NUMBER OF MODULI OF IRREDUCIBLE FAMILIES OF
PLANE CURVES WITH NODES AND CUSPS.
CONCETTINA GALATI
Abstract. Let Σn
n(n+3)
2 be the family of irreducible plane curves of
degree n with d nodes and k cusps as singularities. Let Σ ⊂ Σn
be an
irreducible component. We consider the natural rational map
ΠΣ : Σ 99K Mg,
from Σ to the moduli space of curves of genus g =
−d−k. We define the
number of moduli of Σ as the dimension dim(ΠΣ(Σ)). If Σ has the expected
dimension equal to 3n+ g − 1− k, then
(1) dim(ΠΣ(Σ)) ≤ min(dim(Mg), dim(Mg) + ρ− k),
where ρ := ρ(2, g, n) = 3n − 2g − 6 is the Brill-Neother number of the linear
series of degree n and dimension 2 on a smooth curve of genus g. We say that
Σ has the expected number of moduli if the equality holds in (1). In this paper
we construct examples of families of irreducible plane curves with nodes and
cusps as singularities having expected number of moduli and with non-positive
Brill-Noether number.
1. Introduction
In this paper we compute the number of moduli of certain families of irreducible
plane curves with nodes and cusps as singularities. Let Σnk,d ⊂ P(H
0(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 Σ
k,d. We denote by
Σ0 the open set of Σ of points [Γ] ∈ Σ such that Σ is smooth at [Γ] and such
that [Γ] corresponds to a reduced and irreducible plane curve of degree n with d
nodes, k cusps and no further singularities. Since the tautological family S0 → Σ0,
parametrized by Σ0, is an equigeneric family of curves, by normalizing the total
space, we get a family
// S0
// P2 × Σ0
of smooth curves of genus g =
−k−d. Because of the functorial properties of
the moduli space Mg of smooth curves of genus g, we get a regular map Σ0 → Mg,
sending every point [Γ] ∈ Σ0 to the isomorphism class of the normalization of the
plane curve Γ corresponding to the point [Γ]. This map extends to a rational map
ΠΣ : Σ 99K Mg.
We say that ΠΣ is the moduli map of Σ and we set
number of moduli of Σ := dim(ΠΣ(Σ)).
Date: 06 September 2005.
1991 Mathematics Subject Classification. 14H15; 14H10; 14B05.
Key words and phrases. families of plane curves, number of moduli, nodes and cusps.
http://arxiv.org/abs/0704.0618v1
2 CONCETTINA GALATI
Notice that, when Σnk,d is reducible, two different irreducible components of Σ
can have different number of moduli. We say that Σ has general moduli if ΠΣ is
dominant. Otherwise, we say that Σ has special moduli.
Definition 1.1. When Σ has the expected dimension equal to 3n+ g − 1 − k and
g ≥ 2, we say that Σ has the expected number of moduli if
dim(ΠΣ(Σ)) = min(dim(Mg), dim(Mg) + ρ− k),
where ρ := ρ(2, g, n) = 3n−2g−6 is the number of Brill-Noether of the linear series
of degree n and dimension 2 on a smooth curve of genus g.
As we shall see in the next section, when g ≥ 2 and when Σ has the expected
dimension equal to 3n+ g − 1 − k, the number of moduli of Σ is at most equal to
the expected one. This happens in particular if k < 3n. If k ≥ 3n, in general we
have not an upper-bound for the dimension of Σ and we cannot provide an upper
bound for the number of moduli of Σ, (see lemma 2.2 and remark 2.3). Moreover,
by classical Brill-Neother theory when ρ is positive and by a well know result of
Sernesi when ρ ≤ 0 (see [18]), we have that Σn0,d, (which is irreducible by [8]), has
the expected number of moduli for every d ≤
. When k > 0 there are known
results giving sufficient conditions for the existence of irreducible components Σ of
Σnk,d with general moduli, (see propositions 2.5 and 2.6 and corollary 2.7). In this
article we construct examples of families of irreducible plane curves with nodes and
cusps with finite and expected number of moduli. A large part of this paper is
obtained working out the main ideas and techniques that Sernesi uses in [18].
In section 2.1 we introduce the varieties Σnk,d and we recall their main properties.
In section 2.2 we discuss on definition 1.1 and we summarize known results on the
number of moduli of families of irreducible plane curves with nodes and cusps. In
theorem 3.5 we prove the existence of plane curves with nodes and cusps as singular-
ities whose singular points are in sufficiently general position to impose independent
linear conditions to a linear system of plane curves of a certain degree. This result
is related to the moduli problem by lemma 3.2, remark 3.4 and proposition 4.1,
where we find sufficient conditions in order that an irreducible component Σ ⊂ Σnk,d
has the expected number of moduli. If Σ verifies the hypotheses of proposition 4.1,
then the Brill-Neother number ρ is not positive and Σ has finite number of moduli.
Moreover, by lemma 4.6 and corollary 4.7, for every k′ ≤ k and d′ ≤ d+k−k′, there
is at least an irreducible component Σ′ ⊂ Σnk′,d′ , such that Σ ⊂ Σ
′ and the general
element [D] ∈ Σ′ corresponds to a plane curve D verifying hypotheses of propo-
sition 4.1 and so having the expected number of moduli. Finally, the main result
of this paper is contained in theorem 4.9, where, by using induction on the degree
n and on the genus g of the general curve of the family, we construct examples of
families of irreducible plane curves with nodes and cusps verifying the hypotheses
of proposition 4.1. In particular, we prove that, if k ≤ 6 and ρ ≤ 0, then Σnk,d has
at least an irreducible component which is not empty and which has the expected
number of moduli. This result may be improved and examples of families of curves
showing that the condition k ≤ 6 is not sharp are given in remark 4.10. Notice
that the previous theorem provides only examples of families of plane curves with
nodes and cusps with expected number of moduli, when ρ is not positive. When the
number of cusps k is very small, we expect it is possible to prove the existence of
irreducible components of Σnk,d with expected number of moduli, for every value of
ρ. For example, from a result of Eisenbud and Harris, it follows that Σn1,d, (which is
irreducible by [16]), has general moduli if ρ ≥ 2, (see corollary 2.7). In theorem 4.11,
by using induction on n we find that Σn1,d has general moduli also when ρ = 1. By
recalling that, by theorem 4.9, Σn1,d has expected number of moduli when ρ ≤ 0, we
conclude that Σn1,d has the expected number of moduli for every ρ or, equivalently,
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 3
for every d ≤
− 1. We still don’t know examples of irreducible components of
Σnk,d having number of moduli smaller that the expected.
2. Preliminaries
2.1. On Severi-Enriques varieties. We shall denote by PN = P
n(n+3)
2 the Hilbert
scheme of plane curves of degree n, by [Γ] ∈ PN the point parametrizing a plane
curve Γ ⊂ P2 and by Σnk,d ⊂ P
N the closure, in the Zariski topology, of the locally
closed set parametrizing reduced and irreducible plane curves of degree n with d
nodes and k cusps as singularities. These varieties have been introduced at the
beginning of the last century by Severi and Enriques. In particular, the case k = 0
has been studied first by Severi and for this reason the varieties Σn0,d are usually
called Severi varieties, while for k > 0 the varieties Σnk,d are called Severi-Enriques
varieties. We recall that every irreducible component Σ of Σnk,d has dimension at
least equal to
N − d− 2k = 3n+ g − 1− k,
where g =
−k−d. When the equality holds we say that Σ has expected dimen-
sion. Moreover, it is well known that if k < 3n then every irreducible component
Σ of Σnk,d has expected dimension, (see for example [23] or [25]). On the contrary,
when k ≥ 3n, there exist examples of irreducible components of Σnk,d having di-
mension greater than the expected, (see [25]). Moreover, we recall that Σn0,d is not
empty for every d ≤
and it contains in its closure all points parameterizing
irreducible plane curves of degree n and genus g =
− d, (see [24], [25] and
[1]). Often, we shall denote Σn0,d by Vn,g. While the proof of the existence of Vn,g
is quite elementary and it is due to Severi, the irreducibility of Vn,g remained an
open problem for a long time and it has been proved by Harris only in 1986. Later,
by using the same techniques of Harris, Kang has proved the irreducibility of Σnk,d
with k ≤ 3, see [8] and [16]. However, in general, Σnk,d is reducible and there exist
values of n, d and k such that Σnk,d is empty, (see [25], [12], [20], [11] or chapter 2
of [7] and related references). Finally, we recall that, if Σ ⊂ Σnk,d is a non-empty
irreducible component of the expected dimension equal to 3n + g − 1 − k, then,
for every k′ ≤ k and d′ ≤ d + k − k′, there exists a non-empty irreducible compo-
nent Σ′ ⊂ Σnk′,d′ such that Σ ⊂ Σ
′. This happens in particular if k < 3n. More
precisely, it is true that, if Γ ⊂ P2 is a reduced (possibly reducible) plane curve of
degree n with k < 3n cusps at points q1, . . . , qk, nodes at points p1, . . . , pd and no
further singularities, then, chosen arbitrarily k1 cusps, say q1, . . . , qk1 among the
k cusps of Γ, k2 cusps qk1+1, . . . , qk2 among qk1+1, . . . , qk and d1 nodes p1, . . . , pd1
among the nodes of Γ, there exists a family of reduced plane curves D → B ⊂ PN
of degree n, whose special fibre is D0 = Γ and whose general fibre Dt = D has a
node in a neighborhood of every marked node of Γ, a cusp in a neighborhood of
each point q1, . . . , qk1 , a node in a neighborhood of each point qk1+1, . . . , qk2 and
no further singularities, (see [25], corollary 6.3 of [11] or lemma 3.17 of chapter 2
of [7]). To save space, we shall say that the family D → B is obtained from Γ by
preserving the singularities q1, . . . , qk1 and p1, . . . , pd1 , by deforming in a node each
cusp qk1+1, . . . , qk2 and by smoothing the other singularities.
2.2. Known results on the number of moduli of Σnk,d. In order to explain
the definition 1.1, we need to recall some basics of Brill-Noether theory. Given a
smooth curve C of genus g, the set G2n(C) of linear series g
n on C of dimension 2
and degree n, is a projective variety which verifies the following properties:
(1) G2n(C) is not empty of dimension at least ρ, if ρ(2, n, g) = 3n− 2g − 6 ≥ 0,
(see theorem V.1.1 and proposition IV.4.1 of [4]).
4 CONCETTINA GALATI
(2) Let g2n be a given linear series, letH ∈ g
n be a divisor and letW ⊂ H
0(C,H)
be the three dimensional vector space corresponding to g2n. Denoting by
ωC = OC(KC) the canonical sheaf of C and by
µo,C :W ⊗H
0(C, ωC(−H)) → H
0(C, ωC)
the natural multiplication map, also called the Brill-Noether map of the
pair (C,W ), we have that the dimension of the tangent space to G2n(C) at
the point [g2n],corresponding to g
n, is equal to
dim(T[g2
n(C)) = ρ+ dim(ker(µ0,C)),
(see [2] or proposition IV.4.1 of [4] for a proof).
(3) Moreover, if C is a curve with general moduli (i.e. if [C] varies in an open
set of Mg), the variety G
n(C) is empty if ρ < 0, it consists of a finite
number of points if ρ = 0 and it is reduced, irreducible, smooth and not
empty variety of dimension exactly ρ, when ρ ≥ 1, (see theorem V.1.5 and
theorem V.1.6 of [4]). In the latter case, the general g2n on C defines a local
embedding on C and it maps C to P2 as a nodal curve, (see theorem 3.1 of
[1] or lemma 3.43 of [9]).
From (3), we deduce that, the Severi variety Σn0,d = Vn,g of irreducible plane curves
of genus g =
− d, has general moduli when ρ ≥ 0 and it has special moduli
when ρ < 0. When ρ < 0, and then g ≥ 3, by definition 1.1, we expect that the
image of Vn,g into Mg has codimension exactly −ρ. Equivalently, recalling that, in
this case,
dim(Vn,g) = 3n+ g − 1 = 3g − 3 + ρ+ 8 = dim(Mg) + ρ+ dim(Aut(P
we expect that on the smooth curve C, obtained by normalizing the plane curve
corresponding to the general element of Vn,g, there is only a finite number of g
mapping C to the plane as a nodal curve. This is a well known result proved by
Sernesi in [18].
Theorem 2.1 (Sernesi, [18]). The Severi variety Vn,g = Σ
0,d of irreducible plane
curves of degree n and genus g =
− d has number of moduli equal to
min(dim(Mg), dim(Mg) + ρ).
What can we say about the number of moduli of an irreducible component Σ of
Σnk,d, when k > 0? In this case we need to distinguish the two cases k < 3n and
k ≥ 3n. In the first case we have the following result.
Lemma 2.2. For every not empty irreducible component Σ of Σnk,d, with k < 3n
and g =
− k − d ≥ 2, the number of moduli of Σ is at most equal to
min(dim(Mg), dim(Mg) + ρ− k),
where ρ = 3n − 2g − 6 is the Brill-Neother number of moduli of linear series of
dimension 2 and degree n on a smooth curve of genus g.
Proof. We recall that an ordinary cusp P of a plane curve Γ corresponds to a simple
ramification point p of the normalization map φ : C → Γ, i.e. to a simple zero of
the differential map dφ. If we denote by G2n,k(C) ⊂ G
n(C) the set of g
n on C
defining a birational morphism with k simple ramification points, then G2n,k(C) is
a locally closed subset of G2n(C) and every irreducible component G of G
n,k(C) has
dimension at least equal to ρ− k, if it is not empty. In particular, if F 2n,k(C) is the
variety whose points correspond to the pairs ([g2n], {s0, s1, s2}) where [g
n] ∈ G
n,k(C)
and {s0, s1, s2} is a frame of the three dimensional space associated to the linear
series g2n, then every irreducible component of F
n,k(C) has dimension at least equal
min(8, ρ− k + 8).
