arXiv:1609.06551v3 [math.AG] 4 Dec 2017

NUMERICAL INVARIANTS AND MODULI SPACES FOR LINE
ARRANGEMENTS
ALEXANDRU DIMCA1 , DENIS IBADULA, AND DANIELA ANCA MĂCINIC
Abstract. Using several numerical invariants, we study a partition of the space
of line arrangements in the complex projective plane, given by the intersection
lattice types. We offer also a new characterization of the free plane curves using
the Castelnuovo-Mumford regularity of the associated Milnor/Jacobian algebra.

1. Introduction
Line arrangements in the complex projective plane P2 look like being simple objects, but a lot of questions related to them are still unanswered, e.g. Terao’s conjecture saying that the freeness of such an arrangement is determined by the combinatorics, see Conjecture 6.1 below for a statement, [9, Chapter 8] for more information,
and [35] for a survey. Or the conjecture that the monodromy of the associated Milnor
fiber is determined by the combinatorics, see [27] for a recent result and [9] for more
information.
In order to treat such questions, the study of parameter spaces (a.k.a moduli
spaces) of line arrangements has being developed, centered especially on the irreducibility/connectivity questions, see [3], [4], [5], [25], [34].
In this paper, the new idea is to look at the way in which the parameter spaces
A(L) and X(L) of line arrangements with a given intersection lattice L behave when
the lattice L changes. In section 2 we describe two parameter spaces for the line
arrangements A : f = 0 in P2 having d lines, namely A(d) and X(d), which are both
smooth irreducible varieties, see Corollary 2.3. To partition these two varieties A(d)
and X(d) into finer strata, keeping track of the properties of the line arrangements,
we use several numerical invariants and study their semi-continuity properties in
Proposition 2.5. We consider in this section both line arrangements and arbitrary
reduced curves in P2 , in order to point out that the numerical invariants associated
to line arrangements enjoy special properties, see Corollary 2.7.
In section 3 we recall the definition and main properties of free and nearly free plane
curves. Then we prove that a classical invariant in Commutative Algebra, namely
the Castelnuovo-Mumford regularity, coincides, when applied to the Milnor/Jacobian
algebra M(f ) of a reduced plane curve C : f = 0, to a naive invariant st(f ), coming
from the Hilbert function of the graded algebra M(f ), exactly when the curve C is
2010 Mathematics Subject Classification. Primary 32S22; Secondary 14H50, 14B05, 13D02.
Key words and phrases. plane curves; line arrangement; free curves; syzygy; Terao’s conjecture;
intersection lattice, Castelnuovo-Mumford regularity.
1
Partially supported by Institut Universitaire de France.
1

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free, see Theorem 3.4 and Corollaries 3.5 and 3.6. Corollary 3.6 depends on a key
result due to H. Schenck, see [28]. We end this section by noting that our partitions
of the spaces A(d) and X(d) are G−equivariant, where G = Aut(P2 ) is acting in
the obvious way on these parameter spaces. In Proposition 3.12 we describe the
dimension of a line arrangement under this G−action.
In section 4 we fix an integer d ≥ 4 and denote by L(d) the set of all possible
intersection lattices of arrangements with d lines, modulo lattice isomorphisms. For
each (isomorphism class of) lattice L ∈ L(d), we denote by X(L) the subset of X(d)
consisting of line arrangements having an intersection lattice isomorphic to L. Hence
the subsets X(L) for L ∈ L(d) form the strata of a partition of the smooth irreducible
variety X(d). The first properties of these strata are given in Proposition 4.1. Then
we discuss several examples of simple lattices L and of corresponding strata X(L),
e.g. the lattice Lgen corresponding to the generic line arrangement is discussed in
Example 4.6 and an obvious generalization, the lattice L(d, m), is considered in
Proposition 4.7. Other lattices occurs in Proposition 4.9, Example 4.10, Example
4.14, the last two describing free (resp. nearly free) line arrangements. Note that
our results on a stratum X(L) (e.g. dimension, connectivity) easily translate into
properties of the quotient X(L)/G. We also reprove in a new way a result due to
Tohăneanu [33], giving the classification of line arrangements with a Jacobian syzygy
of minimal degree 2, see Theorem 4.11.
In section 5 we point out the complexity of the stratification (X(L))L∈L(d) of the
space X(d). First we describe all the strata when d = 4, 5, 6, and pay particular
attention to the strata formed by (nearly) free arrangements. We explain just after
Proposition 5.7 that these stratification do not satisfy the frontier condition in general, in particular they are not Whitney regular stratifications. In Remark 5.8 and
in the answer to Question 5.9 we show that some nice features of this stratification
noticed when d ≤ 6 do not extend to higher degrees d.
In the final section we discuss Terao’s conjecture in the case of line arrangements,
recall the known results and give a new proof for Theorem 6.3. Finally, in Proposition
6.5 we give a generalization of the result saying that the generic arrangement is not
free when d ≥ 4.
The authors would like to thank the Oberwolfach Research Institute for Mathematics, where the major part on the work on this project was done during a RIP
program.
We also thank Torsten Hoge for his useful remarks on the previous version of this
paper.
2. General facts on plane curves and line arrangements
2.1. Two parameter spaces for line arrangements: A(d) and X(d). Let S =
C[x, y, z] be the graded polynomial ring in the variables x, y, z with complex coefficients, and Sm be the vector space of degree m homogeneous polynomials in S. Fix
an integer d ≥ 1 and regard the projective space C(d) = P(Sd ) as the parameter
space of degree d curves in P2 .

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Proposition 2.2. The subset C(d)0 ⊂ C(d) corresponding to curves having only
isolated singularities is Zariski open and dense in C(d). The subset A(d) ⊂ C(d)0
corresponding to line arrangements A consisting of d distinct lines is a closed Zariski
subset in C(d)0 .
Proof. For the first claim, note that the complement C(d) \ C(d)0 is the union of the
finite family of Zariski closed subsets given by the images of the obvious mappings
φm : P(Sm ) × P(Sd−2m ) → P(Sd ),

(A, B) 7→ A2 B, for m = 1, 2, ..., [d/2]. For the second claim, we consider the map
(2.1)

ψ : P(S1 )d → P(Sd ),

given by (ℓ1 , ..., ℓd ) 7→ ℓ1 · ℓ2 · ... · ℓd . Then A(d) is just the intersection of the set
C(d)0 with the image of the mapping ψ.

Note that
(2.2)

X(d) := ψ −1 (A(d))

is exactly the set of linear forms (ℓ1 , ..., ℓd ) ∈ P(S1 )d such that ℓi 6= ℓj for i 6= j,
and the restriction ψ : X(d) → A(d) is a Galois covering with structure group the
symmetric group σd on d elements.
Corollary 2.3. The parameter spaces A(d) and X(d) are smooth, irreducible algebraic varieties of dimension 2d. The space X(d) is simply connected and the fundamental group of the space A(d) is given by
π1 (A(d)) = σd .
Proof. The only claim that needs some explanation is the fact that X(d) is simply
connected. This follows from the fact that X(d) is obtained from the simply connected variety P(S1 )d by removing the codimension 2 linear subvarieties ∆ij : ℓi = ℓj
for all 1 ≤ i < j ≤ d.

2.4. Some numerical invariants for plane curves and line arrangements.
For a polynomial f ∈ Sd , we denote by Jf the ideal in S generated by the partial
derivatives fx , fy , fz , and call Jf the Jacobian ideal of f . The graded ring M(f ) =
S/Jf is called the Milnor or Jacobian algebra of f . We define
(2.3)

mk (f ) = dim M(f )k .

