ON THE NUMBER OF MODULI OF PLANE SEXTICS WITH
SIX CUSPS
CONCETTINA GALATI
Abstract. Let Σ6
be the variety of irreducible sextics with six cusps as
singularities. Let Σ ⊂ Σ6
be one of irreducible components of Σ6
. Denoting
by M4 the space of moduli of smooth curves of genus 4, we consider the rational
map Π : Σ 99K M4 sending the general point [Γ] of Σ, corresponding to a plane
curve Γ ⊂ P2, to the point of M4 parametrizing the normalization curve of
Γ. The number of moduli of Σ is, by definition the dimension of Π(Σ). We
know that dim(Π(Σ)) ≤ dim(M4) + ρ(2, 4, 6) − 6 = 7, where ρ(2, 4, 6) is the
Brill-Neother number of linear series of dimension 2 and degree 6 on a curve
of genus 4. We prove that both irreducible components of Σ6
have number
of moduli equal to seven.
1. Introduction
Let Σnk,d ⊂ P(H0(P2,OP2(n))) := PN , with N =
n(n+3)
, be the closure, in the
Zariski’s topology, of the locally closed set of reduced and irreducible plane curves
of degree n with k cusps and d nodes. Let Σ ⊂ Σnk,d be an irreducible component
of the variety Σnk,d. Denoting by Mg the moduli space of smooth curves of genus
− k − d, it is naturally defined a rational map
ΠΣ : Σ 99K Mg,
sending the general point [Γ] ∈ Σ0 to the isomorphism class of the normalization
of the curve Γ ⊂ P2 corresponding to [Γ]. We say that ΠΣ is the moduli map of Σ
and we set
number of moduli of Σ := dim(ΠΣ(Σ)).
We say that Σ has general moduli if ΠΣ is dominant. Otherwise, we say that Σ has
special moduli or that Σ has finite number of moduli. By lemma 2.2 of [4], we know
that the dimension of the general fibre of ΠΣ is at least equal to
max(8, 8 + ρ− k),
where ρ := ρ(2, g, n) = 3n− 2g− 6 is the number of Brill-Noether of linear series of
degree n and dimension 2 on a smooth curve of genus g. It follows that, if Σ has
the expected dimension equal to 3n+ g − 1− k and g ≥ 2, then
(1) dim(ΠΣ(Σ)) ≤ min(dim(Mg), dim(Mg) + ρ− k).
Definition 1.1. We say that Σ has the expected number of moduli if equality holds
in (1).
In particular, we expect that, if ρ − k ≤ 0, then on the normalization curve C
of the curve Γ ⊂ P2 corresponding to the general point [Γ] ∈ Σ, there exists only
a finite number of linear series of degree n and dimension 2 mapping C to a plane
curve with nodes and k cusps as singularities and corresponding to a point of Σ,
(see the proof of lemma 2.2 of [4]). For a deeper discussion and a list of known
results about the moduli problem of Σnk,d we refer to sections 1 and 2 of [4] and
related references. In particular, in [4] we have found sufficient conditions in order
Key words and phrases. number of moduli, sextics with six cusps, plane curves, Zariski pairs.
http://arxiv.org/abs/0704.0622v1
2 CONCETTINA GALATI
that an irreducible component Σ of Σnk,d has finite and expected number of moduli.
If Σ verifies these conditions then ρ(2, n, g) ≤ 0. Finally in [4] we constructed
examples of families of plane curves with nodes and cusps with finite and expected
number of moduli. In this paper we consider the particular case of the variety Σ66,0
of irreducible sextics with six cusps.
It was proved by Zariski (see [8]) that Σ66,0 has at least two irreducible com-
ponents. One of them is the parameter space Σ1 of the family of plane curves of
equation
f32 (x0, x1, x2) + f
3 (x0, x1, x2) = 0,
where f2 and f3 are homogeneous polynomials of degree two and three respectively.
The general point of Σ1 corresponds to an irreducible sextic with six cusps on a
conic as singularities. Moreover, Σ66,0 contains at least one irreducible component
Σ2 whose general element corresponds to a sextic with six cusps not on a conic as
singularities and containing in its closure the variety Σ69,0 of elliptic sextics with nine
cusps. Recently, A. Degtyarev has proved that Σ1 and Σ2 are the unique irreducible
components of Σ66,0, (see [1]).
The moduli number of Σ1 and Σ2 can not be calculated by using the result of
[4]. Indeed, in this case ρ(2, 4, 6) = 4 > 0 and then the general element of every
irreducible component of Σ66,0 does not verify the hypotheses of proposition 4.1 of [4].
On the contrary, it is easy to verify that, if Γ ⊂ P2 is the plane curve corresponding
to the general element of one of the irreducible components of Σ66,0 and C is the
normalization curve of Γ, then the map µo,C is injective. But, in contrast with the
nodal case, this information is not useful in order to study the moduli problem of
Σnk,d, (see [6] and remark 4.2 of [4]). In the proposition 2.2 and corollary 2.4, we
prove that Σ2 has the expected number of moduli equal to seven. Moreover, we
show that there exists a stratification
Σ69,0 ⊂ Σ′ ⊂ Σ̃ ⊂ Σ2,
where Σ′ and Σ̃ are respectively irreducible components of Σ68,0 and Σ
7,0 with ex-
pected number of moduli. Finally, in the corollary 2.8, we prove that also Σ1 has
the expected number of moduli by using that every element of Σ1 is the branch
locus of a triple plane.
2. On the number of moduli of complete irreducible families of plane
sextics with six cusps
First of all we want to find sufficient conditions in order that, if an irreducible
component Σ of Σnk,d has the expected number of moduli, then every irreducible
component Σ′ of Σnk′,d′ , containing Σ, has the expected number of moduli. In the
corollary 4.7 of [4] we considered this problem under the hypothesis that Σ has the
expected dimension and ρ(2, n, g) ≤ 0. Now we are interested to the case ρ > 0.
We need the following local result.
= {(a, b, x, y)| y2 = x3 + ax+ b} ⊂ C2 × A2
be the versal deformation family of an ordinary cusp (see [3] for the definition and
properties of the versal deformation family of a plane singularity). We recall that
the general curve of this family is smooth. The locus ∆ of C2 of the pairs (a, b) such
that the corresponding curve is singular, has equation 27b2 = 4a3. For (a, b) ∈ ∆
and (a, b) 6= (0, 0), the corresponding curve has a node and no other singularities,
whereas (0, 0) is the only point parametrizing a cuspidal curve.
ON THE NUMBER OF MODULI OF PLANE SEXTICS WITH SIX CUSPS 3
Lemma 2.1 ([3], page 129.). Let G → C2 be a two parameter family of curves
of genus g ≥ 2, whose general fibre is stable and which is locally given by y2 =
x3 + ax + b, with (a, b) ∈ C2 and let D ⊂ C2 be a curve passing through (0, 0)
and not tangent to the axis b = 0 at (0, 0). Then the j-invariant of the elliptic
tail of the curve which corresponds to the stable limit of G(0,0), with respect to the
curve D, doesn’t depend on D. Otherwise, for every j0 ∈ C, there exists a curve
Dj0 ⊂ C2 passing through (0, 0) and tangent to the axis b = 0 at this point, such that
the elliptic tail of the stable reduction of G(0,0) with respect to Dj0 , has j-invariant
equal to j0.
Proposition 2.2. Let Σ ⊂ Σnk,d, with k < 3n, be an irreducible component of
Σnk,d. Let g be the geometric genus of the plane curve corresponding to the general
element of Σ. Suppose that g ≥ 2, ρ(2, g, n)− k ≤ 0 and Σ has the expected number
of moduli equal to 3g − 3 + ρ− k. Then, every irreducible component Σ′ of Σnk′,d′ ,
with k′ = k− 1 and d = d′ or k = k′ and d′ = d− 1, such that Σ ⊂ Σ′, has expected
number of moduli.