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 5
Now, let Σ be one of the irreducible components of Σnk,d and let [Γ] be a general
point of Σ. Then, if Γ ⊂ P2 is the corresponding plane curve and φ : C → Γ is the
normalization map, then the fibre over the point [C] ∈ Mg of the moduli map
ΠΣ : Σ 99K Mg
consists of an open set in one or more irreducible components of F 2n,k(C). In partic-
ular, every irreducible component of the general fibre of ΠΣ has dimension at least
equal to min(8, ρ− k + 8). Moreover, if k < 3n then Σ has the expected dimension
equal to N−d−2k = 3n+g−1−k, (see [25] or [23]). Finally, if g =
−k−d ≥ 2,
dim(Σ) = 3n+ g − 1− k = 3g − 3 + ρ− k + 8.
This proves the statement. �
Remark 2.3. The proof of the previous lemma still holds if k ≥ 3n but Σ has the
expected dimension. However in general, when k ≥ 3n, we don’t have a bound for
dim(ΠΣ(Σ)). Indeed, in this case the dimension of the general fibre of the moduli
map of Σ is still at least equal to ρ− k + 8, but Σ may have dimension larger than
3n+ g − 1 − k. Anyhow, by the following proposition, every not empty irreducible
component of Σnk,d has special moduli if k ≥ 3n.
Proposition 2.4 (Arbarello-Cornalba, [1]). Let C be a general curve of genus
g ≥ 2 and let φ : C → P2 be a birational morphism, then the degree of the zero
divisor of the differential map of φ is smaller than ρ. In particular, every irreducible
component of Σnk,d has special moduli if ρ = 3n− 2g − 6 < k.
A sufficient condition for the existence of irreducible families of plane curves with
nodes and cusps with general moduli is given by the following result.
Proposition 2.5 (Kang, [15]). Σnk,d is irreducible, not empty and with general
moduli if n > 2g − 1 + 2k, where g =
− d− k.
Actually, in [15], Kang proves that if n > 2g−1+2k, then Σnk,d is not empty and
irreducible. But from his proof it follows that, under the hypothesis of proposition
2.5, Σnk,d has general moduli because the general element of Σ
k,d corresponds to a
curve which is a projection of an arbitrary smooth curve C of genus g in Pn−g, from
a general (n − 3)-plane intersecting the tangent variety of C in k different points.
Another result which may be used to find examples of families of plane curves with
nodes and cusps having general moduli is the following. Let grn be a linear series on
C associated to a (r+1)-spaceW ⊂ H0(C,L), where L is an invertible sheaf on C,
and let {s0, . . . , sr} be a basis of W , then the ramification sequence of the g
n at p
is the sequence b = (b0, . . . , br) with bi = ordpsi − i. Choosing another basis of W ,
the ramification sequence of grn at p doesn’t change. We say that the ramification
sequence of the grn at p is at least equal to b = (b0, . . . , br) if bi ≤ ordpsi − i, for
every i, and we write (ordps0, . . . , ordpsr − r) ≥ (b0, . . . , br).
Proposition 2.6 (Proposition 1.2 of [10]). Let C be a general curve of genus g,
let p be a general point on C and let b = (b0, . . . , br) be any ramification sequence.
There exists a grn on C having ramification at least b at p if and only if
(bi + g − n+ r)+ ≤ g,
where (−)+ := max(−, 0).
From proposition 2.6, we easily deduce the following result.
Corollary 2.7. Suppose that k ≤ 3 and ρ = 3n− 2g − 6 ≥ 2k. Then Σnk,d is not
empty, irreducible and it has general moduli.
6 CONCETTINA GALATI
Proof. By [16], the variety Σnk,d is irreducible for every k ≤ 3 and d ≤
Moreover, by using classical arguments, one can prove that Σnk,d is not empty if
k ≤ 4 and d ≤
− 4, (see, for example, corollary 3.18 of chapter two of [7]).
Finally, by theorem 1.1 of [21], by using the terminology of proposition 2.6, under
the hypothesis k ≤ 3n − 4, in particular if k ≤ 3, the variety Σnk,d contains every
point of PN corresponding to a plane curve Γ of genus g =
− k − d such that
the normalization morphism of Γ has at least a ramification point with ramification
sequence (b0, b1, b2) ≥ (0, k, k). Then, by proposition 2.6, if ρ ≥ 2k and k ≤ 3, the
moduli map of Σnk,d is surjective. �
3. On the existence of certain families of plane curves with nodes
and cusps in sufficiently general position
As we already observed, we don’t have a complete answer for the existence prob-
lem of Σnk,d. In this section we are interested in a little more specific existence
problem. We shall prove the existence of plane curves with nodes and cusps as
singularities whose singular points are in sufficiently general position to impose
independent linear conditions to a linear system of plane curves of a certain degree.
Definition 3.1. A projective curve C ⊂ Pr is said to be geometrically t-normal if
the linear series cut out on the normalization curve C̃ of C by the pull-back to C̃
of the linear system of hypersurfaces of Pr of degree t is complete.
From a geometric point of view, a projective curve C ⊂ Pr is geometrically t-
normal if and only if the image curve νt,r(C) of C by the Veronese embedding
νt,r : P
r → P(
t ) of degree t, is not a projection of a non-degenerate curve living
in a higher dimensional projective space. We shall say that a curve is geometrically
linearly normal (g.l.n. for short) if it is geometrically 1-normal. Every such a curve
C is not a projection of a curve lying in a projective space of larger dimension.
The following result is proved under more general hypotheses in [5], theorem 2.1.
Lemma 3.2. Let Γ ⊂ P2 be an irreducible and reduced plane curve of degree n and
genus g with at most nodes and cusps as singularities. Let t be an integer such that
n− 3 − t < 0, then Γ is geometrically t-normal if and only if it is smooth. On the
contrary, if n − 3 − t ≥ 0, the plane curve Γ is geometrically t-normal if and only
if its singular points impose independent linear conditions to plane curves of degree
n− 3− t.
We recall the following classical definition.
Definition 3.3. Let Γ ⊂ P2 be a plane curve of degree n with d nodes at p1, . . . , pd
and k cusps at q1, . . . , qk as singularities. Let φ : C → Γ be the normalization of
Γ. The adjoint divisor ∆ of φ is the divisor on C defined by ∆ =
i=1 φ
−1(pi) +
j=1 2φ
−1(qj).
Proof of lemma 3.2. Let Γ be a plane curve as in the statement of the lemma. Then,
Γ is geometrically t-normal if and only if, by definition,
h0(C,OC(t)) = h
0(P2,OP2(t))− h
0(P2, IΓ(t))
where IΓ is the ideal sheaf of Γ in P
2 and OC(t) := OC(tφ
∗(H)), where H is the
general line of P2. By Riemann-Roch theorem, Γ is geometrically t-normal if and
only if
(2) h0(C, ωC(−t))) = −nt+ g − 1 +
(t+ 1)(t+ 2)
− h0(P2, IΓ(t)),
where g is the geometric genus of Γ and ωC is the canonical sheaf of C. On the other
hand, it is well known that H0(C, ωC(−t)) = H
0(C,OC(n− 3− t)(−∆)), where ∆
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 7
is the adjoint divisor of φ, (see definition 3.3 and [4], appendix A). If n− 3− t < 0
then h0(C,OC(n− 3− t)) = 0 and Γ is geometrically t-normal if and only if
h0(P2,OP2(t)) − h
0(P2, IΓ(t)) = nt−
n2 − 3n
where δ =
− g = deg(∆)/2. This equality is verified if and only if δ = 0, i.e.
Γ is smooth. If n− 3 ≥ t, h0(P2, IΓ(t)) = 0 and (2) is verified if and only if
h0(C,OC(n− 3− t)(−∆)) = h
0(P2,OP2(n− 3− t))− δ.
On the other hand, if ψ : S → P2 is the blowing-up of the plane at the singular
locus of Γ, denoting by
Ei the pullback of the singular locus of Γ with respect
to ψ and by OS(r) the sheaf OS(rψ
∗(H)), we have that
h0(C,OC(n−3−t)(−∆)) = h
0(S,OS(n−3−t)(−
Ei)) = h
0(P2,OP2(n−3−t)⊗A)
where A is the ideal sheaf of singular points of Γ. �
Remark 3.4. Notice that, if an irreducible and reduced plane curve Γ of degree n
with only nodes and cusps as singularities is geometrically t-normal, with t ≤ n−3,
then it is geometrically r-normal for every r ≤ t. Indeed, if a set of points imposes
independent linear conditions to a linear system S, then it imposes independent
linear conditions to every linear system S′ containing S.
Theorem 3.5. Let Σnk,d be the variety of irreducible and reduced plane curves of
degree n with d nodes and k cusps. Suppose that d, k, n and t are such that
d+ k ≤
n2 − (3 + 2t)n+ 2 + t2 + 3t
= h0(OP2(n− t− 3))(3)
t ≤ n− 3 if k = 0,(4)
k ≤ 6 if t = 1, 2 and(5)
k ≤ 6 + [
], if t = 3,(6)
where [−] is the integer part of −. Then the variety Σnk,d is not empty and there ex-
ists at least an irreducible component W ⊂ Σnk,d whose general element corresponds
to a geometrically t-normal plane curve.
Remark 3.6. As we shall see in the next section, (see proposition 4.1), the geomet-
ric linear normality of the plane curve corresponding to the general element of an
irreducible component Σ of Σnk,d, is related with the number of moduli of Σ. Another
motivation for the previous theorem has been the family of irreducible plane sextics
with six cusps. By [25], we know that Σ66,0 contains at least two irreducible compo-
nents Σ1 and Σ2. The general point of Σ1 corresponds to a sextic with six cusps on
a conic, whereas the general element of Σ2 corresponds to a sextic with six cusps
not on a conic. Note that, by the previous lemma the general element of Σ2 param-
eterizes a geometric linearly normal sextic, unlike the general element of Σ1, which
corresponds to a projection of a canonical curve of genus four. Theorem 3.5, proves
in particular that, under a suitable restriction, (see inequality (3)), on the genus
of the curve corresponding to the general element of the family and, if the number
of the cusps is small, the variety Σnk,d contains a not empty irreducible component
whose general element corresponds to a curve which is not a projection of an other
curve, lying in a projective space of larger dimension. We notice that the inequality
(3) of the previous theorem can’t be improved. Indeed, if g =
− k − d, then
k + d > h0(P2,OP2(n− 3 − t)) if and only if g <
2tn−t2−3t
. On the other hand, by
using the same notation as in theorem (3.5), if g < 2tn−t
, then, by Riemann-
Roch theorem, we have that h0(C,OC(t)) ≥ tn− g+1 >
+1 = h0(P2,OP2(t)).
On the contrary, inequalities (5) and (6) are not sharp, (see example 3.7).
8 CONCETTINA GALATI
In the case of k = 0 and t = 1, theorem 3.5 has been proved by Sernesi in [18],
section 4. The case k = 0 and t ≤ n−3 is already contained in [5]. To show theorem
3.5, we proceed by induction on the degree n and on the number of nodes and cusps
of the curve. The geometric idea at the base of the induction on the degree of the
curve is, mutatis mutandis, the same as that of Sernesi.
Proof of theorem 3.5. Let t be a positive integer such that n−3−t ≥ 0 and letW ⊂
Σnk,d be an irreducible component of Σ
k,d. By standard semicontinuity arguments
it follows that, if there exists a point [C] ∈ W corresponding to a geometrically
t-normal curve with only k cusps and d nodes as singularities, then the general
element of W corresponds to a geometrically t-normal plane curve. Moreover, if the
theorem is true for fixed n, t ≤ n − 3, k as in (5) or in (6) and k + d as in (3),
then the theorem is true for n, t and any k′ ≤ k and d′ ≤ d+ k − k′. Indeed, from
the hypotheses (3), (5) and (6), it follows in particular that k < 3n. By section 2.1,
under this hypothesis, for every k′ ≤ k and for every d′ ≤ d + k − k′, there exists
a family of plane curves C → ∆ of degree n, parametrized by a curve ∆ ⊂ Σnk′,d′ ,
whose special fibre is C0 = C and whose general fibre Cz has d
′ nodes and k′ cusps
as singularities. The statement follows by applying the semicontinuity theorem to
the family C̃ → ∆̃, obtained by normalizing the total space of the pull-back family
of C → ∆ to the normalization curve ∆̃ of ∆. Finally, it’s enough to show the
theorem when the equality holds in (5), (6) and (3).
First of all we consider the case k = 0. We will show the statement for any fixed t
and by induction on n. Let, then t ≥ 1 and n = t+3. In this case the equality holds
in (3) if d = 1 = h0(P2,OP2). Since one point imposes independent linear conditions
to regular functions, by using lemma 3.2, we find that every irreducible plane curve
of degree n = t + 3 with one node and no further singularities is geometrically t-
normal. So, the first step of the induction is proved. Suppose, now, that the theorem
is true for n = t+3+a and let [Γ] ∈ Vn,g be a point corresponding to a geometrically
t-normal curve with a
2+3a+2
nodes. Let D be a line which intersects transversally
Γ and let P1, ..., Pt+1 be t + 1 marked points of Γ ∩ D. If Γ
′ = Γ ∪ D ⊂ P2, then
P1, ..., Pt+1 are nodes for Γ
′. Let C → Γ be the normalization of Γ and C′ → Γ′ the
partial normalization of Γ′, obtained by smoothing all singular points of Γ′, except
P1, ..., Pt+1. We have the following exact sequence of sheaves on C
(7) 0 → OD(t)(−P1 − ...− Pt+1) → OC′(t) → OC(t) → 0,
where OC′(t) := OC′(tH) and H is the pull-back with respect to C
′ → Γ′ of general
line of P2. Since deg(OD(t)(−P1 − ...− Pt+1)) < 0, we get that
h0(D,OD(t)(−P1 − ...− Pt+1)) = 0
and so
(8) h0(C′,OC′(t)) = h
0(C,OC(t)) = h
0(P2,OP2(t)).