Note that one has mk (f ) = τ (f ) for k > 3(d − 2) and f ∈ C(d)0 , where τ (f ) is
the total Tjurina number of the reduced plane curve C(f ) : f = 0, see [6]. We also
denote by τ (C(f )) this number, and note that it is nothing else but the degree of
the Jacobian ideal Jf .
The minimal degree of a Jacobian syzygy for f is the integer mdr(f ) defined to
be the smallest integer r ≥ 0 such that there is a nontrivial relation
(2.4)

afx + bfy + cfz = 0

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among the partial derivatives fx , fy and fz of f with coefficients a, b, c in Sr . We
denote by AR(f ) the graded S-module consisting of all the triples (a, b, c) ∈ S 3
satisfying (2.4). In fact AR(f ) depends only on the class of f ∈ Sd in C(d) = P(Sd ).
Proposition 2.5.
(1) The subset {f ∈ Sd : mk (f ) ≤ m} ⊂ C(d) is Zariski open
and dense in C(d) for any k ≥ 0 and any m ≥ 0. In particular, the following
two sets {f ∈ C(d)0 : τ (f ) ≤ m} ⊂ C(d)0 and {f ∈ A(d) : τ (f ) ≤ m} ⊂
A(d), are Zariski open and dense in C(d)0 (resp. in A(d)) for any k ≥ 0 and
any m ≥ 0.
(2) The subset {f ∈ Sd : mdr(f ) ≤ m} ⊂ C(d) is Zariski closed in C(d) for any
m ≥ 0. In particular, the following two sets {f ∈ C(d)0 : mdr(f ) ≤ m} ⊂
C(d)0 and {f ∈ A(d) : mdr(f ) ≤ m} ⊂ A(d), are Zariski closed in C(d)0
(resp. in A(d)) for any m ≥ 0.
Proof. The first claim is clear by the semicontinuity properties of the rank of a matrix,
whose rows are obtained by taking all the coefficients of the polynomials µg, where
µ runs through the set of monomials of degree k − d + 1 in S and g ∈ {fx , fy , fz }.
3
) × P(Sd )
To prove the second claim, consider the closed subvariety Ym in P(Sm
given by
Ym = {((a, b, c), f ) : afx + bfy + cfz = 0}.
Note that a polynomial f ∈ Sd satisfies mdr(f ) ≤ m if and only if [f ] ∈ P(Sd ) is in
the image of Ym under the second projection.

Definition 2.6. For a polynomial f ∈ C(d)0 , we recall the following invariants.
(i) the coincidence threshold
ct(f ) = max{q : dim M(f )k = dim M(fs )k for all k ≤ q},

with fs a homogeneous polynomial in S of the same degree d as f and such that
Cs : fs = 0 is a smooth curve in P2 .
(ii) the stability threshold st(f ) = min{q : dim M(f )k = τ (f ) for all k ≥ q}.
(iii) the regularity reg(f ) is the Castelnuovo–Mumford regularity of the Milnor algebra M(f ), regarded as a graded S-module, see [18, Chapter 4].
The exact sequences
(2.5)
and
(2.6)

(fx ,fy ,fz )

0 → AR(f )(−(d − 1)) → S(−(d − 1))3 −−−−−→ S → M(f ) → 0
(fs,x ,fs,y ,fs,z )

0 → AR(fs )(−(d − 1)) → S(−(d − 1))3 −−−−−−−−→ S → M(fs ) → 0

and the fact that AR(fs )k = 0 for k < d − 1 imply that
(2.7)

ct(f ) ≥ mdr(f ) + d − 2,

with equality for mdr(f ) < d − 1. To have equality always, it is convenient to introduce the invariant mdre (f ), the minimal degree of an essential Jacobian relation
for f , which is by definition the minimal degree of a relation (2.4), where the triple

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(a, b, c) does not belong to the S-submodule of AR(f ) generated by the Koszul relations (fy , −fx , 0), (fz , 0, −fx ) and (0, fz , −fy ). With this definition we always have
ct(f ) = mdre (f ) + d − 2,

(2.8)
see [13].

Corollary 2.7. With the above notation, the following hold.
(1) Let C : f = 0 be a singular, reduced plane curve of degree d ≥ 3 in P2 . Then
mdr(f ) ≤ d − 1, mdre (f ) ≤ 2(d − 2) and ct(f ) ≤ 3(d − 2).
Moreover, if τ (C) = 1, all these inequalities are equalities.
(2) Let A : f = 0 be an arrangement having d ≥ 2 lines. Then
mdr(f ) = mdre (f ) ≤ d − 2 and ct(f ) ≤ 2(d − 2).
Moreover, both of these inequalities are equalities for a generic arrangement.
Proof. To prove (1), note that mdr(f ) ≤ d − 1 follows from the existence of Koszul
relations. The inequality mdre (f ) ≤ 2(d − 2) follows using (2.8) and the obvious
fact that ct(f ) ≤ T = 3(d − 2). When τ (C) = 1, it follows from [13, Example
4.3] that ct(f ) = T . Moreover, such a curve is nodal and irreducible, and hence
mdr(f ) ≥ d − 1, by [13, Theorem 4.1]. This remark completes the proof of the first
claim.
In view of Proposition 2.5 (2), to prove (2) it is enough to show that mdr(f ) = d−2
when A is a generic arrangement. But this follows from [13, Theorem 4.1].

The invariants st(f ) and reg(f ) are closely related, as the next section shows.
However, they do not seem to satisfy semicontinuity properties similar to those in
Proposition 2.5, see Remark 5.5.
3. Free and nearly free plane curves
3.1. Free plane curves. For the equivalence of the properties in the next definition,
we refer to [32]. See also [9, Chapter 8].
Definition 3.2. The curve C : f = 0 is a free divisor if the following equivalent
conditions hold.
(1) The Milnor algebra M(f ) is a Cohen-Macaulay S-module.
(2) The minimal graded resolution of the Milnor algebra M(f ) as an S-module
has the following form
(fx ,fy ,fz )

0 → S(−d1 − d + 1) ⊕ S(−d2 − d + 1) → S 3 (−d + 1) −−−−−→ S
for some positive integers d1 , d2 .
(3) The graded S-module AR(f ) is free of rank 2, i.e. there is an isomorphism
AR(f ) = S(−d1 ) ⊕ S(−d2 )
for some positive integers d1 , d2 .

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When C is a free divisor, the integers d1 ≤ d2 are called the exponents of C. They
satisfy the relations
(3.1)

d1 + d2 = d − 1 and τ (C) = (d − 1)2 − d1 d2 ,

where τ (C) is the total Tjurina number of C, see for instance [12]. For a free curve,
one has mdr(f ) = d1 , ct(f ) = d1 + d − 2 in view of (2.7), and st(f ) = d2 + d − 3, see
for instance [10].
Definition 3.3. The curve C : f = 0 is a nearly free divisor if the Milnor algebra
M(f ) has a minimal graded resolution of the form
(fx ,fy ,fz )

0 → S(−d − d2 ) → S(−d − d1 + 1) ⊕ S 2 (−d − d2 + 1) → S 3 (−d + 1) −−−−−→ S

for some integers 1 ≤ d1 ≤ d2 , called the exponents of C.

For a nearly free curve, the exponents satisfy d1 +d2 = d, and one has mdr(f ) = d1 ,
ct(f ) = d1 + d − 2 in view of (2.7), and st(f ) = d2 + d − 2 by the results in [10].
Theorem 3.4. Let C : f = 0 be a reduced plane curve. Then
st(f ) − 1 ≤ reg(f ) ≤ st(f ),

and the equality reg(f ) = st(f ) holds if and only if C : f = 0 is a free curve.
Proof. Let HM (f ) (resp. PM (f ) ) be the Hilbert function (resp. the Hilbert polynomial)
of the graded S-module M(f ). Then [18, Theorem 4.2] implies that
HM (f ) (k) = PM (f ) (k)
for any k ≥ reg(f ) + 1. Since for a reduced plane curve the Hilbert polynomial PM (f )
is just the constant τ (C), it follows from the definition of the stability threshold st(f )
that st(f ) ≤ reg(f ) + 1, and hence st(f ) − 1 ≤ reg(f ).
To prove the other inequality, let I be the saturation of the Jacobian ideal Jf , as
discussed in a more general setting in [20]. Consider the exact sequence of graded
S-modules
0 → I/Jf → M(f ) → S/I → 0.
Then [17, Corollary 20.19] implies that
reg(f ) = regM(f ) ≤ max{reg(I/Jf ), reg(S/I)}.

Note the module I/Jf has finite length, so [18, Corollary 4.4] implies that
reg(I/Jf ) = max{k : (I/Jf )k 6= 0} = sat(Jf ) − 1,

in the notation from [8]. Moreover, [8, Corollary 2] says that
sat(Jf ) ≤ max{T − ct(f ), st(f )},

where T = 3(d − 2). On the other hand the quotient S/I is a Cohen-Macaulay
module satisfying depth S/I = dim S/I = 1, and [18, Corollary 4.8] tells us that
s = reg(S/I), where s is the smallest integer such that k ≥ s implies
HS/I (k) = PS/I (k) = τ (C).