Proof. First we consider the case k′ = k − 1 and d = d′. Let q1, . . . , qk be the
cusps of Γ. It is well known that, since k < 3n then [Γ] ∈ Σnk−1,d. In particular,
for every fixed cusp qi of Γ there exists an irreducible analytic branch Si of Σnk−1,d
passing through the point [Γ] and whose general point corresponds to a plane curve
Γ′ of degree n with d nodes and k − 1 cusps specializing to the singular points
of Γ different from qi, as Γ
′ specializes to Γ. Moreover, it is possible to prove
that every Si is smooth at the point [Γ], see [7] or chapter 2 of [5]. Let Σ′ be
one of the irreducible components of Σnk−1,d containing Σ. Notice that the general
element of Σ′ corresponds to a curves of genus g′ = g + 1. Since ρ(2, g′, n) − k′ =
3n− 2g − 2− 6− k + 1 = ρ(2, g, n)− k − 1 < 0, in order to prove the theorem it is
enough to show that the general fibre of the moduli map
ΠΣ′ : Σ
99K Mg+1
has dimension equal to eight. Let us notice that the map ΠΣ′ is not defined at the
general element [Γ] of Σ. More precisely, let γ ⊂ Si ⊂ Σ′ be a curve passing through
[Γ] and not contained in Σ. Let C → γ be the tautological family of plane curves
parametrized by γ. Let C′ → γ be the family obtained from C → γ by normalizing
the total space. The general fibre of C′ → γ is a smooth curve of genus g+1, while
the special fibre C′0 := Γ′ is the partial normalization of Γ obtained by smoothing
all the singular points of Γ, except the marked cusp qi. If we restrict the moduli
map ΠΣ′ to γ, we get a regular map which associates to [Γ] the point corresponding
to the stable reduction of Γ with respect to the family C′ → γ, which is the union
of the normalization curve C of Γ and an elliptic curve, intersecting at the point
q ∈ C which maps to the cusp qi ∈ Γ. Now, let G ⊂ Σ′ × Mg+1 be the graph
of ΠΣ′ , let π1 : G → Σ′ and π2 : G → Mg+1 be the natural projections and let
U ⊂ Σ be the open set parametrizing curves of degree n and genus g with exactly
k cusps and d nodes as singularities. From what we observed before, if we denote
by ΠΣ′(Σ) the Zariski closure in Mg+1 of π2π−11 (U), then ΠΣ′(Σ) is contained in
the divisor ∆1 ⊂ Mg+1, whose points are isomorphism classes of reducible curves
which are union of a smooth curve of genus g and an elliptic curve, meeting at a
point. Denoting by ΠΣ : Σ → Mg the moduli map of Σ, the rational map
∆1 99K Mg
which forgets the elliptic tail, restricts to a rational dominant map
q : ΠΣ′(Σ) 99K ΠΣ(Σ).
The dimension of the general fibre of q is at most two. Since, by hypothesis, the
dimension of the fibre of the moduli map ΠΣ is eight, there exists only a finite
4 CONCETTINA GALATI
number of g2n on C, ramified at k points, which maps C to a plane curve D such
that the associated point [D] ∈ P
n(n+3)
2 belongs to Σ. In particular, the set of points
x of C such that there is a g2n with k simple ramification points, one of which at
x, is finite. So, the dimension of the general fibre of q is at most one. In order to
decide if the general fibre of q has dimension zero or one, we have to understand
how the j-invariant of the elliptic tail of the stable reduction of Γ′ with respect the
family C′ → γ, depends on γ. If C → C2 is the étale versal deformation family of
the cusp. By versality, for every fixed cusp pi of Γ, there exist étale neighborhoods
n(n+3)
2 of [Γ] in P
n(n+3)
2 , V
v→ C2 of (0, 0) in C2 and Ui of pi in the tautological
family U → P
n(n+3)
2 with a morphism f : U → V such that the family Ui → U is
the pullback, with respect to f , of the restriction to V of the versal family. By the
properties of the étale versal deformation family of a plane singularity, (see [2]), we
have that f−1((0, 0)) is an étale neighborhood of [Γ] in the (smooth) analytic branch
Σn1,0 whose general element corresponds to an irreducible plane curves with only one
cusp at a neighborhood of the cusp qi of Γ. So, dim(f
−1((0, 0))) =
n(n+3)
− 2 and
the map f is surjective. Moreover, if g is the restriction of f at u−1(Σ′), then also
g is surjective. Indeed,
g−1((0, 0)) = f−1((0, 0)) ∩ u−1(Σ′) = u−1((Σ))
and, since k < 3n, then dim(Σ) = 3n + g − 1 − k = dim(Σ′) − 2 and g is sur-
jective. By using lemma 2.1, it follows that the general fibre of the natural map
ΠΣ′(Σ) → ΠΣ(Σ) has dimension exactly equal to one. Therefore, dim(ΠΣ′ (Σ)) =
dim(ΠΣ(Σ)) + 1 = 3g − 3 + ρ(2, g, n)− k + 1 = 3(g + 1)− 3 + ρ(2, g + 1, n)− k By
using that
dim(ΠΣ′ (Σ
′)) ≥ dim(ΠΣ′(Σ)) + 1 = 3(g + 1)− 3 + ρ(2, g + 1, n)− k + 1.
and by recalling that, by lemma 2.2 of [4], it is always true that dim(ΠΣ′(Σ
′)) ≤
3(g + 1)− 3 + ρ(2, g + 1, n)− k + 1, the statement is proved in the case k′ = k − 1
and d′ = d.
Suppose, now, that k = k′ and d′ = d − 1. Also in this case Σ is not contained
in the regularity domain of ΠΣ′ . More precisely, if [Γ] ∈ Σ is general, then ΠΣ′([Γ])
consists of a finite number of points, corresponding to the isomorphism classes
of the partial normalizations of Γ obtained by smoothing all the singular points
of Γ, except for a node. Then ΠΣ′(Σ) is contained in the divisor ∆0 of Mg+1
parametrizing the isomorphism classes of the analytic curves of arithmetic genus
g + 1 with a node and no more singularities. The natural map ∆0 99K Mg sending
the general point [C′] of ∆0 to the isomorphism class of the normalization of C
restricts to a rational dominant map q : ΠΣ′(Σ) 99K ΠΣ(Σ). Since we suppose that
Σ has the expected number of moduli and ρ(2, g, n)−k ≤ 0, if C is the normalization
of the plane curve corresponding to the general element of Σ, then the set S of the
linear series of dimension 2 and degree n on C with k simple ramification points,
mapping C to a plane curve D such that the associated point [D] in the Hilbert
Scheme belongs to Σ, is finite. We deduce that also the set S′ of the pairs of points
(p1, p2) of C, such that there is a g
n ∈ S such that the associated morphism maps
p1 and p2 to the same point of the plane, is finite. So, also q
−1([C]) is finite and
dim(ΠΣ′ (Σ)) = dim(ΠΣ(Σ)). It follows that
dim(ΠΣ′ (Σ
′)) ≥ 3g− 3+ 3n− 2g− 6− k+1 = 3(g+1)− 3+ 3n− 2(g+1)− 6− k.
Remark 2.3. Notice that, the arguments used before to prove lemma 2.2 don’t
work if the dimension of the general fibre of the moduli map of Σ has dimension
bigger than eight. Indeed, in this case, the dimension of the general fibre of the map
ON THE NUMBER OF MODULI OF PLANE SEXTICS WITH SIX CUSPS 5
ΠΣ′(Σ) 99K ΠΣ(Σ) could be bigger than one if k
′ = k − 1 and d = d′, or than zero
if k′ = k and d = d′ − 1.