Now, by section 2.1, we can obtain Γ′ as the limit of a 1-parameter family of
irreducible plane curves
ψ : C → ∆ ⊂ P
(n+1)(n+4)
of degree n+ 1 = t+ a+ 4 with
a2 + 3a+ 2
+ n− t− 1 =
(a+ 1)2 + 3(a+ 1) + 2
= h0(P2,OP2(n+ 1− t− 3))
nodes specializing to nodes of Γ′ different from the marked points P1, ..., Pt+1. More-
over, one can prove that ∆ is smooth, (see [24] or [25]). Normalizing C, we obtain a
family whose general fibre is smooth and whose special fibre is exactly C′, and we
conclude the inductive step by (8) and by semicontinuity theorem.
Now we consider the case t = 1, 2 or 3 and k as in (5) and in (6). Suppose the
theorem is true for n and let [Γ] ∈ Σnk,d be a general point in one of the irreducible
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 9
components of Σnk,d. Then, let D be a smooth plane curve of degree t if t = 1, 2 or
an irreducible cubic with a cusp if t = 3. By the generality of Γ, we may suppose
that D intersects Γ transversally. Let P1, ..., Pt2+1 be t
2+1 fixed points of Γ∩D. If
Γ′ = Γ ∪D, then P1, ..., Pt2+1 are nodes for Γ
′. Let C → Γ be the normalization of
Γ and C′ → Γ′ the partial normalization of Γ′, obtained by smoothing all singular
points except P1, ..., Pt2+1. By using the same notation and by arguing as before,
from the following exact sequence of sheaves on C′
0 → OD(t)(−P1 − ...− Pt2+1) → OC′(t) → OC(t) → 0,
we deduce that
(9) h0(C′,OC′(t)) = h
0(C,OC(t)) = h
0(P2,OP2(t)).
Now, by section 2.1, we can obtain Γ′ as limit of a family of irreducible plane curves
φ : C → ∆
of degree n+ t with d+ nt− t2 − 1 =
(n+t)2+(3+2t)(n+t)+t2+3t+2
nodes specializing
to nodes of Γ′ different to P1, ..., Pt2+1, and k +
2−3t+2
cusps specializing to cusps
of Γ. We conclude by (9) and by semicontinuity, as before. Now we have to show
the first step of the induction. For t = 1 the induction begins with the cases
(n, k) = (4, 1), (5, 3), (6, 6). Trivially, if n = 4 and k = 1 one point imposes
independent conditions to the linear system of regular functions. If n = 5 and
k = 3 we have to show that there are irreducible quintics with three cusps not on
a line. A quintic with three cusps is a projection of the rational normal quintic
C5 ⊂ P
5 from a plane generated by three points lying on three different tangent
lines to C5. By Bezout theorem the three cusps of such a plane curve can’t be
aligned. If n = k = 6, one can repeat the classical argument used by Zariski, see
[24] or example 3.20 of chapter 2 of [7]. For t = 2 we have to show the theorem
for (n, k) = (5, 1), (6, 3), (7, 6), (8, 6), while for t = 3 we have to show the theorem
for (n, k) = (6, 1), (7, 3), (8, 6), (9, 6), (10, 6). The case t = 2 and (n, k) = (5, 1) is
trivial. When t = 2, n = 6 and k = 3 we have that n − 3 − t = 1. To show that
there exists an irreducible sextic with three cusps not on a line, consider a rational
quartic C4 with three cusps, (see corollary 3.18 of chapter 2 of [7] for the existence).
By Bezout theorem, the three double points of C4 can’t be aligned. Then consider
a sextic C6 which is union of C4 and a conic C2 which intersects C4 transversally.
By section 2.1, one can smooth the intersection points of C4 and C2 obtaining a
family of sextics with three cusps not on a line. For t = 2, n = 7 and k = 6 we
argue as in the previous case, by using a sextic C6 with six cusps not on a conic
and a line R with intersects C6 transversally. Similarly for t = 2 , n = 8 and k = 6
and t = 3 and (n, k) = (6, 1), (7, 3), (8, 6), (9, 6), (10, 6). �
Example 3.7. Inequalities (5) and (6) are not sharp. To see this, we can consider
the example of curves of degree 10. We recall that we say that a plane curve is
geometrically linearly normal (g.l.n. for short) if it is geometrically 1-normal. The-
orem 3.5 ensures the existence of g.l.n. irreducible plane curves of degree 10 with
k ≤ 6 cusps and nodes as singularities. But, by using the same ideas as we used
in theorem 3.5, one can prove the existence of g.l.n. plane curves of degree 10 with
nodes and k ≤ 9 cusps. It is enough to consider a sextic Γ6 with six cusps not on a
conic and a rational quartic Γ4 with three cusps intersecting Γ6 transversally. We
choose five points P1, . . . , P5 of Γ4 ∩ Γ6. If Γ
6 and Γ
4 are the normalization curves
of Γ6 and Γ4 respectively and C
′ is the partial normalization of Γ6 ∪Γ4 obtained by
normalizing all its singular points except P1, . . . , P5, by considering the following
exact sequence
0 → OΓ′4(1)(−P1 − · · · − P5) → OC′(1) → OΓ′6(1) → 0
10 CONCETTINA GALATI
we find that h0(C′,OC′(1)) = 3. By using terminology of section 2.1, the statement
follows by smoothing the singular points P1, . . . , P5 of Γ6 ∪Γ4, and by semicontinu-
ity, as in the proof of theorem 3.5. The bound on the number of cusps of theorem
3.5 can be improved also for t = 2 or t = 3. For example, theorem 3.5 ensures
the existence of geometrically 3-normal curves of degree 12 with k ≤ 6 and nodes
as further singularities. But, by considering a geometrically 3-normal curve of de-
gree 8 with six cusps and a quartic with 3 cusps and arguing as before, we can find
geometrically 3-normal irreducible plane curves of degree 12 with nodes and k ≤ 9
cusps.
4. Families of plane curves with nodes and cusps with finite and
expected number of moduli.
Let Σ ⊂ Σnk,d be an irreducible component of Σ
k,d. We want to give sufficient
conditions for Σ to have the expected number of moduli. Let [Γ] ∈ Σ be a general
element, corresponding to a plane curve Γ with normalization map φ : C → Γ. We
shall denote by ωC the canonical sheaf of C and by OC(1) the sheaf associated to
the pullback to C of the divisor cut out on Γ from the general line of P2.
Proposition 4.1. Let Σ ⊂ Σnk,d be an irreducible component of Σ
k,d and let [Γ] ∈ Σ
be a general element, corresponding to a plane curve Γ with normalization map
φ : C → Γ. Suppose that Σ is smooth of the expected dimension equal to 3n+g−1−k
at [Γ]. Moreover, suppose that:
(1) Γ is geometrically linearly normal, i.e. h0(C,OC(1)) = 3,
(2) the Brill-Noether map
µo,C : H
0(C,OC(1))⊗H
0(C, ωC(−1)) → H
0(C, ωC)
is surjective.
Then Σ has the expected number of moduli equal to 3g − 3 + ρ− k.
Proof. The case k = 0 has been proved by Sernesi in [18], section 4. We shall
assume k > 0. Let Γ be a plane curve verifying the hypotheses of the proposition.
By lemma 1.5.(b) of [22], the hypothesis that Σ is smooth of the expected dimension
at [Γ] implies the vanishing H1(C,Nφ) = 0, where Nφ if the normal sheaf of φ. We
recall that, denoting by ΘC and ΘP2 the tangent sheaf of C and P
2 respectively,
then the normal sheaf of φ is defined as the cokernel of the differential map φ∗ of φ
(10) 0 → ΘC
→ φ∗ΘP2 → Nφ → 0
By theorem 3.1 of [13], the vanishing H1(C,Nφ) = 0 is a sufficient condition for the
existence of a universal deformation family
of the normalization map φ, whose parameter space B is smooth at the point 0
corresponding to φ, with tangent space at 0 equal to H0(C, Nφ). On the contrary,
by [3], p. 487, the Severi variety Vn,g = Σ
0,k+d of irreducible plane curves of genus
− d − k is singular at the point [Γ] and the universal deformation space
B of φ is a desingularization of Vn,g at [Γ]. Moreover, by corollary 6.11 of [2],
if Bk = F
−1(Σ) is the locus of points of B corresponding to a morphism with k
ramification points, then the tangent space to Bk at 0 is a subspaceW of H
0(C,Nφ)
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 11
of codimension k such that W ∩H0(C,Kφ) = 0, where Kφ is the torsion subsheaf
of Nφ. By [3], p.487, it follows that, if
F : B → Vn,g
is the natural (1 : 1)-map from B to Vn,g, then the differential map
dF : H0(C, Nφ) → T[Γ]Vn,g
restricts to an isomorphism between W and the tangent space T[Γ]Σ to Σ at [Γ].
We can now go back to the number of moduli of Σ. From the exact sequence
(10), by using that H1(C, Nφ) = 0, we get the following long exact sequence
0 → H0(C,ΘC) → H
0(C, φ∗ΘP2) → H
0(C,Nφ)
→ H1(C,ΘC) → H
1(C, φ∗ΘP2) → 0
Recalling that the space H1(C,ΘC) is canonically identified with the tangent space
T[C]Mg to Mg at the point associated to the normalization C of Γ, the coboundary
map δC : H
0(C,Nφ) → H
1(C,ΘC) sends the Horikawa class of an infinitesimal
deformation of φ to the Kodaira-Spencer class of the corresponding infinitesimal
deformation of C. So, δC |W is the differential map at the point 0 ∈ B of the
moduli map ΠΣ ◦ F : Bk = F
−1(Σ) 99K Mg. Since the point [Γ] is general in Σ,
and recalling the isomorphism dF :W
→ T[Γ]Σ, we have that
the number of moduli of Σ = dim(δC(W )).
Now, from the exact sequence (10), we have that
dim(δC(H
0(C,Nφ)) = 3g − 3− h
1(C, φ∗ΘP2).
Moreover, from the pull-back to C of the Euler exact sequence, we deduce the well
known isomorphism
H1(C, φ∗ΘP2) ≃ coker(µ
0,C) ≃ (ker(µ0,C))
and we conclude that
(11) dim(δC(H
0(C,Nφ))) = 3g − 3− dim(ker(µ0,C)).
Notice that the previous equality is always true, even if Γ doesn’t verify the hypoth-
esis (1) or (2) of the statement. Moreover, if Γ is geometrically linearly normal, i.e.
if h0(C,OC(1)) = 3, we have that
ρ = 3n− 2g − 6 = dim(coker(µo,C))− dim(ker(µo,C)).
When µo,C is surjective, ρ = − dim(ker(µo,C)) and
(12) dim(δC(H
0(C,Nφ)) = 3g − 3 + ρ = dim(B)− 8 = dim(Vn,g)− 8.
Since the dimension of the fibre of the moduli map
ΠVn,g ◦ F : B 99K Mg
has dimension at least equal to 8 = dim(Aut(P2)), from (12) we deduce that the
differential map of ΠVn,g ◦F has maximal rank at 0 and, in particular, we have that
dim((ΠVn,g ◦ F )
−1([C])) = 8. Equivalently, there exist only finitely many g2n on C.
It follows that there are only finitely many g2n on C mapping C to the plane as a
curve with k cusps and d nodes. Then,
dim(δc(W )) = dim(ΠΣ(Σ)) = 3g − 3 + ρ− k.
Remark 4.2. Arguing as in the proof of the previous proposition, it has been proved
in [18] that, if Γ is a geometrically linearly normal plane curve with only d nodes
as singularities and the Brill-Noether map µo,C of the normalization morphism of
Γ is injective, then Σ = Σn0,d has general moduli. If Σ ⊂ Σ
k,d and [Γ] ∈ Σ verify
the hypotheses of proposition 4.1 but we assume that µo,C is injective, we may
only conclude that ΠVn,g ◦ F is dominant with surjective differential map at [Γ].
12 CONCETTINA GALATI
So dim(Π−1Vn,g ([C])) = ρ + 8. But this is not useful to compute the dimension of
δC(W ) = δC(T[Γ]Σ). However, in this case we get that
δC(T[Γ]Σ) + δC(H
0(C,Kφ)) = δC(H
0(C,Nφ)) = H
1(C,ΘC).
Then, by using that dim(δC(H
0(C,Kφ))) ≤ k and by recalling that if Σ has the
expected dimension then the number of moduli of Σnk,d is at most the expected one
(see lemma 2.2 and remark 2.3), we find that
3g − 3− k ≤ number of moduli of Σ ≤ 3g − 3 + ρ− k.
Remark 4.3. Notice that, if a plane curve Γ of genus g verifies the hypotheses
(1) and (2) of the previous proposition, then the Brill-Noether number ρ(2, g, n)
is not positive and, in particular, g ≥ 3. We don’t know examples of complete
irreducible families Σ ⊂ Σnk,d with the expected number of moduli whose general
element [Γ] corresponds to a curve Γ of genus g, with ρ(2, g, n) ≤ 0, which doesn’t
verify properties (1) and (2).
Lemma 4.4 ([5], Corollary 3.4). Let Γ be an irreducible plane curve of degree n
with only nodes and cusps as singularities and let φ : C → Γ be the normalization
morphism of Γ. Suppose that Γ is geometrically 2-normal, i.e. h0(C,OC(2)) = 6.
Then the Brill-Noether map
µo,C : H
0(C,OC(1))⊗H
0(C, ωC(−1)) → H
0(C, ωC)
is surjective.