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This integer s is determined in [8, Proposition 2], where it is shown that one has
s = T − ct(f ). It follows that
reg(f ) ≤ max{T − ct(f ), st(f ) − 1}.
When C : f = 0 is free, we have ct(f ) + st(f ) = T by [10], and hence we get
reg(f ) ≤ st(f ). A direct computation using the definition of the CastelnuovoMumford regularity in terms of a resolution, see [17], p. 505, yields reg(f ) = d2 +d−3
when C is free. The equality st(f ) = reg(f ) follows using the above formulas for
st(f ). Note that in the free case, the equality reg(f ) = st(f ) is also a consequence
of Theorem 4.2 in [18], since M(f ) is Cohen-Macaulay in this case.
When C : f = 0 is not free, then it is shown in [10, Corollary 1.7] that ct(f ) +
st(f ) ≥ T + 2, which implies that max{T − ct(f ), st(f ) − 1} = st(f ) − 1, and this
completes the proof.

Corollary 3.5. Let C : f = 0 be a reduced plane curve of degree d ≥ 4. Then the
following hold.
(1) C is free if and only if reg(f ) = 2(d − 2) − mdre (f ).
(2) C is nearly free if and only if reg(f ) = 2(d − 2) − mdre (f ) + 1.
(3) If C is neither free nor nearly free, then reg(f ) ≥ 2(d − 2) − mdre (f ) + 2.
Proof. The claims (1) and (2) then follow from the equalities ct(f ) + st(f ) = T (resp.
ct(f ) + st(f ) = T + 2) which are shown in [10, Corollary 1.7] to characterize the free
(resp. nearly free) curves. The claim in (3) follows from the equality (2.8) and the
inequality ct(f ) + st(f ) ≥ T + 3 which holds in this case by [10, Corollary 1.7]. 
Note that in the cases (1) and (2) above one has mdr(f ) = mdre (f ), while in the
case (3) both mdr(f ) = mdre (f ) and mdr(f ) = d − 1 < mdre (f ) may occur.
Corollary 3.6. If A : f = 0 is any arrangement of d ≥ 4 lines, then
reg(f ) ≤ 2d − 5 and st(f ) ≤ 2d − 4.
When A : f = 0 is a generic arrangement of d ≥ 4 lines, both of the above inequalities
become equalities.
Proof. The inequality reg(f ) ≤ 2d − 5, with equality for a generic arrangement,
follows from [28, Corollary 3.5]. The reader must notice that the regularity there is
the Castelnuovo-Mumford regularity of the graded S-module AR(f ), and not as in
our paper that of the Milnor algebra M(f ). The exact sequence (2.5) allows us to
pass from one regularity to the other, namely one has
reg(f ) = reg(M(f )) = reg(AR(f )) + d − 3.
The formula for st(f ) in the case of a generic arrangement follows from [13, Corollary
1.3]. In fact, using Theorem 3.4, we need only one of these two invariants, since it is
known that such a line arrangement is not free, see for instance [12] or Proposition
6.5 below.

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In the general case, if the arrangement A is not free, then st(f ) = reg(f ) + 1 ≤
2d − 4. And for a free arrangement A : f = 0, one has st(f ) = d + d2 − 3 ≤ 2d − 4
since clearly d2 ≤ d − 1.

For the following result we refer to [11]. For the case of hyperplane arrangements
in Pn see [36].
Theorem 3.7. The set F (d, τ ) of free curves in the variety C(d, τ ) ⊂ C(d)0 of reduced
plane curves of degree d with a fixed global Tjurina number τ is a Zariski open subset
in C(d, τ ).
The set F A(d, τ ) of free line arrangements in the variety A(d, τ ) ⊂ A(d) of line
arrangements consisting of d lines and with a fixed global Tjurina number τ is a
Zariski open subset in A(d, τ ).
Using the Galois covering ψ : X(d) → A(d), we introduce the following notation.
X(d, mk ≤ m) = ψ −1 ({f ∈ A(d) : mk (f ) ≤ m})

(3.2)

and X(d, mk = m) = X(d, mk ≤ m) \ X(d, mk ≤ m − 1).

(3.3)

X(d, τ ≤ m) = ψ −1 ({f ∈ A(d) : τ (f ) ≤ m}) and X(d, τ ) = ψ −1 (A(d, τ )).
X(d, mdr ≤ m) = ψ −1 ({f ∈ A(d) : mdr(f ) ≤ m})

(3.4)

and X(d, mdr = m) = X(d, mdr ≤ m) \ X(d, mdr ≤ m − 1).

F X(d, τ ) = ψ −1 (F A(d, τ )) and F X(d) = ∪τ F X(d, τ ).

(3.5)

Then we have the following obvious consequence of Proposition 2.5 and Theorem
3.7.
Corollary 3.8. The sets X(d, mk ≤ m) and X(d, τ ≤ m) are Zariski open in X(d)
for any positive integers k and m. The set X(d; mdr ≤ m) is Zariski closed in X(d)
for any positive integer m. Moreover, the set F X(d, τ ) is Zariski open in the variety
X(d, τ ).
It is known that F A(d, τ ) 6= ∅ implies that there is an integer r ∈ [0, (d − 1)/2]
such that
(3.6)

F X(d, τ ) ⊂ X(d, mdr = r), where τ = τ (d, r) = (d − 1)2 − r(d − 1 − r),

see [10], [14]. This implies the following result.

Corollary 3.9. Let A : f = 0 be a free line arrangement in A(d, τ ) and define
3
τ (d)min = (d − 1)2
4

for d odd, and
3
τ (d)min = ⌊ (d − 1)2 ⌋ + 1
4
for d even. Then τ = τ (f ) ≥ τ (d)min .

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Proof. The inequalities follow from the formula for τ (d, r) given above. Here ⌊x⌋
denotes the integral part of the real number x.

Remark 3.10. It is shown in [10] that a line arrangement A : f = 0 with d = |A|
is nearly free with exponents d1 ≤ d2 = d − d1 if and only if
τ (A) = τ (d, d1 ) − 1.

3.11. Three group actions on parameter spaces. Consider the connected algebraic group G = Aut(P2 ) = P GL(3, C) of dimension 8. This group acts naturally
on the variety C(d) and all the subsets C(d)0 , A(d), F (d, τ ), F A(d, τ ) are in fact
G-invariant, hence they inherit a natural G-action, and are unions of G-orbits G · f ,
for some f ∈ C(d). Moreover, G acts also on the variety X(d) in a diagonal way,
and such that the map ψ : X(d) → A(d) is G-equivariant. It follows that all the
subsets X(d, mk ≤ m), X(d, τ ≤ m), X(d, τ ), X(d; mdr ≤ m), F X(d, τ ) are also
G-invariant, so they consists of unions of G-orbits, denoted by G·(ℓ1 , ..., ℓd ), for some
(ℓ1 , ..., ℓd ) ∈ X(d).
Proposition 3.12. Let (ℓ1 , ..., ℓd ) ∈ X(d) and denote f = ψ(ℓ1 , ..., ℓd ) ∈ A(d). Then
one has the following.
(1) dim G · (ℓ1 , ..., ℓd ) = dim G · f .
(2) dim G · f = 2 if d = 1, dim G · f = 4 if d = 2 and for d = 3, one has
dim G · f = 5 if A : f = 0 consists of 3 concurrent lines, and dim G · f = 6
if A : f = 0 consists of a triangle.
(3) For d ≥ 4, dim G · f = 5 if mdr(f ) = 0, dim G · f = 7 if mdr(f ) = 1 and
dim G · f = 8 if mdr(f ) > 1.
Proof. The first claim follows since the map ψ has finite fibers. To prove (2) and (3),
note that one has
dim G · f = dim G − dim F ix(f ),
where F ix(f ) is the stabilizer subgroup of f . The Lie algebra of F ix(f ) is exactly
AR(f )1 , i.e. the linear Jacobian syzygies, see [16, Proposition 1.1]. When d = 1, we
can take f = x and it follows that dim AR(f )1 = 6, since in the notation from (2.4)
one takes a = 0 and b, c ∈ S1 arbitrary. When d = 2, we can take f = xy, and it
follows that dim AR(f )1 = 4, since a = λx, b = −λy and c ∈ S1 arbitrary. When
d = 3 there are two possibilities. The first one is f = x3 + y 3, when dim AR(f )1 = 3,
as a = b = 0 and c ∈ S1 . The second case is f = xyz and then dim AR(f )1 = 2,
since AR(f )1 is spanned in this case by (x, −y, 0) and (x, 0, −z).
Assume now that d ≥ 4. Then, if r = mdr(f ) = 0, this means that dim AR(f )0 =
1, which implies dim AR(f )1 = 3. If r = 1, then it follows from [16, Proposition 2.2]
that dim AR(f )1 = 1. When r > 1, one has AR(f )1 = 0, so the claims in (3) are
now proved.

