. Not. IMRN **2004** (2004), 211–237; MR 2038713.

ST05 B. Siebert and G. Tian, *On the holomorphicity of genus two Lefschetz fibrations*, Ann. of Math. (2) **161** (2005), 959–1020; MR 2153404.

Sik03 J.-C. Sikorav, *The gluing construction for normally generic $J$-holomorphic curves*, in *Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001)*, Fields Institute Communications, vol. 35 (American Mathematical Society, Providence, RI, 2003), 175–199; MR 1969276.

Sta15 L. Starkston, *Symplectic fillings of Seifert fibered spaces*, Trans. Amer. Math. Soc. **367** (2015), 5971–6016; MR 3347194.

SSzW08 A. I. Stipsicz, Z. Szabó, and J. Wahl, *Rational blowdowns and smoothing of surface singularities*, J. Topol. **1** (2008), 447–517.

Tau96 C. H. Taubes, SW ⇒ Gr: *from the Seiberg-Witten equations to pseudo-holomorphic curves*, J. Amer. Math. Soc. **9** (1996), 845–918; MR 1362874.

Ton05 K. Tono, *On the number of the cusps of cuspidal plane curves*, Math. Nachr. **278** (2005), 216–221; MR 2111810.

Wah81 J. Wahl, *Smoothings of normal surface singularities*, Topology **20** (1981), 219–246.

Wal04 C. T. C. Wall, *Singular points of plane curves*, London Mathematical Society Student Texts, vol. 63 (Cambridge University Press, 2004).

Wen10 C. Wendl, *Strongly fillable contact manifolds and $J$-holomorphic foliations*, Duke Math. J. **151** (2010), 337–384.
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

OSz04b P. S. Ozsváth and Z. Szabó, *Holomorphic disks and topological invariants for closed three-manifolds*, Ann. of Math. (2) **159** (2004), 1027–1158.

OSz05 P. S. Ozsváth and Z. Szabó, *Heegaard Floer homology and contact structures*, Duke Math. J. **129** (2005), 39–61.

Pal19 K. Palka, *Cuspidal curves, minimal models and Zaidenberg's finiteness conjecture*, J. Reine Angew. Math. **747** (2019), 147–174; MR 3905132.

PP17 K. Palka and T. Pěłka, *Classification of planar rational cuspidal curves I. $C^{**}$-fibrations*, Proc. Lond. Math. Soc. (3) **115** (2017), 638–692; MR 3694295.

PP20 K. Palka and T. Pěłka, *Classification of planar rational cuspidal curves. II. Log del Pezzo models*, Proc. Lond. Math. Soc. (3) **120** (2020), 642–703.

Pio07 J. Piontkowski, *On the number of cusps of rational cuspidal plane curves*, Exp. Math. **16** (2007), 251–255; MR 2339280.

PV10 O. Plamenevskaya and J. Van Horn-Morris, *Planar open books, monodromy factorizations and symplectic fillings*, Geom. Topol. **14** (2010), 2077–2101; MR 2740642.

PY07 G. Prasad and S.-K. Yeung, *Fake projective planes*, Invent. Math. **168** (2007), 321–370; MR 2289867.

Rie74 O. Riemenschneier, *Deformationen von Quotientensingularitäten (nach zyklischen Gruppen)*, Math. Ann. **209** (1974), 211–248.

RS19 D. Ruberman and L. Starkston, *Topological realizations of line arrangements*, Int. Math. Res. Not. IMRN **2019** (2019), 2295–2331.

Ryb11 G. L. Rybnikov, *On the fundamental group of the complement of a complex hyperplane arrangement*, Funktsional. Anal. i Priložhen. **45** (2011), 71–85; MR 2848779.

Sch07 S. Schönenberger, *Determining symplectic fillings from planar open books*, J. Symplectic Geom. **5** (2007), 19–41.

She00 V. V. Shevchishin, *Pseudoholomorphic curves and the symplectic isotopy problem*, Preprint (2000), arXiv:math/0010262.

She04 V. V. Shevchishin, *On the local Severi problem*, Int. Math. Res. Not. IMRN **2004** (2004), 211–237; MR 2038713.