Corollary 2.4. There exists at least one irreducible component Σ2 of Σ
6,0 having
the expected number of moduli equal to dim(M4) − 2 and whose general element
corresponds to a sextic with six cusps not on a conic.
Remark 2.5. As we already observed in the previous section, Σ2 is the only com-
ponent of Σ66,0 parametrizing sextics with six cusps not on a conic by [1].
Proof. Let Σ69,0 be the variety of elliptic plane curves of degree six with nine cusps
and no more singularities. It is not empty and irreducible, because, by the Plücker
formulas, the family of dual curves is Σ30,0 ≃ P9, which is irreducible and not empty.
Moreover, if we compose an holomorphic map φ : C → P2 from a complex torus C
to a smooth plane cubic with the natural map φ(C) → φ(C)∗, where we denoted by
the dual curve of φ(C), we get a morphism from C to a plane sextic with nine
cusps. Therefore, the number of moduli of Σ69,0 is equal of the number of moduli of
Σ30,0, equal to one. Since 6 < 3n = 18, there is at least one irreducible component
Σ′ of Σ68,0 containing Σ
9,0. Let ΠΣ′ : Σ
99K M2 be the moduli map of Σ′ and let
G ⊂ Σ′ × M2 be its graph. If we denote by π1 : G → Σ′ and π2 : G → M2 the
natural projection, by U the open set of Σ69,0 parametrizing cubics of genus one
with nine cusps and by ΠΣ′(Σ
9,0) the Zariski closure in M2 of π2π−11 (U), then, by
arguing as in the first part of the proof of the lemma 2.2, we have a dominant map
ΠΣ′(Σ
9,0) 99K M1, whose general fibre has dimension one. We conclude that
dim(πΣ′ (Σ
′)) ≥ dim(πΣ′(Σ69,0)) + 1 = 3
and so, the moduli map of Σ′ is dominant, as one expects, because ρ(2, 2, 6)− 8 =
18− 4− 6− 8 = 0. Let D be the plane sextic corresponding to the general point of
Σ′. By Bezout theorem, the height cusps P1, . . . , P8 of D don’t belong to a conic
and, however we choose five cusps of D, no four of them lie on a line. Then, let
C2 be the unique conic containing P1, . . . , P5. There exists at least a cusp, say P6,
which does not belong to C2. Since 8 < 3n = 18, there exists a family of plane
sextics D → ∆, whose special fibre is D and whose general fibre has a cusp at
a neighborhood of every cusp of D different from P7 and no further singularities.
By lemma 2.2, the curve ∆ is contained in an irreducible component of Σ67,0 with
expected number of moduli. By repeating the same argument for the general fibre
of the family D → ∆ we get an irreducible component Σ2 of Σ66,0 with the expected
number of moduli and whose general element parametrizes a sextic with six cusps
not on a conic. �
Now we consider the irreducible component Σ1 of Σ
6,0 parametrizing plane curves
of equation f23 (x0, x1, x2) + f
2 (x0, x1, x2) = 0, where f2 is an homogeneous poly-
nomial of degree two and f3 is an homogeneous polynomial of degree three. The
general element of Σ1 corresponds to an irreducible plane curve of degree six with
six cusps on a conic. We want to show that Σ1 has the expected number of moduli
equal to 12− 3 + ρ(2, 4, 6)− 6 = 7 = dim(M4)− 2. Equivalently, we want to show
that the general fibre of the moduli map
Σ1 99K M4
has dimension equal to eight.
Lemma 2.6. Let Γ2 : f2(x0, x1, x2) = 0 and Γ3 : f3(x0, x1, x2) = 0 be a smooth
conic and a smooth cubic intersecting transversally. Then, the plane curve
Γ : f23 (x0, x1, x2)− f32 (x0, x1, x2) = 0
is an irreducible sextic of genus four with six cusps at the intersection points of Γ2
and Γ3 as singularities. The curve Γ is projection of a canonical curve C ⊂ P3 from
6 CONCETTINA GALATI
a point p ∈ P3 which is contained in six tangent lines to C. Moreover, for every
point q ∈ P3 −C such that the projection plane curve πq(C) of C from q is a sextic
with six cusps on a conic of equation g23(x0, x1, x2) − g32(x0, x1, x2) = 0, where g3
and g2 are two homogeneous polynomials of degree three and two respectively, there
exists a cubic surface S3 ∈ |IC|P3(3)|, containing C, such that the plane curve πq(C)
is the branch locus of the projection πq : S3 → P2.
Remark 2.7. Notice that, by [1], every irreducible sextic with six cusps on a conic
as singularities has equation given by (f2(x0, x1, x2))
3 + (f3(x0, x1, x2))
2 = 0, with
f2 and f3 homogeneous polynomials of degree two and three. In order words, all the
sextics with six cusps on a conic as singularities are parametrized by points of Σ1.
An other proof of this result as been provided to us by G. Pareschi.
Proof of lemma 2.6. Let f(x0, x1, x2) = f
3 (x0, x1, x2) − f32 (x0, x1, x2) = 0 be the
equation of Γ. From the relation f3(x) = ±f2(x)
f2(x), we deduce that
(x) =
±2∂f2
f2(x) and hence
(x) = 2
(x)f3(x)− 3f2(x)2
(x) = −f2(x)2
By using that the conic Γ2 : f2 = 0 is smooth, it follows that, if a point x ∈ Γ is
singular, then x ∈ Γ2 and hence x ∈ Γ3∩Γ2. On the other hand, always from (2), if
x ∈ Γ2∩Γ3, then x is a singular point of Γ. Hence, the singular locus of Γ coincides
with Γ3 ∩ Γ2. Let x be a singular point of Γ. If
p1(x, y) + terms of degree two = 0
q1(x, y) + terms of degree ≥ two = 0
are respectively affine equations of Γ2 and Γ3 at x, then, the affine equation of Γ at
x is given by
q1(x, y)
2 − p1(x, y)3 + terms of degree ≥ four = 0.
Since Γ2 and Γ3 intersect transversally, we have that q1(x, y) does not divide p1(x, y)
and hence Γ has an ordinary cusp at x. Let now φ : C → Γ be the normalization of
Γ. We recall that the cubics passing through the six cusps of Γ cut out on C the
complete canonical series |ωC |. Since the cusps of Γ is contained in the conic Γ2 ⊂ P2
of equation f2 = 0, the lines of P
2 cut out on C a subseries g ⊂ |ωC | of dimension
two of the canonical series. Moreover, if we still denote by C a canonical model of
C in P3, then the linear series g is cut out on C in P3 from a two dimensional family
of hyperplanes passing through a point p ∈ P3−C. If we project C from p we get a
plane curve projectively equivalent to Γ. Since Γ has six cusps as singularities, we
deduce that there are six tangent lines to C passing through p. To see that Γ is the
branch locus of a triple plane, let S3 ⊂ P3 be the cubic surface of equation
F3(x0, . . . , x3) = x
3 − 3f2(x0, x1, x2)x3 + 2f3(x0, x1, x2) = 0.
If p = [0, 0, 0, 1], then, by using Implicit Function Theorem, the ramification locus
of the projection πp : S3 → P2, is given by the intersection of S3 with the quadric
S2 of equation
= x23 − f2(x0, x1, x2) = 0. Now, if x = [x0, x1, x2] ∈ S3 ∩ S2,
then x3 = ±
f2(x0, x1, x2). By substituting in the equation of S3, we find that the
branch locus of the projection πp : S3 → P2 coincides with the plane curve Γ. From
what we proved before, it follows that the ramification locus of the projection map
πp : S3 → P2 is the normalization curve C of Γ. Finally, if q ∈ P3 − C is an other
point such that the plane projection πq(C) is an irreducible sextic with six cusps on a
conic parametrized by a point xq ∈ Σ1 ⊂ P27, then, up to projective motion, we may
always assume that q = [0 : 0 : 0 : 1] and hence, if g23(x0, x1, x2)− g32(x0, x1, x2) = 0
ON THE NUMBER OF MODULI OF PLANE SEXTICS WITH SIX CUSPS 7
is the equation of the plane curve πq(C), then C is the locus of ramification of the
projection from q to the plane of the cubic surface of equation
x33 − 3g2(x0, x1, x2)x3 + 2g3(x0, x1, x2) = 0.