Let G be the Galois group of C over Q. Then G acts on the parameter spaces
A(d) and X(d) by acting on the coefficients of the defining equations. It follows that
all the subsets X(d, mk ≤ m), X(d, τ ≤ m), X(d, τ ), X(d; mdr ≤ m), F X(d, τ ),

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as well as A(d, mk ≤ m), A(d, τ ≤ m), A(d, τ ), A(d; mdr ≤ m), F A(d, τ ) are also
G-invariant.
The symmetric group σd also acts on X(d) by permuting the linear factors of the
defining equation f = 0 of a line arrangement, and this is the reason why some strata
in X(d) are not irreducible, while their images in A(d) have this property, see for
instance Proposition 4.7 (1).
3.13. On rigid plane curves and line arrangements. We say that a plane curve
C : f = 0 is algebraically rigid if (I/Jf )d = 0, where I denotes as above the saturation
of the Jacobian ideal Jf . Indeed, the vector space (I/Jf )d is naturally identified to
the space of first order locally trivial deformation of C in P2 , modulo the above Gaction, see [30], [31]. These deformations preserve the analytic isomorphism type of
each singular point of C.
Example 3.14. It is known that a curve C : f = 0 is free if and only if I = Jf , see
[32]. In particular, any free curve is algebraically rigid. A generic line arrangement
of 4 lines in P2 is not free, but it is algebraically rigid by Proposition 3.12 (3) since
mdr(f ) > 1 in this case. In fact, in this case one has dim(I/Jf )3 = 1 and (I/Jf )k = 0
for k 6= 3.
We say that a reduced plane curve C : f = 0 is topologically rigid if any deformation of C preserving the number of irreducible components of C, their degrees and
the topological type of each singularity of C is trivial modulo the above G-action.
For more on this type of rigidity see [24].
Example 3.15. A line arrangement A consisting in d ≥ 4 lines passing through one
point satisfies mdr(f ) = 0, and it is free. Hence A is an algebraically rigid curve. On
the other hand, we can modify the cross-ratio of a subset of 4 lines in A by moving
one line, without changing the topology of the singularity, and hence such a family
will not be contained in one G−orbit. Hence A is not topologically rigid.
Remark 3.16. If a reduced plane curve has only simple singularities of type Ak , Dk
and E6 , E7 , E8 , then C is algebraically rigid if and only if C is topologically rigid.
Indeed, for a simple singularity, a topologically constant deformation is the same as
an analytically constant deformation. In particular, for a line arrangement A having
only double and triple points, the two rigidity notions coincide. In such a case we
will simply say that A is rigid. For examples of this situation, see Remark 4.2 and
the stratum A(L(∆)) in Proposition 5.7 below.
For more on the interest of rigidity in the study of line arrangements, see [2].
4. A partition of the parameter space X(d)
From now on in this paper we assume that d ≥ 4. For a fixed integer d ≥ 4, we
denote by L(d) the set of all possible intersection lattices L(A), for line arrangements
in P2 consisting of d distinct lines. For a lattice L ∈ L(d), we denote by X(L) the
set of all elements (ℓ1 , ..., ℓd ) ∈ X(d) such that the line arrangement
(4.1)

A : ℓ1 = ... = ℓd = 0

NUMERICAL INVARIANTS AND MODULI SPACES FOR LINE ARRANGEMENTS

11

has an intersection lattice L(A) isomorphic to L, see [26] for more on intersection
lattices. We also set A(L) = ψ(X(L)). Such a lattice gives in particular information
on the multiple points p in the arrangement A, and about their multiplicities, denoted
by mp ≥ 2. In particular, we define
X
(4.2)
τ (L) =
(mp − 1)2 = τ (A).
p

By definition, we have the following partitions
X(d) = ∪L∈L(d) X(L) and X(d, τ ) = ∪L∈L(d),τ (L)=τ X(L)

(4.3)
and similarly
(4.4)

A(d) = ∪L∈L(d) A(L) and A(d, τ ) = ∪L∈L(d),τ (L)=τ A(L).

One has the following.

Proposition 4.1. For any lattice L ∈ L(d), the following hold.
(1) The sets X(L) and A(L) are constructible; they are also G-invariant and
G-invariant.
(2) X(L) ⊂ X(d, τ (L)) and A(L) ⊂ A(d, τ (L)).
(3) The function mdr : A(L) → N attains its minimal value on a Zariski closed
subset F of A(L), and in general F 6= A(L).
(4) The function mk : A(L) → N attains its minimal value on a Zariski open
subset Uk of A(L), and in general Uk 6= A(L).
Proof. The claim about the constructibility in (1) can be settled as follows. A point
p of multiplicity k ≥ 3 will give rise to a set E(L)p of k − 2 equations to be satisfied
by the set of coefficients (ai , bi , ci ) ∈ P(S1 ), where ℓi = ai x + bi y + ci z. Indeed,
if the lines passing through p are for instance Li : ℓi = 0 for i = 1, 2, ..., k, then
the fact that all these lines pass through p is expressed by the vanishing of k − 2
determinants D(1, 2, j) of 3 × 3 matrices A(1, 2, j), constructed using the coefficients
of ℓ1 , ℓ2 and ℓj to define the corresponding three rows, where j = 3, 4, ..., k. Note that
such determinants really define hypersurfaces in the product P(S1 )d . Moreover, when
three lines Lu , Lv and Lw are not concurrent, we should add the condition that the
corresponding determinant D(u, v, w) is not zero. More details on this construction
can be found in [25], see however Remark 4.4 below. The G-invariance and the
G-invariance of X(L) is obvious.
The claim (2) is clear. For the first part in claim (3), use Proposition 2.5, (2). For
the second part of claim (3), one may consider the example of two line arrangements
A : f = xy(x − y − z)(x − y + z)(2x + y − 2z)(x + 3y − 3z)(3x + 2y + 3z)
and

(x + 5y + 5z)(7x − 4y − z) = 0
A′ : f ′ = xy(x + y − z)(5x + 2y − 10z)(3x + 2y − 6z)(x − 3y + 15z)
(2x − y + 10z)(6x + 5y + 30z)(3x − 4y − 24z) = 0,

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A. DIMCA, D. IBADULA, AND A. D. MĂCINIC

having isomorphic intersection lattices and constructed by Ziegler in [37]. A picture
of these arrangements can be found in [9, Chapter 8]. They consists both of nine
lines, and have only double and triple points. More precisely, they have n2 = 18
double points and n3 = 6 triple points, and hence τ (A) = τ (A′ ) = 42. In the
case of A, the six triple points are on a conic, and a direct computation shows that
mdr(f ) = 5. For A′ , the six triple points are not on a conic, i.e. the arrangement A′
is a small deformation of the arrangement A, and a direct computation shows that
mdr(f ) = 6. See also [29, Example 13]. The above example settles also the claim
(4) by taking k = 13, since
m13 (f ′ ) = τ (f ′ ) = 42 < m13 (f ).