Proof. By lemma 3.2, the curve Γ is geometrically 2-normal if and only if the scheme
N of the singular points of Γ imposes independent linear conditions to the linear
systemH0(P2,OP2(n−5)) of plane curves of degree n−5. SinceH
0(P2,OP2(n−5)) ⊂
H0(P2,OP2(n−4)), N imposes independent linear conditions plane curves of degree
n−4, and, by using lemma 3.2, we get that h0(C,OC(1)) = 3, i.e. Γ is geometrically
linearly normal. Now, denote by IN |P2 the ideal sheaf of N . Notice that the curve
Γ is geometrically 2-normal if and only if the ideal sheaf IN |P2(n− 4) is 0-regular,
(in the sense of Castelnuovo-Mumford). Indeed, since h2(P2, IN |P2(n− 6)) = 0, the
ideal sheaf IN |P2(n− 4) is 0-regular if and only if h
1(P2, IN |P2(n− 5)) = 0. Because
of the 0-regularity of IN |P2(n− 4), we have the surjectivity of the natural map
H0(P2, IN |P2(n− 4))⊗H
0(P2,OP2(1)) → H
0(P2, IN |P2(n− 3)),
(see [17]). Finally, by the geometric linear normality of Γ, the vertical maps of the
following commutative diagram
H0(P2,OP2(1))⊗H
0(P2, IN |P2(n− 4)) //
H0(P2, IN |P2(n− 3))
H0(C,OC(1))⊗H
0(C, ωC(−1))
// H0(C, ωC)
are surjective and, hence, the Brill-Noether map µo,C is surjective too. �
Corollary 4.5. Let Σ ⊂ Σnk,d be an irreducible component of Σ
k,d of dimension
equal to 3n + g − 1 − k, such that the general point [Γ] ∈ Σ corresponds to a
geometrically 2-normal plane curve. Then Σ has the expected number of moduli
equal to 3g − 3 + ρ− k.
Proof. It follows from proposition 4.1 and lemma 4.4. �
In order to produce examples of families of irreducible plane curves with nodes
and cusps with the expected number of moduli, we study how increases the rank of
the Brill-Noether map by smoothing a node or a cusp of the general curve of the
family, (in the sense of section 2.1).
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 13
Let Σ ⊂ Σnk,d, with n ≥ 5, be an irreducible component of Σ
k,d, let [Γ] ∈ Σ be a
general point of Σ and let φ : C → Γ be the normalization of Γ. Choose a singular
point P ∈ Γ and denote by φ′ : C′ → Γ the partial normalization of Γ obtained by
smoothing all singular points of Γ, except the point P . If ωC′ is the dualizing sheaf
of C′ and
µo,C′ : H
0(C′,OC′(1))⊗H
0(C′, ωC′(−1)) → H
0(C′, ωC′),
is the natural multiplication map, we have the following result.
Lemma 4.6. If h0(C,OC(1)) = 3 and the geometric genus g of C is such that g >
n−2, with n ≥ 5, then rk(µo,C′) ≥ rk(µo,C)+1. In particular, if h
0(C,OC(1)) = 3,
n ≥ 5 and µo,C is surjective, then µo,C′ is also surjective.
Proof. Let ψ : C → C′ be the normalization map.
We recall that, if we set φ∗(P ) := p1 + p2 when P is a node and φ
∗(P ) = 2φ−1(P )
when P is a cusp, then the dualizing sheaf of C′ is a subsheaf of ψ∗(ωC(φ
∗(P ))),
(see for example [10], p.80). In particular we have the following exact sequence
(13) 0 → ωC′ → ψ∗ωC(φ
∗(P )) → CP → 0
where CP is the skyscraper sheaf on C with support at P . From this exact sequence,
we deduce that
H0(C′, ωC′) ≃ H
0(C, ωC(φ
∗(P ))).
Moreover, tensoring (13) by OC′(−1), we find the exact sequence
(14) 0 → ωC′(−1) → ψ∗ωC(φ
∗(P ))(−1) → CP → 0
from which we get an injective map H0(C′, ωC′(−1)) → H
0(C, ωC(φ
∗(P ))(−1)).
On the other hand
(15) h0(C′, ωC′(−1)) = h
0(C, ωC(φ
∗(P ))(−1)) = g − n+ 3
and so
H0(C′, ωC′(−1)) ≃ H
0(C, ωC(φ
∗(P ))(−1)).
Moreover, from the hypothesis h0(C,OC(1)) = 3, we have that H
0(C,OC(1)) ≃
H0(C′,OC′(1)) ≃ H
0(P2,OP2(1)). Therefore, in the following commutative dia-
H0(C′,OC′(1))⊗H
0(C′, ωC′(−1))
µo,C′
H0(C′, ωC′)
H0(C,OC(1))⊗H
0(C, ωC(−1)(φ
∗(P )))
// H0(C, ωC(φ
∗(P )))
where we denoted by µ′o,C the natural multiplication map, the vertical maps are
isomorphisms. In particular,
rk(µo,C′) = rk(µ
o,C).
In order to compute the rank of µ′o,C , we consider the following commutative dia-
14 CONCETTINA GALATI
H0(C,OC(1))⊗H
0(C, ωC(−1))
H0(C, ωC)
H0(C,OC(1))⊗H
0(C, ωC(−1)(φ
∗(P )))
// H0(C, ωC(φ
∗(P )))
where the vertical maps are injections. Notice that, since we supposed n ≥ 5,
h0(C,OC(1)) = 3 and g > n−2 ≥ 3, the sheaf OC(1) is special. We deduce that C is
not hyperelliptic and, chosen a basis of H0(C, ωC), the associated map C → P
g−1 is
an embedding. On the contrary, the sheaf ωC(φ
∗(P )) does not define an embedding
on C. Choosing a basis of H0(C, ωC(φ
∗(P )) and denoting by Φ : C → Pg the
associated map, this will be an embedding outside φ∗(P ). If P is a node of C
and φ∗(P ) = p1 + p2, the image of C to P
g, with respect to Φ, will have a node
at the image point Q of p1 and p2. If P ∈ Γ is a cusp, then Φ(C) will have a
cusp at the image point Q of φ−1(P ). The hyperplanes of Pg passing through Q
cut out on C the canonical linear series |ωC |. Moreover, if we denote by B ⊂ P
the subspace which is the base locus of the hyperplanes of Pg corresponding to
Im(µ′o,C), then Q /∈ B. Indeed, B intersects the curve C in the image of the base
locus of |OC(1)|+ |ωC(φ
∗(P ))(−1)| := P(Im(µ′0,C)), which coincides with the base
locus of |ωC(φ
∗(P ))(−1)|, since |OC(1)| is base point free. Now, by (15),
h0(ωC(φ
∗(P ))(−1)) = 3 + g − n = h0(C, ωC(−1)) + 1.
Then φ∗(P ) does not belong to the base locus of |ωC(φ
∗(P ))(−1)|, and so
dim(< Q,B >Pg) = dim(B) + 1.
Finally, we find that
rk(µo,C) = rk(Gµo,C) ≤ dim(Im(G) ∩ Im(µ
o,C))
≤ g + 1− dim(< B,Q >Pg)− 1
= g − 1− dim(B)
= rk(µ′o,C)− 1.
Corollary 4.7. Let Σ ⊂ Σnk,d be a non-empty irreducible component of the expected
dimension of Σnk,d, with n ≥ 5. Suppose that Σ has the expected number of moduli
and that the general element [Γ] ∈ Σ corresponds to a g.l.n. plane curve Γ of
geometric genus g such that, if C → Γ is the normalization of Γ, then the map
µo,C is surjective. Then, for every k
′ ≤ k and d′ ≤ d + k − k′, there is at least an
irreducible component Σ′ ⊂ Σnk′,d′ , such that Σ ⊂ Σ
′, the general element [D] ∈ Σ′
corresponds to a g.l.n. plane curve D of geometric genus g′ with normalization
Dν → D and the Brill-Noether map µ0,Dν surjective. In particular, Σ
′ has the
expected number of moduli.
Proof. Let Γ be the curve corresponding to the general element [Γ] of Σ ⊂ Σnk,d.
Since by hypothesis Σ is smooth of the expected dimension at [Γ], by section 2.1,
for every k′ ≤ k and for every d′ ≤ d+ k− k′ there exists an irreducible component
Σ′ of Σnk′,d′ containing Σ. In order to prove the statement, it is enough to show it
under the hypotheses k′ = k− 1 and d′ = d+1, k = k′ and d′ = d− 1 or d = d′ and
k′ = k − 1. If k′ = k − 1 and d′ = d + 1, then the statement follows by standard
semiconinuity arguments. If k = k′ and d′ = d − 1 or d = d′ and k′ = k − 1, the
statement follows by lemma 4.6 and by standard semicontinuity arguments. �
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 15
The following lemma has been stated and proved by Sernesi in [18]. Actually,
Sernesi supposes that Γ has only nodes as singularities. But, since his proof works
for plane curves Γ with any type of singularities and, since we need it for curves
with nodes and cusps, we state the lemma in a more general form.
Lemma 4.8. ([18], lemma 2.3) Let Γ be an irreducible and reduced plane curve of
degree n ≥ 5 with any type of singularities. Denote by C the normalization of Γ.
Suppose that h0(C,OC(1)) = 3 and the Brill-Noether map
µo,C : H
0(C,OC(1))⊗H
0(C, ωC(−1)) → H
0(C, ωC),
has maximal rank. Let R be a general line and let P1, P2 and P3 be three fixed points
of Γ ∩ R. We denote by C′ the partial normalization of Γ′ = Γ ∪ R, obtained by
smoothing all the singular points, except P1, P2 and P3. Then h
0(C′,OC′(1)) = 3
and, denoting by ωC′ the dualizing sheaf of C
′, the multiplication map
µo,C′ : H
0(C′,OC′(1))⊗H
0(C′, ωC′(−1)) → H
0(C′, ωC′),
has maximal rank.
Theorem 4.9. Let Σnk,d be the algebraic system of irreducible plane curves of degree
n ≥ 4 with k cusps, d nodes and geometric genus g =
− k − d. Suppose that:
(16) n− 2 ≤ g equivalently k + d ≤ h0(P2,OP2(n− 4))
(17) k ≤ 6 +
if 3n− 9 ≤ g and n ≥ 6,
(18) k ≤ 6 otherwise.
Then Σnk,d has at least an irreducible component Σ which is not empty and such
that, if Γ ⊂ P2 is the curve corresponding to the general element of Σ and C is the
normalization curve of Γ, then h0(C,OC(1)) = 3 and the map µo,C has maximal
rank. In particular, when ρ ≤ 0, the algebraic system Σ has the expected number of
moduli equal to 3g − 3 + ρ− k.
Proof. Suppose that (17) holds. Then, by observing that
g ≥ 3n− 9 if and only if k + d ≤ h0(P2,OP2(n− 6))
and by using theorem 3.5 for t = 3, we have that there exists an irreducible com-
ponent Σ of Σnk,d whose general element is a geometrically 3-normal plane curve Γ.
By remark 3.4, it follows that also the linear systems cut out on C by the conics
and the lines are complete. The statement follows from corollary 4.5.
In order to prove the theorem under the hypothesis (18), we consider the following
subcases:
(1) 2n− 5 ≤ g ≤ 3n− 9, i.e. h0(OP2(n− 6)) ≤ k+ d ≤ h
0(OP2(n− 5)) and
n ≥ 5,
(2) n− 2 ≤ g ≤ 2n− 7 and n ≥ 5,
(3) g = 2n− 6 and n ≥ 4.
Suppose that (1) holds. By theorem 3.5 for t = 2, we know that, under this
hypothesis, there exists a nonempty component Σ ⊂ Σnk,d, whose general element is
geometrically 2-normal. We conclude as in the previous case, by corollary 4.5.
Now, suppose that g and n verify (2). We shall prove the theorem by induction
on n and g. Set g = 2n − 7 − a, with a ≥ 0 fixed. Suppose that the theorem is
true for the pair (n, g), with n ≥ 7. We shall prove the theorem for (n + 1, g + 2),
observing that g + 2 = 2(n + 1) − 7 − a. Let Γ be a g.l.n. irreducible plane curve
16 CONCETTINA GALATI
of degree n and genus g = 2n − 7 − a with k ≤ 6 cusps, d nodes and no more
singularities. Let C be the normalization of Γ. Suppose that the Brill-Noether map
µo,C has maximal rank. Let R ⊂ P
2 be a general line and let P1, P2 and P3 be three
fixed points of Γ∩R. By section 2.1, since k ≤ 6 < 3n, one can smooth the singular
points P1, P2, P3 and preserve the other singularities of Γ ∪ R ⊂ P
2, obtaining a
family of plane curves C → ∆ whose general fibre is irreducible, has degree n+1 and
genus g+2. We conclude by lemma 4.8 and by standard semicontinuity arguments.
Now we prove the first step of the induction for n ≥ 7. If n = 7, we get 0 ≤ a ≤ 2.
Let a = 0, i.e. g = 2n−7−a= 7. Let Γ be a g.l.n. irreducible plane curve of degree
n = 7, of genus g = n = 7 with k ≤ 6 cusps and nodes as singularities, such that
no seven singular points of Γ lie on an irreducible conic. To prove that there exists
such a plane curve, notice that, by applying theorem 3.5 for t = 1, we get that, for
any fixed k ≤ 6, there exists a g.l.n. irreducible sextic D of genus four with k cusps
and d = 6 − k nodes. Let R1, . . . , R6 be the singular points of D. Since the points
R1, . . . , R6 of D impose independent linear conditions to the conics, however we
choose five singular points Ri1 , . . . , Ri5 of D, with I = (i1, . . . , i5) ⊂ (1, . . . , 6), there
exists only one conic CI , passing through these points. Let us set S =
CI ∩D
and let R be a line intersecting D transversally at six points out of S. By Bezout
theorem, no seven singular points of Γ′ = D ∪ R belong to an irreducible conic.
Moreover, if D̃ is the normalization of D, if Q1, . . . , Q4 are four fixed points of
D∩R and D′ is the partial normalization of Γ′ obtained by smoothing the singular
points except Q1, . . . , Q4, then, by the following exact sequence
(19) 0 → OR(1)(−Q1 − · · · −Q4) → OD′(1) → OD̃(1) → 0
we find that h0(D′, OD′(1)) = 3. By section 2.1, one can smooth the singulari-
ties Q1, . . . , Q4 and preserve the other singularities of D ∪ R, getting a family of
irreducible septics G → ∆ whose general fibre Γ is a geometrically linearly normal
irreducible septic with k cusps and 8 − k nodes such that no seven singular point
of Γ belong to an irreducible conic. Let, now, C be the normalization of Γ and let
∆ ⊂ C be the adjoint divisor of the normalization map φ : C → Γ. We shall prove
that ker(µo,C) = 0. Since Γ is geometrically linearly normal, we have that
h0(C, ωC(−1)) = h
0(C,OC(3)(−∆))) = g − n+ 2 = 2.