Corollary 2.8. The irreducible component Σ1 of Σ
6,0 parametrizing plane curves of
equation f23 (x0, x1, x2)+ f
2 (x0, x1, x2) = 0, where f2 is an homogeneous polynomial
of degree two and f3 is an homogeneous polynomial of degree three, has the expected
number of moduli equal to 7 = dim(M4) + ρ(2, 4, 6)− 6.
Proof. Let [Γ] ⊂ P2 be a plane sextic of equation f23 (x0, x1, x2) − f32 (x0, x1, x2) =
0, where the conic f2 = 0 and the cubic f3 = 0 are smooth and they intersect
transversally. Let C ⊂ P3 be the normalization curve of Γ and let SC be the set of
points v = [v0 : · · · : v3] ∈ P3 such that there exists a cubic surface S3 ∈ |IC|P3(3)|,
containing C, such that the curve C is the ramification locus of the projection πv :
S3 → P2. By the former lemma, in order to prove that Σ1 has the expected number
of moduli, it is enough to find a point [Γ] of Σ1 corresponding to an irreducible
plane sextic Γ ⊂ P2 with six cusps of a conic such that the set SC is finite. Let Γ2
be the smooth conic of equation f2(x0, x1, x2) = x
1 − x22 = 0 and let Γ3 be the
smooth cubic of equation f3(x0, x1, x2) = x
0+x0x
2−x21x2 = 0. If a1, a2 and a3 are
the three different solutions of the polynomial x3 + x2 + x− 1 = 0, then Γ2 and Γ3
intersect transversally at the points [ai,
ai, 1], [ai,−
ai, 1], with i = 1, 2, 3. By
the former lemma, the plane sextic Γ of equation f32 − f23 = 0 is irreducible and
it has six cusps at the intersection points of Γ2 and Γ3 as singularities. Moreover,
the normalization curve C of Γ is the canonical curve of genus 4 in P3 which is
intersection of the cubic surface S3 ⊂ P3 of equation
F3(x0, x1, x2, x3) = x
3 + (x
0 + x
1 − x22)x3 + x30 + x0x22 − x21x2 = 0
and the quadric S2 of equation
= 3x23 + x
0 + x
1 − x22 = 0.
We want to show that SC is finite. To see this we observe that, since
h0(P3, IC|P3(2)) = 1 and h0(P3, IC|P3(3)) = 5,
the equation of every cubic surface containing C and which is not the union of S2
and an hyperplane is given by
G(x0, . . . , x3;β0, . . . , β3) = F3(x0, x1, x2, x3) +
∂F3(x0, x1, x2, x3)
with βj ∈ C, for i = 0, . . . , 3. Now, a point [v] = [v0, . . . , v3] ∈ SC if and only if
there exist β0, . . . , β3 such that C is contained in the intersection of
G(x0, . . . , x3;β0, . . . , β3) = 0 and
∂G(x0, . . . , x3;β0, . . . , β3)
Still using that h0(P3, IC|P3(2)) = 1, a point [v] ∈ P3 belongs to SC if and only if
∂G(x0, . . . , x3;β0, . . . , β3)
∂F3(x0, . . . , x3)
for some λ ∈ R− 0, or, equivalently,
βjxj)
∂x3∂xi
= (λ−
viβi − v3)
8 CONCETTINA GALATI
The previous equality of polynomials is equivalent to the following bilinear system
of ten equations in the variables v0, . . . , v3 and β0, . . . , β3
(1 + β3)v0 + 3β0v3 = 0 (x0x3)
(1 + β3)v1 + 3β1v3 = 0 (x1x3)
(1 + β3)v2 − 3β2v3 = 0 (x2x3)
β1v0 + β0v1 = 0 (x0x1)
β2v0 + (1− β0)v2 = 0 (x0x2)
(1 − β2)v1 + β1v2 = 0 (x1x2)
2β1v1 − v2 = λ−
j=0 βjvj − v3 (x21)
−v0 + 2β2v2 = λ−
j=0 βjvj − v3 (x22)
(3 + 2β0)v0 = λ−
j=0 βjvj − v3 (x20)
2β3v3 = λ−
j=0 βjvj − v3 (x23)
The points of SC are the solutions v of the previous linear system, as a linear
system whose coefficients depend on β0, . . . , β3. It is easy to prove that it has only
a solution equal to (v0, v1, v2, v3) = (0, 0, 0, λ) if β0 = β1 = β2 = β3 = 0 and it has
not solutions otherwise, (see [5], page 98). By the previous lemma, we conclude
that the point [0 : 0 : 0 : 1] ∈ P3 is the only point which belongs to six tangent lines
to the canonical curve C ⊂ P3 which is intersection of the cubic surface of equation
F3(x0, x1, x2, x3) = x
3 + (x
0 + x
1 − x22)x3 + x30 + x0x22 − x21x2 = 0
and the quadric of equation
= 3x23 + x
0 + x
1 − x22 = 0.
It follows that, on the normalization curve D of the plane curve Γ′ corresponding
to the general point of Σ1 ⊂ Σ66,0 there exists only a finite number of linear series
of dimension two with six ramification points. �
Remark 2.9. By using the notation introduced in the proof of corollary 2.8, we
observe that in this corollary we have proved that if C is a general canonical curve of
genus four such that the set SC is not empty, then SC is finite. Actually, C. Ciliberto
pointed out to our attention that it is possible to show, with a very simple argument,
that for every canonical curve C of genus four such that SC is not empty, we have
that SC is finite. Finally, we observe that, by remark 2.7, for every canonical curve
C of genus four, the set SC coincides with the set of points of P3 which are contained
in six tangent lines to C.
Acknowledgment. The results of this paper are contained in my PhD-thesis. I
would like to thank my advisor C. Ciliberto for introducing me into the subject and
for providing me very useful suggestions. I have also enjoyed and benefited from
conversation with G. Pareschi and my college M. Pacini.
References
[1] A. Degtyarev: On deformations of singular plane sextics, math.AG/0511379, appearing on
Journal of Algebraic Geometry.
[2] S. Diaz and J. Harris: Ideals associated to deformations of singular plane curves, Transactions
of the American Mathematical Society, vol. 309, n. 2, 433–468 (1988).
[3] J. Harris and I. Morrison: Moduli of curves, Graduate texts in mathematics, vol. 187, Springer,
New York, 1988.
[4] C. Galati: Number of moduli of irreducible families of plane curves with nodes and cusps,
Collect. Math. 57,3 (2006), 319-346.
[5] C. Galati: Number of moduli of plane curves with nodes and cusps, PhD the-
sis, Università degli Studi di Tor Vergata, 2004-2005, available on the homepage
http://dspace.uniroma2.it/dspace/items-by-author?author=Galati
http://arxiv.org/abs/math/0511379
http://dspace.uniroma2.it/dspace/items-by-author?author=Galati
ON THE NUMBER OF MODULI OF PLANE SEXTICS WITH SIX CUSPS 9
[6] E. Sernesi: On the existence of certain families of curves, curves. Invent. Math. vol. 75 (1984),
no. 1, 25–57.
[7] O. Zariski: Dimension theoretic characterization of maximal irreducible sistems of plane nodal
curves, Amer. J. Math. vol. 104 (1982), no. 1, 209–226.
[8] O. Zariski: Algebraic surfaces, Classics in mathematics, Springer.