Remark 4.2. In fact, it is clear that there is a topologically constant 1-parameter
family of line arrangements At such that A0 = A and At for t 6= 0 has the same
numerical invariants as A′ . This family is obtained by moving the sixth triple point
till it gets onto the conic determined by the first 5 triple points. It follows that A is
not rigid, and one can check that dim(I/Jf )9 = 4. A direct computation shows that
for A′ one has dim(I/Jf )9 = 4 as well, i.e. A′ is not rigid either.
Remark 4.3. The above result says that the invariants mdr and mk are not determined by the combinatorics in general. However, if A is a free arrangement, both
mdr(f ) and mk (f ) are determined by the lattice L(A). The claim for mdr(f ) follows
from the formula (3.6). The claim for mk (f ) follows from the fact that the exponents
d1 = mdr(f ) and d2 = d − 1 − mdr(f ) determine the Hilbert function HM (f ) via the
resolution given in Definition 3.2 (2).
Remark 4.4. The set of equations E(L) = ∪p E(L)p defined above is smaller than the
set of equations constructed in [25], and which we call E ′ (L) here. Indeed, any point
p of multiplicity k ≥ 3 contributes k − 2 equations to our set E(L), and k3 equations
to the set E ′ (L). The two ideals I(E(L)) and I(E ′(L)) are distinct. Indeed, the
equations in E ′ (L) are linearly independent degree 3-forms, as each of them involves
monomials in distinct set of variables. For instance the monomial a1 b2 c3 occurs only
in the equation associated to the triple of lines (L1 , L2 , L3 ), supposed to pass through
a multiple point p.
On the other hand, it is clear that the two ideals I(E(L)) and I(E ′(L)) both have
Y (L) = X(L) as zero set, and hence one has in particular
X
(4.5)
codim X(L) = codim Y (L) ≤
(mp − 2),
p

where codim X(L) means the codimension of X(L) in the corresponding X(d). For
lattices L coming from line arrangements with few lines, or of a reduced complexity,
the above inequality is an equality, see for an example Proposition 4.7 (1) below.
However, the monomial arrangement
A(m, m, 3) : f = (xm − y m )(xm − z m )(y m − z m ) = 0,

NUMERICAL INVARIANTS AND MODULI SPACES FOR LINE ARRANGEMENTS

13

has d = 3m, 3 points of multiplicity m and m2 points of multiplicity 3. It follows
that
X
(mp − 2) = 3(m − 2) + m2 > 6m = dim X(3m)
p

for m ≥ 5. Hence for these values of m, the inequality (4.5) is strict.
Remark 4.5. The variety X(L) corresponds exactly to the variety of all ordered
complex realizations Σord (C) of the ordered combinatorics C ord considered in [5],
where C ord is the ordered combinatorial type associated to the lattice L with a fixed
numbering of the lines. The quotient X(L)/G is the ordered moduli space Mord (C)
considered in [5]. The variety A(L) corresponds exactly to the variety of all complex
realizations Σ(C) of the combinatorics C as considered in [5], while A(L)/G is the
moduli space M(C) of the combinatorics C. If L is the lattice corresponding to the
MacLane line arrangement, it follows from [5, Example 1.7] that X(L) is the union
of two G-orbits and in particular is not connected, while A(L) is just one G-orbit,
and hence it is irreducible.
Example 4.6. For any d, we denote by Lgen the lattice of the generic line arrangement of d lines. Then by the above description X(Lgen ) is a Zariski open
 subset of
d
X(d), and hence dim X(Lgen ) = dim X(d) = 2d. Moreover τ (Lgen ) = 2 and in fact
one has X(d, τ (Lgen )) = X(Lgen ), i.e. any lattice L ∈ L(d) with τ (L) = τ (Lgen ) is
in fact isomorphic to the lattice Lgen . To prove this claim, recall the formula
X mp  d
,
=
(4.6)
2
2
p
valid for any line arrangement, see for instance [21]. Since
 
mp
≤ (mp − 1)2
2
for any mp ≥ 2, and the equality holds if and only if mp = 2, the claim follows using
the formula (4.2). This argument implies also that τ (L) > τ (Lgen ) for any lattice
L ∈ L(d), L 6= Lgen .
Moreover, in this case it follows that the function mdr ◦ ψ is constant on X(Lgen ),
and it takes the value d − 2, see [13, Theorem 4.1], as well as all the functions mk ,
since one has ct(f ) = st(f ) = 2d−4 in this case, recall Corollary 2.7 (2) and Corollary
3.6.
This example can be generalized as follows. For any m satisfying 2 ≤ m ≤ d,
let L(d, m) denote the intersection lattice of a line arrangement in A(d) having one
point of multiplicity m and only double points in rest. Note that L(d, 2) = Lgen for
any integer d.
Proposition 4.7. Assume that d ≥ 4. Then the following hold.
(1) The sets X(L(d, m)) and A(L(d, m)) are smooth of dimension (2d − m + 2).
Moreover A(L(d, m)) is irreducible.


(2) τ (L(d, m)) = (m − 1)2 + m + (m + 1) + .... + (d − 1) = d2 + m−1
.
2

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A. DIMCA, D. IBADULA, AND A. D. MĂCINIC

(3) For 2m > d, the function mdr is constant on A(L(d, m)), and it takes the
values d − m.
(4) For m = d and m = d − 1, any arrangement in A(L(d, m)) is free. Any
arrangement in A(L(d, d − 2)) is nearly free. For 2 ≤ m ≤ d − 3, any
arrangement in A(L(d, m)) is neither free, nor nearly free.
(5) Any line arrangement A : f = 0 with mdr(f ) = 0 satisfies L(A) = L(d, d).
Any line arrangement A : f = 0 with mdr(f ) = 1 satisfies L(A) = L(d, d−1).
(6) A(L(d, d)) = A(d, τ (L(d, d)) and A(L(d, d − 1)) = A(d, τ (L(d, d − 1))).
Proof. To get an arrangement in A(L(d, m)), we have first to fix a point p ∈ P2 ,
and then m distinct lines passing through p. These choices are parametrized by
B = P2 ×U, where U is an open subset in (P1 )m . Note that B is smooth of dimension
m + 2 and irreducible. The remaining d − m lines are to be chosen in a Zariski open
set F ⊂ (P(S1 ))d−m , which is smooth of dimension 2d − 2m and irreducible. In this
way we have constructed a fibration F → A(L(d, m)) → B, proving the first claim
(1).
Note that X(L(d, m)) is not connected in general. Indeed, for d = 4 and m = 3,
we cannot continuously deform within X(L(4, 3)) an element (ℓ1 , ℓ2 , ℓ3 , ℓ4 ) where the
lines Lj : ℓj = 0 are concurrent for j = 1, 2, 3 to an element (ℓ′1 , ℓ′2 , ℓ′3 , ℓ′4 ) where the
lines L′j : ℓ′j = 0 are concurrent for j = 2, 3, 4.
The second claim follows from the formula (4.2). The third claim follows from [11,
Theorem 1.2]. The claim (4) follows from the formula for τ (L(d, m)) given in (2).
Indeed, if a line arrangement A : f = 0 in A(d) is free, then one has τ (A) = τ (d, r)
where r = mdr(f ), as explained in (3.6).
Suppose first that 2m > d, and hence r = d − m. The formula for τ (L(d, m)) given
in (2) shows that
(4.7)

δ = τ (d, r) − τ (A) =

(d − m)(d − m − 1)
.
2

Hence we have the equality δ = 0 only for m = d or for m = d − 1. Assume now
that r ≤ d − m − 1. Then [11, Theorem 1.2] implies that either r = m − 1 or m ≤ r.
In the first case the arrangement is free with exponents d1 = m − 1 and d2 = d − m,
and the equation (4.7) shows that this is possible only if m = d − 1 or m = d, which
is impossible. In the second case, [11, Theorem 1.1] shows that τ (A) ≤ τ (d, m) and
equality holds exactly when r = m and A is free. A direct computation shows that
(4.8)

δ ′ = τ (d, m) − τ (A) =

(d − m − 2)(d − m − 1)
+ (m − 1) > 0
2

for 2 ≤ m ≤ d − 2. For claim involving the nearly free arrangements, use the above
and Remark 3.10.
The first part in claim (5) is clear, since mdr(f ) = 0 if and only if f does not
depend on the variable z after a coordinate change. If mdr(f ) = 1, then let m ≥ 2
be the maximal multiplicity of an intersection point in A. Using [11, Theorem 1.2],
we deduce that 3 cases are possible.
(a) The case mdr(f ) = d − m = 1, which clearly settles our claim.