Then, by the base point free pencil trick, we find that
ker(µo,C) = H
0(C, ω∗C(B)⊗OC(2)),
where B is the base locus of |ωC(−1) = OC(3)(−∆)|. Let F be the pencil of plane
cubics passing through the eight double points P1, . . . , P8 of Γ and let BF be the
base locus of the pencil F . Let Γ3 be the general element of F . Suppose that BF
has dimension one. If BF contains a line l, then, by Bezout theorem, at most three
points among P1, . . . , P8, say P1, . . . , P3 can lie on l and the other points have to be
contained in the base locus of a pencil of conics F ′. Using again Bezout theorem,
we find that the curves of F ′ are reducible and the base locus of F ′ contains a line
l′. But also l′ contains at most three points of P4, . . . , P6. It follows that there is
only one cubic through P1, . . . , P8. This is not possible by construction. Suppose
that BF contains an irreducible conic Γ2. By Bezout theorem, at most seven points
among P1, . . . , P8 may lie on Γ2. On the other hand, since dim(F) = 1, there
are exactly seven points of P1, . . . , P8, say P1, . . . , P7, on Γ2 and the general cubic
Γ3 of F is union of Γ2 and a line passing through P8. Since, by construction, no
seven singular points of Γ lie on an irreducible conic, also in this case we get a
contradiction. So the general element Γ3 of F is irreducible. Using again Bezout
theorem, we find that Γ3 is smooth and F has only one more base point Q. We
consider the following cases:
a) Q doesn’t lie on Γ;
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 17
b) Q lies on Γ, but Q 6= P1, . . . , P8;
c) Q is infinitely near to one of the points P1, . . . , P8, say Pî, i.e. the cubics of F
have at P
the same tangent line l, but l is not contained in the tangent cone to Γ
d) Q is like in the case c), but l is contained in the tangent cone to Γ at P
Suppose that the case a) or c) holds. Thus B = 0 and
ker(µo,C) = H
0(C, ω∗C ⊗OC(2)) = H
0(C,OC(−2)(∆)).
By Riemann-Roch theorem, h0(C,OC(−2)(∆)) = h
0(C,OC(6)(−2∆)) − 4. One
sees that h0(C,OC(6)(−2∆)) = 4, by blowing-up the plane at P1, . . . , P8 and by
using some standard exact sequences. Suppose now that the case b) holds. Thus
B = Q and
dim(ker(µo,C)) = h
0(C,OC(−2)(∆ +Q)) = h
0(C,OC(6)(−2∆−Q))− 3.
Also in this case one sees that h0(C,OC(6)(−2∆ − Q)) = 3 by blowing-up at
P1, . . . , P8 and Q and by using standard exact sequences. Finally, we analyze the
case d). Let Φ : S → P2 be the blow-up of the plane at P1, . . . , P8 with exceptional
divisors E1, . . . , E8. Let Q ∈ Eî be the intersection point of Eî and the strict
transform C3 of the general cubic Γ3 of the pencil F . We denote by Φ̃ : S̃ → S the
blow-up of S at Q and by Ψ : S̃ → P2 the composition map of the maps Φ and Φ̃.
We still denote by E1, . . . , E8 their strict transforms on S̃, by C and C3 the strict
transforms of Γ and Γ3 and by EQ the new exceptional divisor of S̃. In this case we
have that Ψ∗(Γ) = C+2
Ei+3EQ, Ψ
∗(Γ3) = C3+
Ei+2EQ. Moreover, the
divisor ∆ is cut out on C by
iEi + EQ and the base locus B of the linear series
|ωC(−1)| coincides with the intersection point of EQ and C. So, we have that
dim(ker(µo,C)) = h
0(C,OC(−2)(
Ei+2EQ)) = h
0(C,OC(6)(−2
Ei−3EQ))−3.
Moreover, from the following exact sequence
0 → O
(−1) → O
(6)(−2
Ei − 3EQ) → OC(6)(−2
Ei − 3EQ) → 0
we find that H0(C,OC(6)(−2
Ei− 3EQ)) = H
0(S̃,O
(6)(−2
Ei− 3EQ)). In
order to show that h0(S̃,O
(6)(−2
Ei − 3EQ)) = 3, we consider the following
exact sequence
0 → O
(3)(−
Ei − EQ) → OS̃(6)(−2
Ei − 3EQ) →(20)
→ OC3(6)(−2
Ei − 3EQ) → 0
By Riemann-Roch theorem, we have that
h0(C3,OC3(6)(−2
Ei − 3EQ)) = 1 and h
1(C3,OC3(6)(−2
Ei − 3EQ) = 0.
Moreover, by Serre duality we have that
H1(S̃,O
(3)(−
Ei − EQ))) = H
1(S̃,O
(−6)(2
Ei + 3EQ))).
From the exact sequence
(21) 0 → O
(−6)(+2
Ei + 3EQ)) → OS̃(1) → OC(1) → 0
18 CONCETTINA GALATI
by using that the map H0(S̃,O
(1)) → H0(C,OC(1)) is surjective and that
h1(S̃,O
(1)) = 0, we find that
H1(S̃,O
(−6)(+2
Ei + 3EQ))) = H
1(S̃,O
(3)(−
Ei − EQ))) = 0.
Then, by (20), h0(S̃,O
(6)(−2
Ei − 3EQ)) = h
0(S̃,O
(3)(−
Ei − EQ)) +
h0(C3,OC3(6)(−2
Ei−3EQ)) = 3 and ker(µo,C) = 0. The first step of induction
for g = n = 7 and k ≤ 6 is proved.
We complete the proof of the first step of the induction, for n and g verifying
(2). When n = 7 and 1 ≤ a ≤ 2, the existence of a g.l.n. plane curve Γ follows
from theorem 3.5. Using the above notation, h0(C, ωC(−1)) = 1 if a = 1 and
h0(C, ωC(−1)) = 0 if a = 2. In any case µo,C is injective. When n ≥ 8 and
a ≤ n − 6 the theorem follows by induction from the case n = 7. For n ≥ 8 and
a = n− 5, we find that g = n− 2, or, equivalently, k + d = h0(P2,OP2(n− 4)). In
theorem 3.5, we proved the existence of geometrically linearly normal plane curves
of degree n ≥ 8 and genus g = n− 2, with nodes and k ≤ 6 cusps. For every such
plane curve Γ, using the above notation, the Brill-Noether map µo,C is injective
since h0(C, ωC(−1)) = 0. The cases n = 5 and n = 6 are similar.
Suppose now that n and g verify (3). First of all we prove the theorem for
(n, g) = (4, 2), (5, 4), (6, 6). For n = 4 and g = 2, we find n = g + 2 and we argue
as in the case n ≥ 8 and g = n − 2. Similarly, for (n, g) = (5, 4). For n = 6
and g = 6 in theorem 3.5 we proved the existence of geometrically linearly normal
plane curves Γ with k ≤ 4 cusps and nodes as singularities. For every such a plane
curve Γ, denoting by C its normalization, we get that h0(C, ωC(−1)) = 2, i.e. the
linear system F of conics passing through the four singular points P1, . . . , P4 of Γ
is a pencil which cuts out on C the complete linear series |ωC(−1)|. We have two
possibilities: either the general element of this pencil is irreducible or it consists of
a line containing exactly three singular points P1, P2, P3 of Γ and a line passing
through P4. In any case the base locus of F intersects Γ only at P1, . . . , P4 and
the linear series |ωC(−1)| has no base points. Then, by the base point free pencil
trick , we find that ker(µo,C) = H
0(C, ω∗C ⊗ O(2)) = H
0(C,OC(−1)(∆)), where
∆ ⊂ C is the adjoint divisor of the normalization map C → Γ. By Riemann-Roch
theorem, we have that h0(C,OC(−1)(∆)) = h
0(C,OC(4)(−2∆))−3. By blowing-up
at P1, . . . , P4, one can see that h
0(C,OC(4)(−2∆)) = 3, as we wanted.
Finally, we show the theorem under the hypothesis (3) for n ≥ 7, by using
induction on n. In order to prove the inductive step we may use lemma 4.8, exactly
as we did in the case (2). We prove the first step of induction. If n = 7 we have
that g = 8. On pages 15 and 16 we proved the existence of geometrically linearly
normal plane curves Γ of degree 7 and genus 7 with k ≤ 6, such that, if P1, . . . , P8
are the singular points of Γ, then no seven points among P1, . . . , P8 lie on a conic.
In particular, we proved that, for every such a plane curve Γ, the general element
of the pencil of cubics passing through P1, . . . , P8 is irreducible and, if φ : C → Γ
is the normalization of Γ, then the Brill-Noether map µo,C is injective. Let C
the partial normalization of Γ which we get by smoothing all the singular points of
Γ except a node, say P8. By using the same notation and by arguing exactly as in
the proof of lemma 4.6, we get the following commutative diagram
H0(C′,OC′(1))⊗H
0(C′, ωC′(−1))
µo,C′
H0(C′, ωC′)
H0(C,OC(1))⊗H
0(C, ωC(−1)(φ
∗(P8)))
// H0(C, ωC(φ
∗(P8)))
where µ′o,C is the multiplication map and the vertical maps are isomorphisms. We
want to prove that the map µo,C′ is surjective. By the previous diagram it is enough
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 19
to prove that µ′o,C is surjective. Since h
0(C, ωC(φ
∗(P8))) = 8 and
h0(C,OC(1))h
0(C, ωC(−1)(φ
∗(P8))) = 3(7− 7 + 3) = 9,
we have that dim(ker(µo,C′)) ≥ 1 and µo,C′ is surjective if dim(ker(µo,C′)) = 1. By
recalling that Γ is geometrically linearly normal, we have that, if Z is the scheme of
the points P1, . . . , P7 and IZ|P2 is the ideal sheaf of Z in P
2, then in the following
commutative diagram
H0(C,OC(1))⊗H
0(C, ωC(−1)(φ
∗(P8)))
H0(C, ωC(φ
∗(P8)))
H0(P2,OP2(1))⊗H
0(P2, IZ|P2(3))
// H0(P2, IZ|P2(4))
the vertical maps are isomorphisms. Hence, it is enough to prove that the kernel
of the multiplication map µ has dimension one. Let {f0, f1, f2} be a basis of the
vector space H0(P2, IZ|P2(3)). Since the general cubic passing through P1, . . . , P8
is irreducible, we may assume that f0, f1 and f2 are irreducible. Suppose, by
contradiction, that there exist at least two linearly independent vectors in the kernel
of µ. Then, there exist sections u0, u1, u2 and v0, v1, v2 of H
0(P2,OP2(1)) such that
the sections
ui⊗ fi and
vi⊗ fi are linearly independent in H
0(P2,OP2(1))⊗
H0(P2, IZ|P2(3)) and
i=0 uifi = 0
i=0 vifi = 0.
We can look at (22) as a linear system in the variables f0, f1, f2. The space of
solutions of (22) is generated by the vector
(u1v2 − u2v1, u3v0 − u0v3, u0v1 − u1v0).
In particular, if we set qi = (−1)
1+iuivj − viuj, we find that fjqi = fiqj , for every
i 6= j. But this is not possible since f1, f2 and f3 are irreducible. We deduce that
dim(ker(µ)) = dim(ker(µo,C′)) = 1
and µo,C′ is surjective. The existence of a plane septic of genus 8 with k ≤ 6
cusps and nodes as singularities, with injective Brill-Neother map, follows now by
smoothing the node P8 (in the sense of section 2.1) and by standard semicontinuity
arguments. �
Remark 4.10. Notice that the conditions which we found in theorem 4.9 in order
that Σnk,d has at least an irreducible component with the expected number of moduli,
are not sharp, even if we suppose ρ ≤ 0. To see this, notice that in remark 3.6 we
proved the existence of an irreducible component Σ of Σ129,0 whose general element
corresponds to a 3-normal plane curve. By remark 3.4 and corollary 4.5, we have
that Σ has the expected number of moduli.
Theorem 4.11. Σn1,d has the expected number of moduli, for every d ≤
Proof. First of all, we recall that, by [16], Σn1,d is irreducible for every d ≤
Moreover, from theorem 4.9 and from corollary 2.7, we know that Σn1,d is not empty
and it has the expected number of moduli if either ρ ≤ 0 or ρ ≥ 2. Next we shall
prove that, if ρ = 1, then the algebraic system
Σn1,d = Σ
(n−3)2
has general moduli. Equivalently, we will show that, if [Γ] ∈ Σn1,d is a general point
and g =
− 1 − d = 3n−7
, then, on the normalization curve C of Γ there are
only finitely many linear series g2n with at least a ramification point. Notice that,
20 CONCETTINA GALATI
if g =
− 1 − d = 3n−7
, then n is odd and n ≥ 5. We prove the statement by
induction on n.
If n = 5 then g = 4. Let C ⊂ P3 be the canonical model of a general curve of
genus four and let 2P + Q, with P 6= Q be a divisor in a g13 on C. This divisor is
cut out on C by the tangent line to C at P . The projection of C from Q is a plane
quintic of genus four with a cusp. This proves that Σ51,1 has general moduli.
Now we suppose that the theorem is true for n and we prove the theorem for
n+2. Let Γ ⊂ P2 be the plane curve with a cusp and
(n−3)2
−1 nodes corresponding
to a general point [Γ] ∈ Σn
(n−3)2
and let C2 be an irreducible conic intersecting
Γ transversally. By section 2.1, the point [C2 ∪ Γ] belongs to Σ
(n+2−3)2
particular, however we choose four points P1, . . . , P4 of intersection between Γ and
C2, there exists an analytic branch SP1,..., P4 of Σ
(n−1)2
, passing through [C2∪Γ]
and whose general point corresponds to an irreducible plane curve of degree n + 2
with a cusp in a neighborhood of the cusp of Γ and a node at a neighborhood of
every node of C2 ∪ Γ different from P1, . . . , P4. Moreover, S := SP1,..., P4 is smooth
at the point [C2 ∪ Γ], (see [7], chapter 2). Let
Π : Σn+2
(n−1)2
99K M 3(n+2)−7
be the moduli map of Σn+2
(n−1)2
. In order to prove that Π is dominant it is
sufficient to show that Π(S) = M 3n−1
. By section 2.1, there exist an analytic open
sets Si ⊂ Σn+2
(n−3)2
−1+2n−i
, with i = 1, 2, 3, such that
S0 := S ∩ (P5 × Σn
(n−3)2
) ⊂ S1 ⊂ S2 ⊂ S3 ⊂ S.