Dipartimento di Matematica, Università della Calabria, Arcavacata di Rende (CS)
E-mail address: galati@mat.unical.it
	1. Introduction
	2. On the number of moduli of complete irreducible families of plane sextics with six cusps
	Acknowledgment
	References

  Let S be the variety of irreducible sextics with six cusps as singularities.
Let W be one of irreducible components of W. Denoting by M_4 the space of
moduli of smooth curves of genus 4, the moduli map of W is the rational map
from W to M_4 sending the general point of W, corresponding to a plane curve D,
to the point of M_4 parametrizing the normalization curve of D. The number of
moduli of W is, by definition the dimension of the image of W with respect to
the moduli map. We know that this number is at most equal to seven. In this
paper we prove that both irreducible components of S have number of moduli
exactly equal to seven.

Introduction
Let Σnk,d ⊂ P(H0(P2,OP2(n))) := PN , with N =
n(n+3)
, be the closure, in the
Zariski’s topology, of the locally closed set of reduced and irreducible plane curves
of degree n with k cusps and d nodes. Let Σ ⊂ Σnk,d be an irreducible component
of the variety Σnk,d. Denoting by Mg the moduli space of smooth curves of genus
− k − d, it is naturally defined a rational map
ΠΣ : Σ 99K Mg,
sending the general point [Γ] ∈ Σ0 to the isomorphism class of the normalization
of the curve Γ ⊂ P2 corresponding to [Γ]. We say that ΠΣ is the moduli map of Σ
and we set
number of moduli of Σ := dim(ΠΣ(Σ)).
We say that Σ has general moduli if ΠΣ is dominant. Otherwise, we say that Σ has
special moduli or that Σ has finite number of moduli. By lemma 2.2 of [4], we know
that the dimension of the general fibre of ΠΣ is at least equal to
max(8, 8 + ρ− k),
where ρ := ρ(2, g, n) = 3n− 2g− 6 is the number of Brill-Noether of linear series of
degree n and dimension 2 on a smooth curve of genus g. It follows that, if Σ has
the expected dimension equal to 3n+ g − 1− k and g ≥ 2, then
(1) dim(ΠΣ(Σ)) ≤ min(dim(Mg), dim(Mg) + ρ− k).
Definition 1.1. We say that Σ has the expected number of moduli if equality holds
in (1).
In particular, we expect that, if ρ − k ≤ 0, then on the normalization curve C
of the curve Γ ⊂ P2 corresponding to the general point [Γ] ∈ Σ, there exists only
a finite number of linear series of degree n and dimension 2 mapping C to a plane
curve with nodes and k cusps as singularities and corresponding to a point of Σ,
(see the proof of lemma 2.2 of [4]). For a deeper discussion and a list of known
results about the moduli problem of Σnk,d we refer to sections 1 and 2 of [4] and
related references. In particular, in [4] we have found sufficient conditions in order
Key words and phrases. number of moduli, sextics with six cusps, plane curves, Zariski pairs.
http://arxiv.org/abs/0704.0622v1
2 CONCETTINA GALATI
that an irreducible component Σ of Σnk,d has finite and expected number of moduli.
If Σ verifies these conditions then ρ(2, n, g) ≤ 0. Finally in [4] we constructed
examples of families of plane curves with nodes and cusps with finite and expected
number of moduli. In this paper we consider the particular case of the variety Σ66,0
of irreducible sextics with six cusps.
It was proved by Zariski (see [8]) that Σ66,0 has at least two irreducible com-
ponents. One of them is the parameter space Σ1 of the family of plane curves of
equation
f32 (x0, x1, x2) + f
3 (x0, x1, x2) = 0,
where f2 and f3 are homogeneous polynomials of degree two and three respectively.
The general point of Σ1 corresponds to an irreducible sextic with six cusps on a
conic as singularities. Moreover, Σ66,0 contains at least one irreducible component
Σ2 whose general element corresponds to a sextic with six cusps not on a conic as
singularities and containing in its closure the variety Σ69,0 of elliptic sextics with nine
cusps. Recently, A. Degtyarev has proved that Σ1 and Σ2 are the unique irreducible
components of Σ66,0, (see [1]).
The moduli number of Σ1 and Σ2 can not be calculated by using the result of
[4]. Indeed, in this case ρ(2, 4, 6) = 4 > 0 and then the general element of every
irreducible component of Σ66,0 does not verify the hypotheses of proposition 4.1 of [4].
On the contrary, it is easy to verify that, if Γ ⊂ P2 is the plane curve corresponding
to the general element of one of the irreducible components of Σ66,0 and C is the
normalization curve of Γ, then the map µo,C is injective. But, in contrast with the
nodal case, this information is not useful in order to study the moduli problem of
Σnk,d, (see [6] and remark 4.2 of [4]). In the proposition 2.2 and corollary 2.4, we
prove that Σ2 has the expected number of moduli equal to seven. Moreover, we
show that there exists a stratification
Σ69,0 ⊂ Σ′ ⊂ Σ̃ ⊂ Σ2,
where Σ′ and Σ̃ are respectively irreducible components of Σ68,0 and Σ
7,0 with ex-
pected number of moduli. Finally, in the corollary 2.8, we prove that also Σ1 has
the expected number of moduli by using that every element of Σ1 is the branch
locus of a triple plane.
2. On the number of moduli of complete irreducible families of plane
sextics with six cusps
First of all we want to find sufficient conditions in order that, if an irreducible
component Σ of Σnk,d has the expected number of moduli, then every irreducible
component Σ′ of Σnk′,d′ , containing Σ, has the expected number of moduli. In the
corollary 4.7 of [4] we considered this problem under the hypothesis that Σ has the
expected dimension and ρ(2, n, g) ≤ 0. Now we are interested to the case ρ > 0.
We need the following local result.
= {(a, b, x, y)| y2 = x3 + ax+ b} ⊂ C2 × A2
be the versal deformation family of an ordinary cusp (see [3] for the definition and
properties of the versal deformation family of a plane singularity). We recall that
the general curve of this family is smooth. The locus ∆ of C2 of the pairs (a, b) such
that the corresponding curve is singular, has equation 27b2 = 4a3. For (a, b) ∈ ∆
and (a, b) 6= (0, 0), the corresponding curve has a node and no other singularities,
whereas (0, 0) is the only point parametrizing a cuspidal curve.
ON THE NUMBER OF MODULI OF PLANE SEXTICS WITH SIX CUSPS 3
Lemma 2.1 ([3], page 129.). Let G → C2 be a two parameter family of curves
of genus g ≥ 2, whose general fibre is stable and which is locally given by y2 =
x3 + ax + b, with (a, b) ∈ C2 and let D ⊂ C2 be a curve passing through (0, 0)
and not tangent to the axis b = 0 at (0, 0). Then the j-invariant of the elliptic
tail of the curve which corresponds to the stable limit of G(0,0), with respect to the
curve D, doesn’t depend on D. Otherwise, for every j0 ∈ C, there exists a curve
Dj0 ⊂ C2 passing through (0, 0) and tangent to the axis b = 0 at this point, such that
the elliptic tail of the stable reduction of G(0,0) with respect to Dj0 , has j-invariant
equal to j0.
Proposition 2.2. Let Σ ⊂ Σnk,d, with k < 3n, be an irreducible component of
Σnk,d. Let g be the geometric genus of the plane curve corresponding to the general
element of Σ. Suppose that g ≥ 2, ρ(2, g, n)− k ≤ 0 and Σ has the expected number
of moduli equal to 3g − 3 + ρ− k. Then, every irreducible component Σ′ of Σnk′,d′ ,
with k′ = k− 1 and d = d′ or k = k′ and d′ = d− 1, such that Σ ⊂ Σ′, has expected
number of moduli.