NUMERICAL INVARIANTS AND MODULI SPACES FOR LINE ARRANGEMENTS

15

(b) The case mdr(f ) = m − 1, impossible, since this would imply that A is a
generic arrangement, for which mdr(f ) = d − 2 ≥ 2.
(c) The case m ≤ mdr(f ), which is clearly impossible.
For claim (6), let B : g = 0 be a line arrangement in A(d, τ (L(d, d)) (resp. in
A(d, τ (L(d, d − 1))). Then mdr(g) ≥ 0 (resp. mdr(g) ≥ 1) and the claim follows
using [11, Theorem 1.1], which, though not stated there, holds for mdr(g) ≥ 0 as
well.

Corollary 4.8. For d ≥ 4 and m ∈ {2, d − 1, d}, one has
A(L(d, m)) = A(d, τ (L(d, m)) = A(d, mdr = d − m)
and
X(L(d, m)) = X(d, τ (L(d, m)) = X(d, mdr = d − m).
Here is another example of a simple combinatorics for line arrangements. Let
L̃(m1 , m2 ) be the lattice of a projective line arrangement A : f = 0 obtained by the
generic intersection of two pencils of m1 , respectively m2 lines, with m2 ≥ m1 ≥ 2.
Hence, a corresponding arrangement has d = (m1 + m2 ) lines, m1 m2 double points,
one point of multiplicity m1 and one point of multiplicity m2 . We will prove that
such an arrangement A is never free.
Proposition 4.9. With this notation, one has the following.
(1) The set A(L̃(m1 , m2 )) is smooth, irreducible of dimension d + 4.
(2) τ (L̃(m1 , m2 )) = (d − 1)2 − m1 m2 + 1.
(3) The function mdr is constant on A(L̃(m1 , m2 )) and takes the value m1 .
(4) The intersection F A(d) ∩ A(L̃(m1 , m2 )) is empty.
Proof. The first claim can be proved by a similar argument as that used in the proof
of Proposition 4.7 (1). The second claim is obvious.
By [11, Theorem 1.2] (applied for m = m2 ), either mdr(f ) = m1 , or mdr(f ) ≤
m1 −1. In the second case, either mdr(f ) ≤ m2 −1 and 2m2 < m1 +m2 , contradiction
with m1 ≤ m2 , or m2 ≤ mdr(f ) ≤ m1 − 1, again contradiction with m1 ≤ m2 . In
conclusion, mdr(f ) = m1 and this proves (3). To prove (4), note that the formula
for τ (L̃(m1 , m2 )) can be rewritten in the form
τ (L̃(m1 , m2 )) = (m1 − 1)2 + (m2 − 1)2 + m1 m2 = (d − 1)2 − m1 (m2 − 1) + 1 − m1 .
Then, by (3.6), the arrangement A is not free, since m2 − 1 = d − m1 − 1 and m1 > 1.

Example 4.10. A slight variation of the previous configuration yields examples of
free arrangements. Let L̂(m1 , m2 ) be the lattice of a projective line arrangement
A : f = 0 having exactly one line containing one point of multiplicity m1 and one
point of multiplicity m2 , m2 ≥ m1 ≥ 3, only double points apart from that, and
d = |A| = m1 + m2 − 1.

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By [11, Theorem 1.2] (applied for m = m2 ), either mdr(f ) = m1 − 1, or mdr(f ) ≤
m1 − 2. In the second case, it can only happen that mdr(f ) ≤ m1 − 2 ≤ m2 − 1 and
2m2 < m1 + m2 , contradiction with the assumption m1 ≤ m2 . So, mdr(f ) = m1 − 1
and τ (L̂(m1 , m2 )) = (m1 − 1)2 + (m2 − 1)2 + (m1 − 1)(m2 − 1). We already know that
such an arrangement A is free, since it is supersolvable (see [26, Prop 5.114] and [22,
Theorem 4.2]). See also [15] for this family of line arrangements.
In general one has X(L̂(m1 , m2 )) 6= X(m1 + m2 , τ (L̂(m1 , m2 ))). Indeed, by [15]
any exponents 2 ≤ d1 ≤ d2 of a free line arrangement can be obtained by such an
arrangement. But there are free arrangements B which are not of this type, e.g. the
monomial arrangements A(m, m, 3) for m ≥ 2 considered in Remark 4.4.
The classification of the line arrangements A : f = 0 with mdr(f ) = 2 is given by
the following theorem, which is one of the main results in [33]. We give a proof of
this classification from a new viewpoint.
Theorem 4.11. Let A : f = 0 be a line arrangement in P2 , with mdr(f ) = 2. Then
d = |A| ≥ 4 and A is one of the following type of line arrangements, described by
their intersection lattices.
(1) A ∈ A(L(d, d − 2)), or
(2) A ∈ A(L̂(3, d − 2)) with d ≥ 5, or
(3) A is linear equivalent to the monomial arrangement A(2, 2, 3).
Proof. Let m be the maximal multiplicity of an intersection point in A. If we denote
d = |A| ≥ 2 and we assume mdr(f ) = 2, then [11, Theorem 1.2] implies that only
the following cases are possible.
CASE 1. mdr(f ) = d − m, in other words m = d − 2. This case covers the two
cases (i) and (ii) in [33, Theorem 2]. Indeed, the case (1) corresponds to the case
when A has a triple point except the point of multiplicity m = d − 2 (e.g. for d = 5
we have two triple points in A), while the case (ii) corresponds to the case when
A has only double points except the point of multiplicity m = d − 2. In fact, the
line arrangements of type (ii) are exactly the line arrangement in A(L(d, d − 2))
considered in Proposition 4.7 (4) above, in particular they are all nearly free. When
d ≥ 5, the arrangements of type (i) are exactly the line arrangements with the
intersection lattice of type L̂(3, d − 2) which is discussed in Example 4.10 above, in
particular they are all free.
CASE 2. mdr(f ) = m−1 and A is free. In particular, this implies that the exponents
of A are d1 = 2 ≤ d2 , and hence d ≥ 5. Moreover m = 3, and hence A has n2 double
points, n3 triple points and no points of multiplicity > 3. Using the formulas (3.1)
and (4.6), we get the equations
n2 + 4n3 = d2 − 4d + 7 and 2n2 + 6n3 = d2 − d.

They imply that n2 = 10d − d2 − 21 ≥ 0 which yields d ≤ 7. It is easy to classify the
free line arrangements with 5 ≤ d ≤ 7 and only double and triple points and we get
in this way the lattice L̂(3, 3) already seen above, and the case (3) in [33, Theorem
2], which is essentially the monomial arrangement A(2, 2, 3).

NUMERICAL INVARIANTS AND MODULI SPACES FOR LINE ARRANGEMENTS

17

CASE 3. m = mdr(f ), in other words A has only double points, as in Example 4.6
above. But one knows that in this case mdr(f ) = d − 2, see [13, Theorem 4.1], and
hence we get again the case (ii) from [33, Theorem 2] for d = 4.

The above Theorem and Proposition 4.7 imply the following.
Corollary 4.12. Any line arrangement A : f = 0 with mdr(f ) ≤ 2 is either free or
nearly free. Moreover, when mdr(f ) ≤ 2, the lattice L(A) determines the values of
mdr(f ) and whether A is free or nearly free. In fact one has the following, where
d = |A|.
(1) mdr(f ) = 0 if and only if L(A) = L(d, d);
(2) mdr(f ) = 1 if and only if L(A) = L(d, d − 1);
(3) mdr(f ) = 2 if and only if L(A) is one of the lattices L(d, d − 2), L̂(3, d − 2)
or L(A(2, 2, 3)).
Remark 4.13. The line arrangements A : f = 0 with mdr(f ) = 3 can be classified
using the same approach, but the number of possibilities is much higher. Moreover,
there are line arrangements with mdr(f ) = 3 which are neither free, nor nearly free,
for instance the generic arrangement of 5 lines.
Example 4.14. We introduce a final lattice type. For two integers i ≤ j we define
a homogeneous polynomial in C[u, v] of degree j − i + 1 by the formula
(4.9)

gi,j (u, v) = (u − iv)(u − (i + 1)v) · · · (u − jv).