Every Si, with i = 1, 2, 3, has
irreducible components, passing through [C2∪Γ]
and intersecting transversally at [C2 ∪Γ], (see [7], chapter 2 or [25]). Moreover, the
general point of every irreducible component of Si, with i = 1, 2, 3, corresponds
to an irreducible plane curve Γi of degree n + 2 with a cusp in a neighborhood of
the cusp of Γ, a node in a neighborhood of every node of C2 ∪ Γ different from
P1, . . . , P4 and 4 − i nodes specializing to 4 − i fixed points among P1, . . . , P4, as
Γi specializes to C2 ∪ Γ. Now, notice that the moduli map Π is not defined at the
point [C2 ∪ Γ], but, if S is sufficiently small, then the restriction of Π to S extends
to a regular function on S. More precisely, let C → ∆ be any family of curves,
parametrized by a projective curve ∆ ⊂ S, passing through the point [C2 ∪ Γ] and
whose general point corresponds to an irreducible plane curve of degree n+2 of genus
3(n+2)−7
with a cusp and nodes as singularities. If we denote by C′ → ∆
the family of curves obtained from C → ∆ by normalizing the total space, we have
that the general fibre of C′ → ∆ is a smooth curve of genus 3n−1
, corresponding
to the normalization of the general fibre of C → ∆, whereas the special fibre C′0 is
the partial normalization of C2∪Γ, obtained by normalizing all the singular points,
except P1, . . . , P4. Then, the map Π|S is defined at [C2 ∪Γ] and it associates to the
point [C2∪Γ] the isomorphism class of C
0. Similarly, if [Γi] is a general point in one
of the irreducible components of Si, with i = 1, 2, 3, then Π|S ([Γi]) is the partial
normalization of Γi obtained by smoothing all the singular points except for the 4−i
nodes of Γi tending to 4−i fixed points among P1, . . . , P4 as Γi specializes to C2∪Γ.
It follows that, if we denote by M
the locus of M 3n−1
parametrizing j-nodal
curves, then ΠS(S
i) ⊆ M4−i3n−1
, for every i = 0, . . . , 4, and ΠS(S
i)  ΠS(S
i+1). In
particular, we find that
dim(Π|S (S)) ≥ dim(Π|S (S
0)) + 4.
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 21
In order to compute the dimension of Π|S (S
0) we consider the rational map
F : Π|S (S
0) 99K M 3n−7
forgetting the rational tail. By the hypothesis that Σn
(n−3)2
has general moduli
and hence F is dominant. Moreover, if C is the normalization curve of Γ, by the
generality of [Γ] in Σn
(n−3)2
, we may assume that C is general in M 3n−7
want to show that dim(F−1([C])) = 5. In order to see this, we recall that, by the
hypothesis that Σn
(n−3)2
has general moduli, on C there exist only finitely many
linear series of degree n and dimension two, mapping C to the plane as curve with a
cusp and nodes as singularities. Let g2n be one of these linear series, let {s0, s1, s2}
be a basis of g2n and φ
′ : C → Γ′ ⊂ P2 the associated morphism. If Q1, . . . , Q4
are four general points of Γ′, then the linear system of conics through Q1, . . . , Q4
is a pencil F(Q1, . . . , Q4). Let C2 and D2 be two general conics of F(Q1, . . . , Q4).
We claim that, if η : P1 → C2 and β : P
1 → D2 are isomorphisms between P
1 and
C2 and D2 respectively, then the points η
−1(Q1), . . . , η
−1(Q4) are not projectively
equivalent to the points β−1(Q1), . . . , β
−1(Q4). In order to prove this, it is enough to
prove that there are at least two conics in the pencil F(Q1, . . . , Q4) which verify the
claim. LetD ⊂ P2 be a conic. If we choose two sets of points p1, . . . , p4 and q1, . . . , q4
ofD not projectively equivalent onD, we may always find projective automorphisms
A : P2 → P2 and A′ : P2 → P2 such that A(pi) = Qi and A
′(qi) = (Qi), for every
i. By construction, the conics C2 = A(D) and D2 = A
′(D) belong to the pencil
F (Q1, . . . , Q4) and verify the claim. This implies that the partial normalizations C
and D′ of Γ′∪C2 and Γ
′∪D2, obtained by smoothing all the singular points except
Q1, . . . , Q4, are not isomorphic. Now, let C
2 be a general conic of F(Q1, . . . , Q4)
and let R1, . . . , R4 be four general points of Γ
′, different from Q1, . . . , Q4. If D
is a general conic of the pencil F(R1, . . . , R4), then the partial normalization C
and D′ of Γ′ ∪C′2 and Γ
′ ∪D′2 obtained, respectively, by smoothing all the singular
points except Q1, . . . , Q4 and R1, . . . , R4, are not isomorphic. Indeed, since C is a
general curve of genus 3n−7
≥ 7, the only automorphism of C is the identity. This
proves that dim(F−1([C])) = 5. In particular, we deduce that
dim(Π|S (S
0)) = 3
3n− 7
− 3 + 5
dim(Π|S (S) ≥ 3
3n− 7
− 3 + 9 = 3
3(n+ 2)− 7
Remark 4.12. We expect that it is possible to prove that Σnk,d has expected number
of moduli for every ρ also when k = 2 or k = 3. By corollary 2.7 and theorem 4.9,
Σnk,d is not empty, irreducible and it has expected number of moduli for ρ ≤ 0 and
ρ ≥ 2k. In order to extend theorem 4.11 to the case k = 2 and k = 3 one needs to
consider a finite number of cases.
Acknowledgment. The results of this paper are part of my PhD-thesis. I would
like to express my gratitude to my advisor Prof. C. Ciliberto who initiated me into
the subject of algebraic geometry and who provided me many invaluable suggestions.
I have also enjoyed and benefited from conversation with many people including F.
Flamini, E. Sernesi, L. Chiantini, L. Caporaso and G. Pareschi. Finally, I would
like to thank the referee for useful remarks which allowed me to improve the finale
version of this paper.
22 CONCETTINA GALATI
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87–99 (1981).
[2] E. Arbarello and M. Cornalba: Su una congettura di Petri., Comment. Math. Helv., vol. 56
(1981), no. 1, 1–38.
[3] E. Arbarello and M. Cornalba: A few remarks about the variety of irreducible plane curves of
given degree and genus., Ann. Sci. École Norm. Sup. (4), vol. 16 (1983), 467–488 (1984).
[4] E. Arbarello and M. Cornalba, P.A. Griffiths, J. Harris: Geometry of algebraic curves., vol. 1,
Springer-Verlag.
[5] A. Arsie and C. Galati: Geometric k-normality of curves and applications, Le Matematiche,
Vol. LVIII (2003), Fasc. II, 179–199.
[6] 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).
[7] C. Galati: Number of moduli of plane curves with nodes and cusps., PhD thesis, Università
degli Studi di Tor Vergata, 2004-2005.
[8] J. Harris: On the Severi problem, Invent. Math., vol. 84 (1986), no. 3, 445–461.
[9] J. Harris and I. Morrison: Moduli of curves, Graduate texts in mathematics, vol. 187, Springer,
New York, 1988.
[10] D. Eisenbud and J. Harris: The Kodaira dimension of the moduli space of curves of genus
≥ 23. Invent. Math. vol. 90 (1987), no. 2, 359–387.
[11] G.M Greuel and U. Karras: Families of varieties with prescribed singularities, Compositio
Math. vol. 69 (1989), no. 1, 83–110.
[12] G.M. Greuel, C. Lossen, and E. Shustin: Castelnuovo function, zero-dimensional schemes
and singular plane curves. J. Algebraic Geom. vol. 9 (2000), no. 4, 663–710.
[13] E. Horikawa: On the deformations of the holomorphic maps I, J. Math. Soc. Japan, vol. 25
(1973), 372–396.
[14] E. Horikawa: On the deformations of the holomorphic maps II, J. Math. Soc. Japan, vol. 26
(1974), 647–667.
[15] P. Kang: A note on the variety of plane curves with nodes and cusps, Proc. Amer. Math.
Soc. vol. 106 (1989), no. 2, 309–312.
[16] P. Kang: On the variety of plane curves of degree d with δ nodes and k cusps, Trans. Amer.
Math. Soc. vol. 316 (1989), no. 1, 165–192.
[17] D. Mumford: Lectures on curves on an algebraic surface. , Princeton University Press, 1966.
[18] E. Sernesi: On the existence of certain families of curves, curves. Invent. Math. vol. 75 (1984),
no. 1, 25–57.
[19] F. Severi: Vorlesungen über algebraische Geometrie, Teuner, Leipzig, 1921.
[20] E. Shustin: Smoothness and irreducibility of varieties of plane curves with nodes and cusps,
Bull. Soc. Math. France vol. 122 (1994), no. 2, 235–253.
[21] E. Shustin: Equiclassical deformations of plane algebraic curves, Singularities (Oberwolfach,
1996), 195–204, Progr. Math., vol. 162, Birkhuser, Basel, 1998.
[22] A. Tannenbaum: On the classical characteristic linear series of plane curves with nodes and
cuspidal points: two examples of Beniamino Segre, Compositio Mathematica vol. 51 (1984),
169–183.
[23] J. Wahl: Deformations of plane curves with nodes and cusps, Amer. J. Math. vol. 96 (1974),
529–577.
[24] O. Zariski: Dimension theoretic characterization of maximal irreducible sistems of plane
nodal curves, Amer. J. Math. vol. 104 (1982), no. 1, 209–226.
[25] O. Zariski: Algebraic surfaces, Classics in mathematics, Springer.
Dipartimento di Matematica, Università degli Studi della Calabria, via P. Bucci,
cubo 30B, Arcavacata di Rende (CS)
E-mail address: galati@mat.unical.it
	1. Introduction
	2. Preliminaries
	2.1. On Severi-Enriques varieties
	2.2. Known results on the number of moduli of nk,d
	3. On the existence of certain families of plane curves with nodes and cusps in sufficiently general position
	4. Families of plane curves with nodes and cusps with finite and expected number of moduli.
	Acknowledgment
	References

  Consider the family S of irreducible plane curves of degree n with d nodes
and k cusps as singularities. Let W be an irreducible component of S. We
consider the natural rational map from W to the moduli space of curves of genus
g=(n-1)(n-2)/2-d-k. We define the "number of moduli of W" as the dimension of
the image of W with respect to this map. If W has the expected dimension equal
to 3n+g-1-k, then the number of moduli of W is at most equal to the min(3g-3,
3g-3+\rho-k), dove \rho is the Brill-Neother number of the linear series of
degree n and dimension 2 on a smooth curve of genus g. We say that W has the
expected number of moduli if the equality holds. In this paper we construct
examples of families of irreducible plane curves with nodes and cusps as
singularities having expected number of moduli and with non-positive
Brill-Noether number.

Introduction
In this paper we compute the number of moduli of certain families of irreducible
plane curves with nodes and cusps as singularities. Let Σnk,d ⊂ P(H
0(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 Σ
k,d. We denote by
Σ0 the open set of Σ of points [Γ] ∈ Σ such that Σ is smooth at [Γ] and such
that [Γ] corresponds to a reduced and irreducible plane curve of degree n with d
nodes, k cusps and no further singularities. Since the tautological family S0 → Σ0,
parametrized by Σ0, is an equigeneric family of curves, by normalizing the total
space, we get a family
// S0
// P2 × Σ0
of smooth curves of genus g =
−k−d. Because of the functorial properties of
the moduli space Mg of smooth curves of genus g, we get a regular map Σ0 → Mg,
sending every point [Γ] ∈ Σ0 to the isomorphism class of the normalization of the
plane curve Γ corresponding to the point [Γ]. This map extends to a rational map
ΠΣ : Σ 99K Mg.
We say that ΠΣ is the moduli map of Σ and we set
number of moduli of Σ := dim(ΠΣ(Σ)).
Date: 06 September 2005.
1991 Mathematics Subject Classification. 14H15; 14H10; 14B05.
Key words and phrases. families of plane curves, number of moduli, nodes and cusps.
http://arxiv.org/abs/0704.0618v1
2 CONCETTINA GALATI
Notice that, when Σnk,d is reducible, two different irreducible components of Σ
can have different number of moduli. We say that Σ has general moduli if ΠΣ is
dominant. Otherwise, we say that Σ has special moduli.
Definition 1.1. When Σ has the expected dimension equal to 3n+ g − 1 − k and
g ≥ 2, we say that Σ has the expected number of moduli if
dim(ΠΣ(Σ)) = min(dim(Mg), dim(Mg) + ρ− k),
where ρ := ρ(2, g, n) = 3n−2g−6 is the number of Brill-Noether of the linear series
of degree n and dimension 2 on a smooth curve of genus g.
As we shall see in the next section, when g ≥ 2 and when Σ has the expected
dimension equal to 3n+ g − 1 − k, the number of moduli of Σ is at most equal to
the expected one. This happens in particular if k < 3n. If k ≥ 3n, in general we
have not an upper-bound for the dimension of Σ and we cannot provide an upper
bound for the number of moduli of Σ, (see lemma 2.2 and remark 2.3). Moreover,
by classical Brill-Neother theory when ρ is positive and by a well know result of
Sernesi when ρ ≤ 0 (see [18]), we have that Σn0,d, (which is irreducible by [8]), has
the expected number of moduli for every d ≤
. When k > 0 there are known
results giving sufficient conditions for the existence of irreducible components Σ of
Σnk,d with general moduli, (see propositions 2.5 and 2.6 and corollary 2.7). In this
article we construct examples of families of irreducible plane curves with nodes and
cusps with finite and expected number of moduli. A large part of this paper is
obtained working out the main ideas and techniques that Sernesi uses in [18].