Proof. First we consider the case k′ = k − 1 and d = d′. Let q1, . . . , qk be the
cusps of Γ. It is well known that, since k < 3n then [Γ] ∈ Σnk−1,d. In particular,
for every fixed cusp qi of Γ there exists an irreducible analytic branch Si of Σnk−1,d
passing through the point [Γ] and whose general point corresponds to a plane curve
Γ′ of degree n with d nodes and k − 1 cusps specializing to the singular points
of Γ different from qi, as Γ
′ specializes to Γ. Moreover, it is possible to prove
that every Si is smooth at the point [Γ], see [7] or chapter 2 of [5]. Let Σ′ be
one of the irreducible components of Σnk−1,d containing Σ. Notice that the general
element of Σ′ corresponds to a curves of genus g′ = g + 1. Since ρ(2, g′, n) − k′ =
3n− 2g − 2− 6− k + 1 = ρ(2, g, n)− k − 1 < 0, in order to prove the theorem it is
enough to show that the general fibre of the moduli map
ΠΣ′ : Σ
99K Mg+1
has dimension equal to eight. Let us notice that the map ΠΣ′ is not defined at the
general element [Γ] of Σ. More precisely, let γ ⊂ Si ⊂ Σ′ be a curve passing through
[Γ] and not contained in Σ. Let C → γ be the tautological family of plane curves
parametrized by γ. Let C′ → γ be the family obtained from C → γ by normalizing
the total space. The general fibre of C′ → γ is a smooth curve of genus g+1, while
the special fibre C′0 := Γ′ is the partial normalization of Γ obtained by smoothing
all the singular points of Γ, except the marked cusp qi. If we restrict the moduli
map ΠΣ′ to γ, we get a regular map which associates to [Γ] the point corresponding
to the stable reduction of Γ with respect to the family C′ → γ, which is the union
of the normalization curve C of Γ and an elliptic curve, intersecting at the point
q ∈ C which maps to the cusp qi ∈ Γ. Now, let G ⊂ Σ′ × Mg+1 be the graph
of ΠΣ′ , let π1 : G → Σ′ and π2 : G → Mg+1 be the natural projections and let
U ⊂ Σ be the open set parametrizing curves of degree n and genus g with exactly
k cusps and d nodes as singularities. From what we observed before, if we denote
by ΠΣ′(Σ) the Zariski closure in Mg+1 of π2π−11 (U), then ΠΣ′(Σ) is contained in
the divisor ∆1 ⊂ Mg+1, whose points are isomorphism classes of reducible curves
which are union of a smooth curve of genus g and an elliptic curve, meeting at a
point. Denoting by ΠΣ : Σ → Mg the moduli map of Σ, the rational map
∆1 99K Mg
which forgets the elliptic tail, restricts to a rational dominant map
q : ΠΣ′(Σ) 99K ΠΣ(Σ).
The dimension of the general fibre of q is at most two. Since, by hypothesis, the
dimension of the fibre of the moduli map ΠΣ is eight, there exists only a finite
4 CONCETTINA GALATI
number of g2n on C, ramified at k points, which maps C to a plane curve D such
that the associated point [D] ∈ P
n(n+3)
2 belongs to Σ. In particular, the set of points
x of C such that there is a g2n with k simple ramification points, one of which at
x, is finite. So, the dimension of the general fibre of q is at most one. In order to
decide if the general fibre of q has dimension zero or one, we have to understand
how the j-invariant of the elliptic tail of the stable reduction of Γ′ with respect the
family C′ → γ, depends on γ. If C → C2 is the étale versal deformation family of
the cusp. By versality, for every fixed cusp pi of Γ, there exist étale neighborhoods
n(n+3)
2 of [Γ] in P
n(n+3)
2 , V
v→ C2 of (0, 0) in C2 and Ui of pi in the tautological
family U → P
n(n+3)
2 with a morphism f : U → V such that the family Ui → U is
the pullback, with respect to f , of the restriction to V of the versal family. By the
properties of the étale versal deformation family of a plane singularity, (see [2]), we
have that f−1((0, 0)) is an étale neighborhood of [Γ] in the (smooth) analytic branch
Σn1,0 whose general element corresponds to an irreducible plane curves with only one
cusp at a neighborhood of the cusp qi of Γ. So, dim(f
−1((0, 0))) =
n(n+3)
− 2 and
the map f is surjective. Moreover, if g is the restriction of f at u−1(Σ′), then also
g is surjective. Indeed,
g−1((0, 0)) = f−1((0, 0)) ∩ u−1(Σ′) = u−1((Σ))
and, since k < 3n, then dim(Σ) = 3n + g − 1 − k = dim(Σ′) − 2 and g is sur-
jective. By using lemma 2.1, it follows that the general fibre of the natural map
ΠΣ′(Σ) → ΠΣ(Σ) has dimension exactly equal to one. Therefore, dim(ΠΣ′ (Σ)) =
dim(ΠΣ(Σ)) + 1 = 3g − 3 + ρ(2, g, n)− k + 1 = 3(g + 1)− 3 + ρ(2, g + 1, n)− k By
using that
dim(ΠΣ′ (Σ
′)) ≥ dim(ΠΣ′(Σ)) + 1 = 3(g + 1)− 3 + ρ(2, g + 1, n)− k + 1.
and by recalling that, by lemma 2.2 of [4], it is always true that dim(ΠΣ′(Σ
′)) ≤
3(g + 1)− 3 + ρ(2, g + 1, n)− k + 1, the statement is proved in the case k′ = k − 1
and d′ = d.
Suppose, now, that k = k′ and d′ = d − 1. Also in this case Σ is not contained
in the regularity domain of ΠΣ′ . More precisely, if [Γ] ∈ Σ is general, then ΠΣ′([Γ])
consists of a finite number of points, corresponding to the isomorphism classes
of the partial normalizations of Γ obtained by smoothing all the singular points
of Γ, except for a node. Then ΠΣ′(Σ) is contained in the divisor ∆0 of Mg+1
parametrizing the isomorphism classes of the analytic curves of arithmetic genus
g + 1 with a node and no more singularities. The natural map ∆0 99K Mg sending
the general point [C′] of ∆0 to the isomorphism class of the normalization of C
restricts to a rational dominant map q : ΠΣ′(Σ) 99K ΠΣ(Σ). Since we suppose that
Σ has the expected number of moduli and ρ(2, g, n)−k ≤ 0, if C is the normalization
of the plane curve corresponding to the general element of Σ, then the set S of the
linear series of dimension 2 and degree n on C with k simple ramification points,
mapping C to a plane curve D such that the associated point [D] in the Hilbert
Scheme belongs to Σ, is finite. We deduce that also the set S′ of the pairs of points
(p1, p2) of C, such that there is a g
n ∈ S such that the associated morphism maps
p1 and p2 to the same point of the plane, is finite. So, also q
−1([C]) is finite and
dim(ΠΣ′ (Σ)) = dim(ΠΣ(Σ)). It follows that
dim(ΠΣ′ (Σ
′)) ≥ 3g− 3+ 3n− 2g− 6− k+1 = 3(g+1)− 3+ 3n− 2(g+1)− 6− k.
Remark 2.3. Notice that, the arguments used before to prove lemma 2.2 don’t
work if the dimension of the general fibre of the moduli map of Σ has dimension
bigger than eight. Indeed, in this case, the dimension of the general fibre of the map
ON THE NUMBER OF MODULI OF PLANE SEXTICS WITH SIX CUSPS 5
ΠΣ′(Σ) 99K ΠΣ(Σ) could be bigger than one if k
′ = k − 1 and d = d′, or than zero
if k′ = k and d = d′ − 1.
Corollary 2.4. There exists at least one irreducible component Σ2 of Σ
6,0 having
the expected number of moduli equal to dim(M4) − 2 and whose general element
corresponds to a sextic with six cusps not on a conic.