Consider the line arrangement A : f = 0 of d = m1 + m2 ≥ 4 lines in P2 given by
f (x, y, z) = x(y − z)g1,m1 −1 (x, y)g2,m2 (x, z) = 0

for 2 ≤ m1 ≤ m2 . Denote by L′ (m1 , m2 ) the corresponding intersection lattice L(A).
One can show that the following hold, see for instance [15].
(1) The line arrangement A has one point of multiplicity m1 , one point of multiplicity m2 , in addition to (m1 − 2) points of multiplicity 3 and m1 (m2 − 3) + 6
nodes;
(2) mdr(f ) = m1 ;
(3) τ (f ) = (d − 1)2 − m1 (m2 − 1) − 1.
Remark 4.15. We say that a lattice L is rigid if the corresponding constructible
set A(L) is the disjoint union of finitely many G−orbits. It is clear that if A : f = 0
corresponds to a point in A(L) with L rigid, any topologically constant deformation
of A is in fact a path in the connected component of A(L) containing A, which is by
definition a G−orbit. It follows that any such line arrangement A is topologically
rigid. Notice that the lattice L(d, m) is rigid for d ≥ 4 if and only if either (d, m) =
(4, 2) or (d, m) = (4, 3). This follows from Proposition 3.12 and Proposition 4.7.
Other examples of rigid lattices L are given in the next section.
A case of special interest is when the Galois group G acts transitively on the set
of orbits in A(L) for a rigid lattice L, see [2], [3], [5].

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5. On the partition A(d) = ∪L∈L(d) A(L)
In this section we describe the partition A(d) = ∪L∈L(d) A(L) for 4 ≤ d ≤ 6, and
show that the complexity of this partition increases rapidly with d.
5.1. The case d = 4. For d = 4, the list L(4) consists of 3 lattices, namely L(4, 2),
L(4, 3) and L(4, 4) in the notation from Proposition 4.7. Hence we have the following
partition
A(4) = A(L(4, 2)) ∪ A(L(4, 3)) ∪ A(L(4, 4)),
where dim A(L(4, 2)) = 8, dim A(L(4, 3)) = 7 and dim A(L(4, 4)) = 6. Moreover,
the sets A(L(4, 2)) and A(L(4, 3)) are G-orbits, i.e. the corresponding arrangements
are rigid, while A(L(4, 4)) is the union of a 1-parameter family of G-orbits, as can be
seen using Proposition 3.12 and its proof. Recall also Example 3.15. Note that the
closure of A(L(4, 2)) in A(4) is the whole space A(4), while the closure of A(L(4, 3))
in A(4) is A(L(4, 3)) ∪ A(L(4, 4)), which follows from Corollary 3.8. Moreover the
set A(L(4, 4)) is closed in A(4).
In this case, one has
6 = τ (L(4, 2)) < 7 = τ (L(4, 3)) < 9 = τ (L(4, 4))
and hence the corresponding 3 strata are distinguished by their Tjurina numbers.
Moreover, one has
F A(4) = A(L(4, 3)) ∪ A(L(4, 4)).
Note also that even in this simple case, the set
A(4, st ≤ 3) := {f ∈ A(4) : st(f ) ≤ 3}) = A(L(4, 3))

is neither open nor closed. Hence the invariant st does not have nice semicontinuity
properties as τ or mdr. By inspection of this list, we can state the following result.
Proposition 5.2. With this notation, one has the following complete list of free and
nearly free line arrangements for d = 4.
(1) The set A(4, 6) is open, 8-dimensional, and consists only of nearly free arrangements with exponents d1 = d2 = 2.
(2) The set A(4, 7) is irreducible, 7-dimensional, and consists only of free arrangements with exponents d1 = 1, d2 = 2.
(3) The set A(4, 9) is irreducible, 6-dimensional, and consists only of free arrangements with exponents d1 = 0, d2 = 3.
5.3. The case d = 5. For d = 5, the list L(5) consists of L(5, 2), L(5, 3), L(5, 4),
L(5, 5) and an additional lattice L = L(A) where A : xyz(x + y)(x + z) = 0. Note
that the lattice L is just the lattice L̂(3, 3) from Example 4.10. In this case one has
A(5, mdr = 2) = A(L(5, 3)) ∪ A(L),

with dim A(L(5, 3)) = 9 and dim A(L) = 8. Hence Corollary 4.8 does not hold for
m = d − 2 in this case. One also has
10 = τ (L(5, 2)) < 11 = τ (L(5, 3)) < 12 = τ (L) < 13 = τ (L(5, 4)) < 16 = τ (L(5, 5)).

NUMERICAL INVARIANTS AND MODULI SPACES FOR LINE ARRANGEMENTS

19

Hence again the corresponding 5 strata are distinguished by their Tjurina numbers.
Moreover, one has
F A(5) = A(L) ∪ A(L(5, 4)) ∪ A(L(5, 5)).
Note that
A(L) ∩ A(L(5, 4)) = ∅.
Though this might be obvious for some readers, we prefer to give an argument which
is likely to work in many similar situation. Note that, using the Curve Selection
Lemma, if A(L) ∩ A(L(5, 4)) 6= ∅, then we get a deformation of an ordinary singular
point (Y4 , 0) of multiplicity 4 into two ordinary singular points (Y3 , 0) of multiplicity
3. Such a deformation is impossible, since it would contradict the semicontinuity of
the spectrum on the interval I = (−1/3, 2/3], see for details [23, Theorem (8.9.8)].
Indeed, one has
1 = degI spec(Y4 ) < 2 degI spec(Y3 ) = 2.
It follows that
A(L) ∩ A(L(5, 4)) = A(L(5, 5)).
By inspection of the list of lattices in L(5), we can state the following result.
Proposition 5.4. With this notation, one has the following complete list of free and
nearly free line arrangements for d = 5.
(1) The set A(5, 11) is irreducible, 9-dimensional, and consists only of nearly
free arrangements with exponents d1 = 2, d2 = 3. Any f ∈ X(5, 11) satisfies
st(f ) = 6 and reg(f ) = 5.
(2) The set A(5, 12) is irreducible, 8-dimensional, and consists only of free arrangements with exponents d1 = d2 = 2. Any f ∈ X(5, 12) satisfies st(f ) =
reg(f ) = 4.
(3) The set A(5, 13) is irreducible, 8-dimensional, and consists only of free arrangements with exponents d1 = 1, d2 = 3. Any f ∈ X(5, 13) satisfies
st(f ) = reg(f ) = 5.
(4) The set A(5, 16) is irreducible, 7-dimensional, and consists only of free arrangements with exponents d1 = 0, d2 = 4. Any f ∈ A(5, 16) satisfies
st(f ) = reg(f ) = 6.
Note that among the above sets, only A(5, 12) is a G−orbit and hence the corresponding arrangements are rigid. The arrangements in A(5, 13) are algebraically
rigid, but not topologically rigid, recall Proposition 3.12.
Remark 5.5. It is clear that A(5, 12) is contained in the closure of A(5, 11), and
that A(5, 16) is contained in the closure of A(5, 12). The values given above for st(f )
and reg(f ) show that these invariants do not enjoy simple semicontinuity properties
as in Proposition 2.5.

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A. DIMCA, D. IBADULA, AND A. D. MĂCINIC

5.6. The case d = 6. For d = 6, the list L(6) consists of 10 lattices. We list them
in increasing order of their Tjurina numbers.
• For τ = 15, we have only the lattice L(6, 2) as predicted by the general theory,
recall Example 4.6. Moreover X(L(6, 2)) is an open subset in the 12-dimensional
smooth variety X(6)
• For τ = 16, we have only the lattice L(6, 3) and the corresponding set X(L(6, 3))
has codimension 1 in X(6).
• For τ = 17, we have two lattices, namely L̃(3, 3) and a new lattice, say L̃′ (3, 3).
These two lattices have each 2 triple points and 9 nodes, and the invariant mdr takes
the value 3 in both cases. In the lattice L̃(3, 3) the 2 triple points are not on a line
of the corresponding arrangement, while in the lattice L̃′ (3, 3) the 2 triple points are
on such a line. In conclusion the corresponding two sets X(L̃(3, 3)) and X(L̃′ (3, 3))
are not distinguished by the numerical invariants considered in this paper. Indeed,
since ct(f ) = 7 and st(f ) = 8 in both cases, the invariants mk ’s also coincide for any
k.
Both sets X(L̃(3, 3)) and X(L̃′ (3, 3)) have codimension 2 in X(6)
• For τ = 18, we have again two lattices, namely the lattice L(6, 4), having a point
of multiplicity 4, 9 nodes and mdr = 2 and the lattice L′ (3, 3) introduced in Example
4.14, and having 3 triple points, 6 nodes and mdr = 3.
The set X(L(6, 4)) has codimension 2 in X(6), while the set X(L′ (3, 3)) has codimension 3 in X(d).
Comparing the values of τ (L) ≤ 18 and the corresponding values of the invariant
mdr, we conclude that there are no free arrangements in this range.
• For τ = 19, we have again two lattices, namely the lattice L̂(3, 4), having one
point of multiplicity 4, one triple point and 6 nodes, and the lattice L(∆) corresponding to the arrangement
A : f = (x2 − y 2)(x2 − z 2 )(y 2 − z 2 ) = 0,