In section 2.1 we introduce the varieties Σnk,d and we recall their main properties.
In section 2.2 we discuss on definition 1.1 and we summarize known results on the
number of moduli of families of irreducible plane curves with nodes and cusps. In
theorem 3.5 we prove the existence of plane curves with nodes and cusps as singular-
ities whose singular points are in sufficiently general position to impose independent
linear conditions to a linear system of plane curves of a certain degree. This result
is related to the moduli problem by lemma 3.2, remark 3.4 and proposition 4.1,
where we find sufficient conditions in order that an irreducible component Σ ⊂ Σnk,d
has the expected number of moduli. If Σ verifies the hypotheses of proposition 4.1,
then the Brill-Neother number ρ is not positive and Σ has finite number of moduli.
Moreover, by lemma 4.6 and corollary 4.7, for every k′ ≤ k and d′ ≤ d+k−k′, there
is at least an irreducible component Σ′ ⊂ Σnk′,d′ , such that Σ ⊂ Σ
′ and the general
element [D] ∈ Σ′ corresponds to a plane curve D verifying hypotheses of propo-
sition 4.1 and so having the expected number of moduli. Finally, the main result
of this paper is contained in theorem 4.9, where, by using induction on the degree
n and on the genus g of the general curve of the family, we construct examples of
families of irreducible plane curves with nodes and cusps verifying the hypotheses
of proposition 4.1. In particular, we prove that, if k ≤ 6 and ρ ≤ 0, then Σnk,d has
at least an irreducible component which is not empty and which has the expected
number of moduli. This result may be improved and examples of families of curves
showing that the condition k ≤ 6 is not sharp are given in remark 4.10. Notice
that the previous theorem provides only examples of families of plane curves with
nodes and cusps with expected number of moduli, when ρ is not positive. When the
number of cusps k is very small, we expect it is possible to prove the existence of
irreducible components of Σnk,d with expected number of moduli, for every value of
ρ. For example, from a result of Eisenbud and Harris, it follows that Σn1,d, (which is
irreducible by [16]), has general moduli if ρ ≥ 2, (see corollary 2.7). In theorem 4.11,
by using induction on n we find that Σn1,d has general moduli also when ρ = 1. By
recalling that, by theorem 4.9, Σn1,d has expected number of moduli when ρ ≤ 0, we
conclude that Σn1,d has the expected number of moduli for every ρ or, equivalently,
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 3
for every d ≤
− 1. We still don’t know examples of irreducible components of
Σnk,d having number of moduli smaller that the expected.
2. Preliminaries
2.1. On Severi-Enriques varieties. We shall denote by PN = P
n(n+3)
2 the Hilbert
scheme of plane curves of degree n, by [Γ] ∈ PN the point parametrizing a plane
curve Γ ⊂ P2 and by Σnk,d ⊂ P
N the closure, in the Zariski topology, of the locally
closed set parametrizing reduced and irreducible plane curves of degree n with d
nodes and k cusps as singularities. These varieties have been introduced at the
beginning of the last century by Severi and Enriques. In particular, the case k = 0
has been studied first by Severi and for this reason the varieties Σn0,d are usually
called Severi varieties, while for k > 0 the varieties Σnk,d are called Severi-Enriques
varieties. We recall that every irreducible component Σ of Σnk,d has dimension at
least equal to
N − d− 2k = 3n+ g − 1− k,
where g =
−k−d. When the equality holds we say that Σ has expected dimen-
sion. Moreover, it is well known that if k < 3n then every irreducible component
Σ of Σnk,d has expected dimension, (see for example [23] or [25]). On the contrary,
when k ≥ 3n, there exist examples of irreducible components of Σnk,d having di-
mension greater than the expected, (see [25]). Moreover, we recall that Σn0,d is not
empty for every d ≤
and it contains in its closure all points parameterizing
irreducible plane curves of degree n and genus g =
− d, (see [24], [25] and
[1]). Often, we shall denote Σn0,d by Vn,g. While the proof of the existence of Vn,g
is quite elementary and it is due to Severi, the irreducibility of Vn,g remained an
open problem for a long time and it has been proved by Harris only in 1986. Later,
by using the same techniques of Harris, Kang has proved the irreducibility of Σnk,d
with k ≤ 3, see [8] and [16]. However, in general, Σnk,d is reducible and there exist
values of n, d and k such that Σnk,d is empty, (see [25], [12], [20], [11] or chapter 2
of [7] and related references). Finally, we recall that, if Σ ⊂ Σnk,d is a non-empty
irreducible component of the expected dimension equal to 3n + g − 1 − k, then,
for every k′ ≤ k and d′ ≤ d + k − k′, there exists a non-empty irreducible compo-
nent Σ′ ⊂ Σnk′,d′ such that Σ ⊂ Σ
′. This happens in particular if k < 3n. More
precisely, it is true that, if Γ ⊂ P2 is a reduced (possibly reducible) plane curve of
degree n with k < 3n cusps at points q1, . . . , qk, nodes at points p1, . . . , pd and no
further singularities, then, chosen arbitrarily k1 cusps, say q1, . . . , qk1 among the
k cusps of Γ, k2 cusps qk1+1, . . . , qk2 among qk1+1, . . . , qk and d1 nodes p1, . . . , pd1
among the nodes of Γ, there exists a family of reduced plane curves D → B ⊂ PN
of degree n, whose special fibre is D0 = Γ and whose general fibre Dt = D has a
node in a neighborhood of every marked node of Γ, a cusp in a neighborhood of
each point q1, . . . , qk1 , a node in a neighborhood of each point qk1+1, . . . , qk2 and
no further singularities, (see [25], corollary 6.3 of [11] or lemma 3.17 of chapter 2
of [7]). To save space, we shall say that the family D → B is obtained from Γ by
preserving the singularities q1, . . . , qk1 and p1, . . . , pd1 , by deforming in a node each
cusp qk1+1, . . . , qk2 and by smoothing the other singularities.
2.2. Known results on the number of moduli of Σnk,d. In order to explain
the definition 1.1, we need to recall some basics of Brill-Noether theory. Given a
smooth curve C of genus g, the set G2n(C) of linear series g
n on C of dimension 2
and degree n, is a projective variety which verifies the following properties:
(1) G2n(C) is not empty of dimension at least ρ, if ρ(2, n, g) = 3n− 2g − 6 ≥ 0,
(see theorem V.1.1 and proposition IV.4.1 of [4]).
4 CONCETTINA GALATI
(2) Let g2n be a given linear series, letH ∈ g
n be a divisor and letW ⊂ H
0(C,H)
be the three dimensional vector space corresponding to g2n. Denoting by
ωC = OC(KC) the canonical sheaf of C and by
µo,C :W ⊗H
0(C, ωC(−H)) → H
0(C, ωC)
the natural multiplication map, also called the Brill-Noether map of the
pair (C,W ), we have that the dimension of the tangent space to G2n(C) at
the point [g2n],corresponding to g
n, is equal to
dim(T[g2
n(C)) = ρ+ dim(ker(µ0,C)),
(see [2] or proposition IV.4.1 of [4] for a proof).
(3) Moreover, if C is a curve with general moduli (i.e. if [C] varies in an open
set of Mg), the variety G
n(C) is empty if ρ < 0, it consists of a finite
number of points if ρ = 0 and it is reduced, irreducible, smooth and not
empty variety of dimension exactly ρ, when ρ ≥ 1, (see theorem V.1.5 and
theorem V.1.6 of [4]). In the latter case, the general g2n on C defines a local
embedding on C and it maps C to P2 as a nodal curve, (see theorem 3.1 of
[1] or lemma 3.43 of [9]).
From (3), we deduce that, the Severi variety Σn0,d = Vn,g of irreducible plane curves
of genus g =
− d, has general moduli when ρ ≥ 0 and it has special moduli
when ρ < 0. When ρ < 0, and then g ≥ 3, by definition 1.1, we expect that the
image of Vn,g into Mg has codimension exactly −ρ. Equivalently, recalling that, in
this case,
dim(Vn,g) = 3n+ g − 1 = 3g − 3 + ρ+ 8 = dim(Mg) + ρ+ dim(Aut(P
we expect that on the smooth curve C, obtained by normalizing the plane curve
corresponding to the general element of Vn,g, there is only a finite number of g
mapping C to the plane as a nodal curve. This is a well known result proved by
Sernesi in [18].
Theorem 2.1 (Sernesi, [18]). The Severi variety Vn,g = Σ
0,d of irreducible plane
curves of degree n and genus g =
− d has number of moduli equal to
min(dim(Mg), dim(Mg) + ρ).
What can we say about the number of moduli of an irreducible component Σ of
Σnk,d, when k > 0? In this case we need to distinguish the two cases k < 3n and
k ≥ 3n. In the first case we have the following result.
Lemma 2.2. For every not empty irreducible component Σ of Σnk,d, with k < 3n
and g =
− k − d ≥ 2, the number of moduli of Σ is at most equal to
min(dim(Mg), dim(Mg) + ρ− k),
where ρ = 3n − 2g − 6 is the Brill-Neother number of moduli of linear series of
dimension 2 and degree n on a smooth curve of genus g.
Proof. We recall that an ordinary cusp P of a plane curve Γ corresponds to a simple
ramification point p of the normalization map φ : C → Γ, i.e. to a simple zero of
the differential map dφ. If we denote by G2n,k(C) ⊂ G
n(C) the set of g
n on C
defining a birational morphism with k simple ramification points, then G2n,k(C) is
a locally closed subset of G2n(C) and every irreducible component G of G
n,k(C) has
dimension at least equal to ρ− k, if it is not empty. In particular, if F 2n,k(C) is the
variety whose points correspond to the pairs ([g2n], {s0, s1, s2}) where [g
n] ∈ G
n,k(C)
and {s0, s1, s2} is a frame of the three dimensional space associated to the linear
series g2n, then every irreducible component of F
n,k(C) has dimension at least equal
min(8, ρ− k + 8).
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 5
Now, let Σ be one of the irreducible components of Σnk,d and let [Γ] be a general
point of Σ. Then, if Γ ⊂ P2 is the corresponding plane curve and φ : C → Γ is the
normalization map, then the fibre over the point [C] ∈ Mg of the moduli map
ΠΣ : Σ 99K Mg
consists of an open set in one or more irreducible components of F 2n,k(C). In partic-
ular, every irreducible component of the general fibre of ΠΣ has dimension at least
equal to min(8, ρ− k + 8). Moreover, if k < 3n then Σ has the expected dimension
equal to N−d−2k = 3n+g−1−k, (see [25] or [23]). Finally, if g =
−k−d ≥ 2,
dim(Σ) = 3n+ g − 1− k = 3g − 3 + ρ− k + 8.
This proves the statement. �
Remark 2.3. The proof of the previous lemma still holds if k ≥ 3n but Σ has the
expected dimension. However in general, when k ≥ 3n, we don’t have a bound for
dim(ΠΣ(Σ)). Indeed, in this case the dimension of the general fibre of the moduli
map of Σ is still at least equal to ρ− k + 8, but Σ may have dimension larger than
3n+ g − 1 − k. Anyhow, by the following proposition, every not empty irreducible
component of Σnk,d has special moduli if k ≥ 3n.
Proposition 2.4 (Arbarello-Cornalba, [1]). Let C be a general curve of genus
g ≥ 2 and let φ : C → P2 be a birational morphism, then the degree of the zero
divisor of the differential map of φ is smaller than ρ. In particular, every irreducible
component of Σnk,d has special moduli if ρ = 3n− 2g − 6 < k.
A sufficient condition for the existence of irreducible families of plane curves with
nodes and cusps with general moduli is given by the following result.
Proposition 2.5 (Kang, [15]). Σnk,d is irreducible, not empty and with general
moduli if n > 2g − 1 + 2k, where g =
− d− k.
Actually, in [15], Kang proves that if n > 2g−1+2k, then Σnk,d is not empty and
irreducible. But from his proof it follows that, under the hypothesis of proposition
2.5, Σnk,d has general moduli because the general element of Σ
k,d corresponds to a
curve which is a projection of an arbitrary smooth curve C of genus g in Pn−g, from
a general (n − 3)-plane intersecting the tangent variety of C in k different points.
Another result which may be used to find examples of families of plane curves with
nodes and cusps having general moduli is the following. Let grn be a linear series on
C associated to a (r+1)-spaceW ⊂ H0(C,L), where L is an invertible sheaf on C,
and let {s0, . . . , sr} be a basis of W , then the ramification sequence of the g
n at p
is the sequence b = (b0, . . . , br) with bi = ordpsi − i. Choosing another basis of W ,
the ramification sequence of grn at p doesn’t change. We say that the ramification
sequence of the grn at p is at least equal to b = (b0, . . . , br) if bi ≤ ordpsi − i, for
every i, and we write (ordps0, . . . , ordpsr − r) ≥ (b0, . . . , br).
Proposition 2.6 (Proposition 1.2 of [10]). Let C be a general curve of genus g,
let p be a general point on C and let b = (b0, . . . , br) be any ramification sequence.
There exists a grn on C having ramification at least b at p if and only if
(bi + g − n+ r)+ ≤ g,
where (−)+ := max(−, 0).
From proposition 2.6, we easily deduce the following result.
Corollary 2.7. Suppose that k ≤ 3 and ρ = 3n− 2g − 6 ≥ 2k. Then Σnk,d is not
empty, irreducible and it has general moduli.
6 CONCETTINA GALATI
Proof. By [16], the variety Σnk,d is irreducible for every k ≤ 3 and d ≤
Moreover, by using classical arguments, one can prove that Σnk,d is not empty if
k ≤ 4 and d ≤
− 4, (see, for example, corollary 3.18 of chapter two of [7]).