Remark 2.5. As we already observed in the previous section, Σ2 is the only com-
ponent of Σ66,0 parametrizing sextics with six cusps not on a conic by [1].
Proof. Let Σ69,0 be the variety of elliptic plane curves of degree six with nine cusps
and no more singularities. It is not empty and irreducible, because, by the Plücker
formulas, the family of dual curves is Σ30,0 ≃ P9, which is irreducible and not empty.
Moreover, if we compose an holomorphic map φ : C → P2 from a complex torus C
to a smooth plane cubic with the natural map φ(C) → φ(C)∗, where we denoted by
the dual curve of φ(C), we get a morphism from C to a plane sextic with nine
cusps. Therefore, the number of moduli of Σ69,0 is equal of the number of moduli of
Σ30,0, equal to one. Since 6 < 3n = 18, there is at least one irreducible component
Σ′ of Σ68,0 containing Σ
9,0. Let ΠΣ′ : Σ
99K M2 be the moduli map of Σ′ and let
G ⊂ Σ′ × M2 be its graph. If we denote by π1 : G → Σ′ and π2 : G → M2 the
natural projection, by U the open set of Σ69,0 parametrizing cubics of genus one
with nine cusps and by ΠΣ′(Σ
9,0) the Zariski closure in M2 of π2π−11 (U), then, by
arguing as in the first part of the proof of the lemma 2.2, we have a dominant map
ΠΣ′(Σ
9,0) 99K M1, whose general fibre has dimension one. We conclude that
dim(πΣ′ (Σ
′)) ≥ dim(πΣ′(Σ69,0)) + 1 = 3
and so, the moduli map of Σ′ is dominant, as one expects, because ρ(2, 2, 6)− 8 =
18− 4− 6− 8 = 0. Let D be the plane sextic corresponding to the general point of
Σ′. By Bezout theorem, the height cusps P1, . . . , P8 of D don’t belong to a conic
and, however we choose five cusps of D, no four of them lie on a line. Then, let
C2 be the unique conic containing P1, . . . , P5. There exists at least a cusp, say P6,
which does not belong to C2. Since 8 < 3n = 18, there exists a family of plane
sextics D → ∆, whose special fibre is D and whose general fibre has a cusp at
a neighborhood of every cusp of D different from P7 and no further singularities.
By lemma 2.2, the curve ∆ is contained in an irreducible component of Σ67,0 with
expected number of moduli. By repeating the same argument for the general fibre
of the family D → ∆ we get an irreducible component Σ2 of Σ66,0 with the expected
number of moduli and whose general element parametrizes a sextic with six cusps
not on a conic. �
Now we consider the irreducible component Σ1 of Σ
6,0 parametrizing plane curves
of equation f23 (x0, x1, x2) + f
2 (x0, x1, x2) = 0, where f2 is an homogeneous poly-
nomial of degree two and f3 is an homogeneous polynomial of degree three. The
general element of Σ1 corresponds to an irreducible plane curve of degree six with
six cusps on a conic. We want to show that Σ1 has the expected number of moduli
equal to 12− 3 + ρ(2, 4, 6)− 6 = 7 = dim(M4)− 2. Equivalently, we want to show
that the general fibre of the moduli map
Σ1 99K M4
has dimension equal to eight.
Lemma 2.6. Let Γ2 : f2(x0, x1, x2) = 0 and Γ3 : f3(x0, x1, x2) = 0 be a smooth
conic and a smooth cubic intersecting transversally. Then, the plane curve
Γ : f23 (x0, x1, x2)− f32 (x0, x1, x2) = 0
is an irreducible sextic of genus four with six cusps at the intersection points of Γ2
and Γ3 as singularities. The curve Γ is projection of a canonical curve C ⊂ P3 from
6 CONCETTINA GALATI
a point p ∈ P3 which is contained in six tangent lines to C. Moreover, for every
point q ∈ P3 −C such that the projection plane curve πq(C) of C from q is a sextic
with six cusps on a conic of equation g23(x0, x1, x2) − g32(x0, x1, x2) = 0, where g3
and g2 are two homogeneous polynomials of degree three and two respectively, there
exists a cubic surface S3 ∈ |IC|P3(3)|, containing C, such that the plane curve πq(C)
is the branch locus of the projection πq : S3 → P2.
Remark 2.7. Notice that, by [1], every irreducible sextic with six cusps on a conic
as singularities has equation given by (f2(x0, x1, x2))
3 + (f3(x0, x1, x2))
2 = 0, with
f2 and f3 homogeneous polynomials of degree two and three. In order words, all the
sextics with six cusps on a conic as singularities are parametrized by points of Σ1.
An other proof of this result as been provided to us by G. Pareschi.
Proof of lemma 2.6. Let f(x0, x1, x2) = f
3 (x0, x1, x2) − f32 (x0, x1, x2) = 0 be the
equation of Γ. From the relation f3(x) = ±f2(x)
f2(x), we deduce that
(x) =
±2∂f2
f2(x) and hence
(x) = 2
(x)f3(x)− 3f2(x)2
(x) = −f2(x)2
By using that the conic Γ2 : f2 = 0 is smooth, it follows that, if a point x ∈ Γ is
singular, then x ∈ Γ2 and hence x ∈ Γ3∩Γ2. On the other hand, always from (2), if
x ∈ Γ2∩Γ3, then x is a singular point of Γ. Hence, the singular locus of Γ coincides
with Γ3 ∩ Γ2. Let x be a singular point of Γ. If
p1(x, y) + terms of degree two = 0
q1(x, y) + terms of degree ≥ two = 0
are respectively affine equations of Γ2 and Γ3 at x, then, the affine equation of Γ at
x is given by
q1(x, y)
2 − p1(x, y)3 + terms of degree ≥ four = 0.
Since Γ2 and Γ3 intersect transversally, we have that q1(x, y) does not divide p1(x, y)
and hence Γ has an ordinary cusp at x. Let now φ : C → Γ be the normalization of
Γ. We recall that the cubics passing through the six cusps of Γ cut out on C the
complete canonical series |ωC |. Since the cusps of Γ is contained in the conic Γ2 ⊂ P2
of equation f2 = 0, the lines of P
2 cut out on C a subseries g ⊂ |ωC | of dimension
two of the canonical series. Moreover, if we still denote by C a canonical model of
C in P3, then the linear series g is cut out on C in P3 from a two dimensional family
of hyperplanes passing through a point p ∈ P3−C. If we project C from p we get a
plane curve projectively equivalent to Γ. Since Γ has six cusps as singularities, we
deduce that there are six tangent lines to C passing through p. To see that Γ is the
branch locus of a triple plane, let S3 ⊂ P3 be the cubic surface of equation
F3(x0, . . . , x3) = x
3 − 3f2(x0, x1, x2)x3 + 2f3(x0, x1, x2) = 0.
If p = [0, 0, 0, 1], then, by using Implicit Function Theorem, the ramification locus
of the projection πp : S3 → P2, is given by the intersection of S3 with the quadric
S2 of equation
= x23 − f2(x0, x1, x2) = 0. Now, if x = [x0, x1, x2] ∈ S3 ∩ S2,
then x3 = ±
f2(x0, x1, x2). By substituting in the equation of S3, we find that the
branch locus of the projection πp : S3 → P2 coincides with the plane curve Γ. From
what we proved before, it follows that the ramification locus of the projection map
πp : S3 → P2 is the normalization curve C of Γ. Finally, if q ∈ P3 − C is an other
point such that the plane projection πq(C) is an irreducible sextic with six cusps on a
conic parametrized by a point xq ∈ Σ1 ⊂ P27, then, up to projective motion, we may
always assume that q = [0 : 0 : 0 : 1] and hence, if g23(x0, x1, x2)− g32(x0, x1, x2) = 0
ON THE NUMBER OF MODULI OF PLANE SEXTICS WITH SIX CUSPS 7
is the equation of the plane curve πq(C), then C is the locus of ramification of the
projection from q to the plane of the cubic surface of equation
x33 − 3g2(x0, x1, x2)x3 + 2g3(x0, x1, x2) = 0.