and hence having 4 triple points and 3 nodes. Both of the corresponding sets X(L)
contain only free arrangements with mdr = 2.
• For τ = 21, we have only the lattice L(6, 5).
• For τ = 25, we have only the lattice L(6, 6). The properties of the last two
lattices are discussed in Proposition 4.7 (4), (5) and (6). In particular, the sets
X(d, τ ) for τ = 19, 21, 25 consist only of free arrangements, i.e. the last claim in
Corollary 3.8 holds in a stronger version.
As a conclusion, we can state the following result.
Proposition 5.7. With this notation, one has the following complete list of free and
nearly free line arrangements for d = 6.
(1) The set A(6, 18) has two irreducible components, namely A(L(6, 4)) of dimension 10, and A(L′ (3, 3)) of dimension 9; they consist only of nearly free
arrangements with exponents d1 = 2, d2 = 4, and respectively d1 = d2 = 3.
(2) The set A(6, 19) has two irreducible components, namely A(L̂(3, 4)) of dimension 9, and A(L(∆)) of dimension 8; they consist only of free arrangements
with exponents d1 = 2, d2 = 3.

NUMERICAL INVARIANTS AND MODULI SPACES FOR LINE ARRANGEMENTS

21

(3) The set A(6, 21) is irreducible, 9-dimensional, and consists only of free arrangements with exponents d1 = 1, d2 = 4.
(4) The set A(6, 25) is irreducible, 8-dimensional, and consists only of free arrangements with exponents d1 = 0, d2 = 5.
Note that Y = A(L(∆)) is the only G-orbit in the list above, and hence consists
only of rigid arrangements. Moreover Z = Y \ Y is a closed G-invariant subset of
dimension < 8. Using the same type of argument as in the case d = 5 above, one can
show that Z ⊂ A(L(6, 6). Since A(L(6, 6)) is 8-dimensional by Proposition 4.7 (1), it
follows that Z is not a union of strata in the partition. This shows in particular that
this partition is not Whitney regular, see [7, Chapter 1] for basic facts on regular
stratifications.
Remark 5.8. It is possible to extend this discussion to d = 7. For d < 7 we have
seen that the closed set X(d, τ ≥ τ (d)min ) contains only free arrangements. For
d = 7, one has τ (d)min = 27. The new aspect occurring in this case is that the set
X(7, τ ≥ 27) contain free arrangements and one nearly free arrangement type with
exponents d1 = 2, d2 = 5.
The above leads us to ask the following.
Question 5.9. Is it true that, for any d ≥ 4, the closed set X(d, τ ≥ τ (d)min ) in
X(d) contains only nearly free and free arrangements with d lines?
Unfortunately the answer to this question is negative. To see this, it is enough to
consider the arrangements in X(L(d, d − 3)) for d ≥ 11. Then using Proposition 4.7
it is easy to check that τ (L(d, d − 3)) > τ (d)min . On the other hand, the formula
(4.7) and the characterization of free arrangements (resp. nearly free arrangements)
by the property δ = 0 (resp. δ = 1) given in [10] show that any arrangement in
X(L(d, d − 3)) for d ≥ 11 is neither free nor nearly free.
6. On Terao’s conjecture
With the above notation, this conjecture in the case of line arrangements can be
stated as follows.
Conjecture 6.1. [Terao’s Conjecture for the line arrangement A] Let A be a free
line arrangement with d = |A|. Then
A(L(A)) ⊂ F A(d).

Equivalently, X(L(A)) ⊂ F X(d).

Assume that A is free with exponents d1 ≤ d2 . Then the following are known.

Theorem 6.2. Terao’s conjecture holds for the line arrangement A if one has either
d = |A| ≤ 12, or d1 ≤ 5.

For the proofs of this result, see [1], [2], [19]. The next result was proved in
[11], but we give below a new, clearer proof. Another rapid proof can be obtained
by combining Theorem 2.7 and Lemma 2.10 in [2]. Let m(A) denote the maximal
multiplicity of an intersection point in A.

22

A. DIMCA, D. IBADULA, AND A. D. MĂCINIC

Theorem 6.3. Terao’s conjecture holds for the line arrangement A if m(A) ≥ d1 .
Proof. We apply first [11, Theorem 1.2] to the free arrangement A. It follows that
(6.1)

d1 ∈ {d − m, m − 1, m},

where m = m(A). It follows that m ∈ {d − d1 , d1 , d1 + 1}. Let now B ∈ A(L(A)),
given by g = 0, and note that m(B) = m. Now we apply [11, Theorem 1.2] to the
line arrangement B.
If we are in the case mdr(g) = d − m(B) = d − m ∈ {d1 , d − d1 , d − d1 − 1}, it
follows that mdr(g) ≥ mdr(f ) = d1 . Since τ (B) = τ (A), the result [11, Theorem
1.1] implies that B is free with the same exponents d1 ≤ d2 .
If we are in the case mdr(g) = m(B) − 1, then the arrangement B is free, so there
is nothing to prove. Finally, we have to consider the case m = m(B) ≤ mdr(g) ≤
d − m − 1. For m = d − d1 this implies d − d1 ≤ d1 − 1, which is impossible. For
m = d1 or m = d − d1 − 1 = d2 , we are again in the case mdr(g) ≥ mdr(f ) = d1 ,
and we conclude as above.

As explained in [11], this Theorem implies the following.
Corollary 6.4. With the above notation, one has the following.
(1) Terao’s conjecture holds for the line arrangement A if m(A)√≥ d/2.
(2) Terao’s conjecture holds for the line arrangement A if d1 ≤ 2d + 1 − 1.
We end with a result saying that a free arrangement cannot have too many singularities.
Proposition 6.5. The intersection F X(d) ∩ X(L) is empty if

3
τ (L) < (d − 1)2 .
4
In particular, the inequality (6.2) holds if
X
(d + 3)(d − 1)
(mp − 1) >
4
p
(6.2)

where p runs through the set of multiple points of the lattice L, and mp ≥ 2 denotes
the multiplicity of p.
Proof. The first claim follows from Corollary (3.9).
Then the formula for τ (L) given in (4.2) and the equality (4.6) imply that
 
  X
3
(d + 3)(d − 1)
d
d
= (d − 1)2 .
−
−
(mp − 1) < 2
τ (L) = 2
4
4
2
2
p

Example 6.6. Assume that the line arrangement A is not generic, but has a lot of
− 2 nodes, besides some other multiple points.
nodes, namely it has N > (d+3)(d−1)
4

NUMERICAL INVARIANTS AND MODULI SPACES FOR LINE ARRANGEMENTS

23

Then A is not free by the above result, since there is at least one point p with mp ≥ 3.
When d = 7, this says that an arrangement A having at least 14 nodes satisfies
3
τ (A) < τ (7)min = (d − 1)2 = 27
4

and hence it is not free. A detailed classification of the line arrangements B with
|B| = 7, shows that there is a nearly free arrangement B having 11 nodes and one
point of multiplicity 5 such that τ (B) = 27. Moreover, for all arrangements B′ having
12 nodes (and some other multiple points), one has τ (B′ ) < 27. Hence our bound is
two units apart from the optimal one in this case.
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Université Côte d’Azur, CNRS, LJAD, France
E-mail address: dimca@unice.fr
Ovidius University, Faculty of Mathematics and Informatics, 124 Mamaia Blvd.,
900527 Constanţa, Romania
E-mail address: denis.ibadula@univ-ovidius.ro
Simion Stoilow Institute of Mathematics, P.O. Box 1-764, RO-014700 Bucharest,
Romania
E-mail address: Anca.Macinic@imar.ro