ST05 B. Siebert and G. Tian, *On the holomorphicity of genus two Lefschetz fibrations*, Ann. of Math. (2) **161** (2005), 959–1020; MR 2153404.

Sik03 J.-C. Sikorav, *The gluing construction for normally generic $J$-holomorphic curves*, in *Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001)*, Fields Institute Communications, vol. 35 (American Mathematical Society, Providence, RI, 2003), 175–199; MR 1969276.

Sta15 L. Starkston, *Symplectic fillings of Seifert fibered spaces*, Trans. Amer. Math. Soc. **367** (2015), 5971–6016; MR 3347194.

SSzW08 A. I. Stipsicz, Z. Szabó, and J. Wahl, *Rational blowdowns and smoothing of surface singularities*, J. Topol. **1** (2008), 447–517.

Tau96 C. H. Taubes, SW $\Rightarrow$ Gr: *from the Seiberg-Witten equations to pseudo-holomorphic curves*, J. Amer. Math. Soc. **9** (1996), 845–918; MR 1362874.

Ton05 K. Tono, *On the number of the cusps of cuspidal plane curves*, Math. Nachr. **278** (2005), 216–221; MR 2111810.

Wah81 J. Wahl, *Smoothings of normal surface singularities*, Topology **20** (1981), 219–246.

Wal04 C. T. C. Wall, *Singular points of plane curves*, London Mathematical Society Student Texts, vol. 63 (Cambridge University Press, 2004).

Wen10 C. Wendl, *Strongly fillable contact manifolds and $J$-holomorphic foliations*, Duke Math. J. **151** (2010), 337–384.
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

Wen18 C. Wendl, *Holomorphic curves in low dimensions: from symplectic ruled surfaces to planar contact manifolds*, Lecture Notes in Mathematics, vol. 2216 (Springer, 2018); MR 3821526.

Yau77 S. T. Yau, *Calabi's conjecture and some new results in algebraic geometry*, Proc. Natl. Acad. Sci. USA **74** (1977), 1798-1799; MR 0451180.

ZL83 M. G. Zaïdenberg and V. Y. Lin, *An irreducible, simply connected algebraic curve in $C^2$ is equivalent to a quasihomogeneous curve*, Dokl. Akad. Nauk SSSR **271** (1983), 1048-1052; MR 722017.

ZO96 M. G. Zaïdenberg and S. Y. Orevkov, *On rigid rational cuspidal plane curves*, Uspekhi Mat. Nauk **51** (1996), 149-150; MR 1392680.

Zar29 O. Zariski, *On the problem of existence of algebraic functions of two variables possessing a given branch curve*, Amer. J. Math. **51** (1929), 305-328; MR 1506719.

Marco Golla marco.golla@univ-nantes.fr

CNRS, Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2 rue de la Houssinière, 44322 Nantes, France

Laura Starkston 1starkston@math.ucdavis.edu

Department of Mathematics, UC Davis, One Shields Avenue, Davis, CA 95616, USA
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

Wen18 C. Wendl, *Holomorphic curves in low dimensions: from symplectic ruled surfaces to planar contact manifolds*, Lecture Notes in Mathematics, vol. 2216 (Springer, 2018); MR 3821526.

Yau77 S. T. Yau, *Calabi's conjecture and some new results in algebraic geometry*, Proc. Natl. Acad. Sci. USA **74** (1977), 1798-1799; MR 0451180.

ZL83 M. G. Zaïdenberg and V. Y. Lin, *An irreducible, simply connected algebraic curve in $C^2$ is equivalent to a quasihomogeneous curve*, Dokl. Akad. Nauk SSSR **271** (1983), 1048-1052; MR 722017.

ZO96 M. G. Zaïdenberg and S. Y. Orevkov, *On rigid rational cuspidal plane curves*, Uspekhi Mat. Nauk **51** (1996), 149-150; MR 1392680.

Zar29 O. Zariski, *On the problem of existence of algebraic functions of two variables possessing a given branch curve*, Amer. J. Math. **51** (1929), 305-328; MR 1506719.

Marco Golla marco.golla@univ-nantes.fr

CNRS, Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2 rue de la Houssinière, 44322 Nantes, France

Laura Starkston 1starkston@math.ucdavis.edu

Department of Mathematics, UC Davis, One Shields Avenue, Davis, CA 95616, USA