Corollary 2.8. The irreducible component Σ1 of Σ
6,0 parametrizing plane curves of
equation f23 (x0, x1, x2)+ f
2 (x0, x1, x2) = 0, where f2 is an homogeneous polynomial
of degree two and f3 is an homogeneous polynomial of degree three, has the expected
number of moduli equal to 7 = dim(M4) + ρ(2, 4, 6)− 6.
Proof. Let [Γ] ⊂ P2 be a plane sextic of equation f23 (x0, x1, x2) − f32 (x0, x1, x2) =
0, where the conic f2 = 0 and the cubic f3 = 0 are smooth and they intersect
transversally. Let C ⊂ P3 be the normalization curve of Γ and let SC be the set of
points v = [v0 : · · · : v3] ∈ P3 such that there exists a cubic surface S3 ∈ |IC|P3(3)|,
containing C, such that the curve C is the ramification locus of the projection πv :
S3 → P2. By the former lemma, in order to prove that Σ1 has the expected number
of moduli, it is enough to find a point [Γ] of Σ1 corresponding to an irreducible
plane sextic Γ ⊂ P2 with six cusps of a conic such that the set SC is finite. Let Γ2
be the smooth conic of equation f2(x0, x1, x2) = x
1 − x22 = 0 and let Γ3 be the
smooth cubic of equation f3(x0, x1, x2) = x
0+x0x
2−x21x2 = 0. If a1, a2 and a3 are
the three different solutions of the polynomial x3 + x2 + x− 1 = 0, then Γ2 and Γ3
intersect transversally at the points [ai,
ai, 1], [ai,−
ai, 1], with i = 1, 2, 3. By
the former lemma, the plane sextic Γ of equation f32 − f23 = 0 is irreducible and
it has six cusps at the intersection points of Γ2 and Γ3 as singularities. Moreover,
the normalization curve C of Γ is the canonical curve of genus 4 in P3 which is
intersection of the cubic surface S3 ⊂ P3 of equation
F3(x0, x1, x2, x3) = x
3 + (x
0 + x
1 − x22)x3 + x30 + x0x22 − x21x2 = 0
and the quadric S2 of equation
= 3x23 + x
0 + x
1 − x22 = 0.
We want to show that SC is finite. To see this we observe that, since
h0(P3, IC|P3(2)) = 1 and h0(P3, IC|P3(3)) = 5,
the equation of every cubic surface containing C and which is not the union of S2
and an hyperplane is given by
G(x0, . . . , x3;β0, . . . , β3) = F3(x0, x1, x2, x3) +
∂F3(x0, x1, x2, x3)
with βj ∈ C, for i = 0, . . . , 3. Now, a point [v] = [v0, . . . , v3] ∈ SC if and only if
there exist β0, . . . , β3 such that C is contained in the intersection of
G(x0, . . . , x3;β0, . . . , β3) = 0 and
∂G(x0, . . . , x3;β0, . . . , β3)
Still using that h0(P3, IC|P3(2)) = 1, a point [v] ∈ P3 belongs to SC if and only if
∂G(x0, . . . , x3;β0, . . . , β3)
∂F3(x0, . . . , x3)
for some λ ∈ R− 0, or, equivalently,
βjxj)
∂x3∂xi
= (λ−
viβi − v3)
8 CONCETTINA GALATI
The previous equality of polynomials is equivalent to the following bilinear system
of ten equations in the variables v0, . . . , v3 and β0, . . . , β3
(1 + β3)v0 + 3β0v3 = 0 (x0x3)
(1 + β3)v1 + 3β1v3 = 0 (x1x3)
(1 + β3)v2 − 3β2v3 = 0 (x2x3)
β1v0 + β0v1 = 0 (x0x1)
β2v0 + (1− β0)v2 = 0 (x0x2)
(1 − β2)v1 + β1v2 = 0 (x1x2)
2β1v1 − v2 = λ−
j=0 βjvj − v3 (x21)
−v0 + 2β2v2 = λ−
j=0 βjvj − v3 (x22)
(3 + 2β0)v0 = λ−
j=0 βjvj − v3 (x20)
2β3v3 = λ−
j=0 βjvj − v3 (x23)
The points of SC are the solutions v of the previous linear system, as a linear
system whose coefficients depend on β0, . . . , β3. It is easy to prove that it has only
a solution equal to (v0, v1, v2, v3) = (0, 0, 0, λ) if β0 = β1 = β2 = β3 = 0 and it has
not solutions otherwise, (see [5], page 98). By the previous lemma, we conclude
that the point [0 : 0 : 0 : 1] ∈ P3 is the only point which belongs to six tangent lines
to the canonical curve C ⊂ P3 which is intersection of the cubic surface of equation
F3(x0, x1, x2, x3) = x
3 + (x
0 + x
1 − x22)x3 + x30 + x0x22 − x21x2 = 0
and the quadric of equation
= 3x23 + x
0 + x
1 − x22 = 0.
It follows that, on the normalization curve D of the plane curve Γ′ corresponding
to the general point of Σ1 ⊂ Σ66,0 there exists only a finite number of linear series
of dimension two with six ramification points. �
Remark 2.9. By using the notation introduced in the proof of corollary 2.8, we
observe that in this corollary we have proved that if C is a general canonical curve of
genus four such that the set SC is not empty, then SC is finite. Actually, C. Ciliberto
pointed out to our attention that it is possible to show, with a very simple argument,
that for every canonical curve C of genus four such that SC is not empty, we have
that SC is finite. Finally, we observe that, by remark 2.7, for every canonical curve
C of genus four, the set SC coincides with the set of points of P3 which are contained
in six tangent lines to C.
Acknowledgment. The results of this paper are contained in my PhD-thesis. I
would like to thank my advisor C. Ciliberto for introducing me into the subject and
for providing me very useful suggestions. I have also enjoyed and benefited from
conversation with G. Pareschi and my college M. Pacini.
References
[1] A. Degtyarev: On deformations of singular plane sextics, math.AG/0511379, appearing on
Journal of Algebraic Geometry.
[2] S. Diaz and J. Harris: Ideals associated to deformations of singular plane curves, Transactions
of the American Mathematical Society, vol. 309, n. 2, 433–468 (1988).
[3] J. Harris and I. Morrison: Moduli of curves, Graduate texts in mathematics, vol. 187, Springer,
New York, 1988.
[4] C. Galati: Number of moduli of irreducible families of plane curves with nodes and cusps,
Collect. Math. 57,3 (2006), 319-346.
[5] C. Galati: Number of moduli of plane curves with nodes and cusps, PhD the-
sis, Università degli Studi di Tor Vergata, 2004-2005, available on the homepage
http://dspace.uniroma2.it/dspace/items-by-author?author=Galati
http://arxiv.org/abs/math/0511379
http://dspace.uniroma2.it/dspace/items-by-author?author=Galati
ON THE NUMBER OF MODULI OF PLANE SEXTICS WITH SIX CUSPS 9
[6] E. Sernesi: On the existence of certain families of curves, curves. Invent. Math. vol. 75 (1984),
no. 1, 25–57.
[7] O. Zariski: Dimension theoretic characterization of maximal irreducible sistems of plane nodal
curves, Amer. J. Math. vol. 104 (1982), no. 1, 209–226.
[8] O. Zariski: Algebraic surfaces, Classics in mathematics, Springer.
Dipartimento di Matematica, Università della Calabria, Arcavacata di Rende (CS)
E-mail address: galati@mat.unical.it
	1. Introduction
	2. On the number of moduli of complete irreducible families of plane sextics with six cusps
	Acknowledgment
	References