Finally, by theorem 1.1 of [21], by using the terminology of proposition 2.6, under
the hypothesis k ≤ 3n − 4, in particular if k ≤ 3, the variety Σnk,d contains every
point of PN corresponding to a plane curve Γ of genus g =
− k − d such that
the normalization morphism of Γ has at least a ramification point with ramification
sequence (b0, b1, b2) ≥ (0, k, k). Then, by proposition 2.6, if ρ ≥ 2k and k ≤ 3, the
moduli map of Σnk,d is surjective. �
3. On the existence of certain families of plane curves with nodes
and cusps in sufficiently general position
As we already observed, we don’t have a complete answer for the existence prob-
lem of Σnk,d. In this section we are interested in a little more specific existence
problem. We shall prove the existence of plane curves with nodes and cusps as
singularities whose singular points are in sufficiently general position to impose
independent linear conditions to a linear system of plane curves of a certain degree.
Definition 3.1. A projective curve C ⊂ Pr is said to be geometrically t-normal if
the linear series cut out on the normalization curve C̃ of C by the pull-back to C̃
of the linear system of hypersurfaces of Pr of degree t is complete.
From a geometric point of view, a projective curve C ⊂ Pr is geometrically t-
normal if and only if the image curve νt,r(C) of C by the Veronese embedding
νt,r : P
r → P(
t ) of degree t, is not a projection of a non-degenerate curve living
in a higher dimensional projective space. We shall say that a curve is geometrically
linearly normal (g.l.n. for short) if it is geometrically 1-normal. Every such a curve
C is not a projection of a curve lying in a projective space of larger dimension.
The following result is proved under more general hypotheses in [5], theorem 2.1.
Lemma 3.2. Let Γ ⊂ P2 be an irreducible and reduced plane curve of degree n and
genus g with at most nodes and cusps as singularities. Let t be an integer such that
n− 3 − t < 0, then Γ is geometrically t-normal if and only if it is smooth. On the
contrary, if n − 3 − t ≥ 0, the plane curve Γ is geometrically t-normal if and only
if its singular points impose independent linear conditions to plane curves of degree
n− 3− t.
We recall the following classical definition.
Definition 3.3. Let Γ ⊂ P2 be a plane curve of degree n with d nodes at p1, . . . , pd
and k cusps at q1, . . . , qk as singularities. Let φ : C → Γ be the normalization of
Γ. The adjoint divisor ∆ of φ is the divisor on C defined by ∆ =
i=1 φ
−1(pi) +
j=1 2φ
−1(qj).
Proof of lemma 3.2. Let Γ be a plane curve as in the statement of the lemma. Then,
Γ is geometrically t-normal if and only if, by definition,
h0(C,OC(t)) = h
0(P2,OP2(t))− h
0(P2, IΓ(t))
where IΓ is the ideal sheaf of Γ in P
2 and OC(t) := OC(tφ
∗(H)), where H is the
general line of P2. By Riemann-Roch theorem, Γ is geometrically t-normal if and
only if
(2) h0(C, ωC(−t))) = −nt+ g − 1 +
(t+ 1)(t+ 2)
− h0(P2, IΓ(t)),
where g is the geometric genus of Γ and ωC is the canonical sheaf of C. On the other
hand, it is well known that H0(C, ωC(−t)) = H
0(C,OC(n− 3− t)(−∆)), where ∆
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 7
is the adjoint divisor of φ, (see definition 3.3 and [4], appendix A). If n− 3− t < 0
then h0(C,OC(n− 3− t)) = 0 and Γ is geometrically t-normal if and only if
h0(P2,OP2(t)) − h
0(P2, IΓ(t)) = nt−
n2 − 3n
where δ =
− g = deg(∆)/2. This equality is verified if and only if δ = 0, i.e.
Γ is smooth. If n− 3 ≥ t, h0(P2, IΓ(t)) = 0 and (2) is verified if and only if
h0(C,OC(n− 3− t)(−∆)) = h
0(P2,OP2(n− 3− t))− δ.
On the other hand, if ψ : S → P2 is the blowing-up of the plane at the singular
locus of Γ, denoting by
Ei the pullback of the singular locus of Γ with respect
to ψ and by OS(r) the sheaf OS(rψ
∗(H)), we have that
h0(C,OC(n−3−t)(−∆)) = h
0(S,OS(n−3−t)(−
Ei)) = h
0(P2,OP2(n−3−t)⊗A)
where A is the ideal sheaf of singular points of Γ. �
Remark 3.4. Notice that, if an irreducible and reduced plane curve Γ of degree n
with only nodes and cusps as singularities is geometrically t-normal, with t ≤ n−3,
then it is geometrically r-normal for every r ≤ t. Indeed, if a set of points imposes
independent linear conditions to a linear system S, then it imposes independent
linear conditions to every linear system S′ containing S.
Theorem 3.5. Let Σnk,d be the variety of irreducible and reduced plane curves of
degree n with d nodes and k cusps. Suppose that d, k, n and t are such that
d+ k ≤
n2 − (3 + 2t)n+ 2 + t2 + 3t
= h0(OP2(n− t− 3))(3)
t ≤ n− 3 if k = 0,(4)
k ≤ 6 if t = 1, 2 and(5)
k ≤ 6 + [
], if t = 3,(6)
where [−] is the integer part of −. Then the variety Σnk,d is not empty and there ex-
ists at least an irreducible component W ⊂ Σnk,d whose general element corresponds
to a geometrically t-normal plane curve.
Remark 3.6. As we shall see in the next section, (see proposition 4.1), the geomet-
ric linear normality of the plane curve corresponding to the general element of an
irreducible component Σ of Σnk,d, is related with the number of moduli of Σ. Another
motivation for the previous theorem has been the family of irreducible plane sextics
with six cusps. By [25], we know that Σ66,0 contains at least two irreducible compo-
nents Σ1 and Σ2. The general point of Σ1 corresponds to a sextic with six cusps on
a conic, whereas the general element of Σ2 corresponds to a sextic with six cusps
not on a conic. Note that, by the previous lemma the general element of Σ2 param-
eterizes a geometric linearly normal sextic, unlike the general element of Σ1, which
corresponds to a projection of a canonical curve of genus four. Theorem 3.5, proves
in particular that, under a suitable restriction, (see inequality (3)), on the genus
of the curve corresponding to the general element of the family and, if the number
of the cusps is small, the variety Σnk,d contains a not empty irreducible component
whose general element corresponds to a curve which is not a projection of an other
curve, lying in a projective space of larger dimension. We notice that the inequality
(3) of the previous theorem can’t be improved. Indeed, if g =
− k − d, then
k + d > h0(P2,OP2(n− 3 − t)) if and only if g <
2tn−t2−3t
. On the other hand, by
using the same notation as in theorem (3.5), if g < 2tn−t
, then, by Riemann-
Roch theorem, we have that h0(C,OC(t)) ≥ tn− g+1 >
+1 = h0(P2,OP2(t)).
On the contrary, inequalities (5) and (6) are not sharp, (see example 3.7).
8 CONCETTINA GALATI
In the case of k = 0 and t = 1, theorem 3.5 has been proved by Sernesi in [18],
section 4. The case k = 0 and t ≤ n−3 is already contained in [5]. To show theorem
3.5, we proceed by induction on the degree n and on the number of nodes and cusps
of the curve. The geometric idea at the base of the induction on the degree of the
curve is, mutatis mutandis, the same as that of Sernesi.
Proof of theorem 3.5. Let t be a positive integer such that n−3−t ≥ 0 and letW ⊂
Σnk,d be an irreducible component of Σ
k,d. By standard semicontinuity arguments
it follows that, if there exists a point [C] ∈ W corresponding to a geometrically
t-normal curve with only k cusps and d nodes as singularities, then the general
element of W corresponds to a geometrically t-normal plane curve. Moreover, if the
theorem is true for fixed n, t ≤ n − 3, k as in (5) or in (6) and k + d as in (3),
then the theorem is true for n, t and any k′ ≤ k and d′ ≤ d+ k − k′. Indeed, from
the hypotheses (3), (5) and (6), it follows in particular that k < 3n. By section 2.1,
under this hypothesis, for every k′ ≤ k and for every d′ ≤ d + k − k′, there exists
a family of plane curves C → ∆ of degree n, parametrized by a curve ∆ ⊂ Σnk′,d′ ,
whose special fibre is C0 = C and whose general fibre Cz has d
′ nodes and k′ cusps
as singularities. The statement follows by applying the semicontinuity theorem to
the family C̃ → ∆̃, obtained by normalizing the total space of the pull-back family
of C → ∆ to the normalization curve ∆̃ of ∆. Finally, it’s enough to show the
theorem when the equality holds in (5), (6) and (3).
First of all we consider the case k = 0. We will show the statement for any fixed t
and by induction on n. Let, then t ≥ 1 and n = t+3. In this case the equality holds
in (3) if d = 1 = h0(P2,OP2). Since one point imposes independent linear conditions
to regular functions, by using lemma 3.2, we find that every irreducible plane curve
of degree n = t + 3 with one node and no further singularities is geometrically t-
normal. So, the first step of the induction is proved. Suppose, now, that the theorem
is true for n = t+3+a and let [Γ] ∈ Vn,g be a point corresponding to a geometrically
t-normal curve with a
2+3a+2
nodes. Let D be a line which intersects transversally
Γ and let P1, ..., Pt+1 be t + 1 marked points of Γ ∩ D. If Γ
′ = Γ ∪ D ⊂ P2, then
P1, ..., Pt+1 are nodes for Γ
′. Let C → Γ be the normalization of Γ and C′ → Γ′ the
partial normalization of Γ′, obtained by smoothing all singular points of Γ′, except
P1, ..., Pt+1. We have the following exact sequence of sheaves on C
(7) 0 → OD(t)(−P1 − ...− Pt+1) → OC′(t) → OC(t) → 0,
where OC′(t) := OC′(tH) and H is the pull-back with respect to C
′ → Γ′ of general
line of P2. Since deg(OD(t)(−P1 − ...− Pt+1)) < 0, we get that
h0(D,OD(t)(−P1 − ...− Pt+1)) = 0
and so
(8) h0(C′,OC′(t)) = h
0(C,OC(t)) = h
0(P2,OP2(t)).
Now, by section 2.1, we can obtain Γ′ as the limit of a 1-parameter family of
irreducible plane curves
ψ : C → ∆ ⊂ P
(n+1)(n+4)
of degree n+ 1 = t+ a+ 4 with
a2 + 3a+ 2
+ n− t− 1 =
(a+ 1)2 + 3(a+ 1) + 2
= h0(P2,OP2(n+ 1− t− 3))
nodes specializing to nodes of Γ′ different from the marked points P1, ..., Pt+1. More-
over, one can prove that ∆ is smooth, (see [24] or [25]). Normalizing C, we obtain a
family whose general fibre is smooth and whose special fibre is exactly C′, and we
conclude the inductive step by (8) and by semicontinuity theorem.
Now we consider the case t = 1, 2 or 3 and k as in (5) and in (6). Suppose the
theorem is true for n and let [Γ] ∈ Σnk,d be a general point in one of the irreducible
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 9
components of Σnk,d. Then, let D be a smooth plane curve of degree t if t = 1, 2 or
an irreducible cubic with a cusp if t = 3. By the generality of Γ, we may suppose
that D intersects Γ transversally. Let P1, ..., Pt2+1 be t
2+1 fixed points of Γ∩D. If
Γ′ = Γ ∪D, then P1, ..., Pt2+1 are nodes for Γ
′. Let C → Γ be the normalization of
Γ and C′ → Γ′ the partial normalization of Γ′, obtained by smoothing all singular
points except P1, ..., Pt2+1. By using the same notation and by arguing as before,
from the following exact sequence of sheaves on C′
0 → OD(t)(−P1 − ...− Pt2+1) → OC′(t) → OC(t) → 0,
we deduce that
(9) h0(C′,OC′(t)) = h
0(C,OC(t)) = h
0(P2,OP2(t)).
Now, by section 2.1, we can obtain Γ′ as limit of a family of irreducible plane curves
φ : C → ∆
of degree n+ t with d+ nt− t2 − 1 =
(n+t)2+(3+2t)(n+t)+t2+3t+2
nodes specializing
to nodes of Γ′ different to P1, ..., Pt2+1, and k +
2−3t+2
cusps specializing to cusps
of Γ. We conclude by (9) and by semicontinuity, as before. Now we have to show
the first step of the induction. For t = 1 the induction begins with the cases
(n, k) = (4, 1), (5, 3), (6, 6). Trivially, if n = 4 and k = 1 one point imposes
independent conditions to the linear system of regular functions. If n = 5 and
k = 3 we have to show that there are irreducible quintics with three cusps not on
a line. A quintic with three cusps is a projection of the rational normal quintic
C5 ⊂ P
5 from a plane generated by three points lying on three different tangent
lines to C5. By Bezout theorem the three cusps of such a plane curve can’t be
aligned. If n = k = 6, one can repeat the classical argument used by Zariski, see
[24] or example 3.20 of chapter 2 of [7]. For t = 2 we have to show the theorem
for (n, k) = (5, 1), (6, 3), (7, 6), (8, 6), while for t = 3 we have to show the theorem
for (n, k) = (6, 1), (7, 3), (8, 6), (9, 6), (10, 6). The case t = 2 and (n, k) = (5, 1) is
trivial. When t = 2, n = 6 and k = 3 we have that n − 3 − t = 1. To show that
there exists an irreducible sextic with three cusps not on a line, consider a rational
quartic C4 with three cusps, (see corollary 3.18 of chapter 2 of [7] for the existence).
By Bezout theorem, the three double points of C4 can’t be aligned. Then consider
a sextic C6 which is union of C4 and a conic C2 which intersects C4 transversally.
By section 2.1, one can smooth the intersection points of C4 and C2 obtaining a
family of sextics with three cusps not on a line. For t = 2, n = 7 and k = 6 we
argue as in the previous case, by using a sextic C6 with six cusps not on a conic
and a line R with intersects C6 transversally. Similarly for t = 2 , n = 8 and k = 6
and t = 3 and (n, k) = (6, 1), (7, 3), (8, 6), (9, 6), (10, 6). �
Example 3.7. Inequalities (5) and (6) are not sharp. To see this, we can consider
the example of curves of degree 10. We recall that we say that a plane curve is
geometrically linearly normal (g.l.n. for short) if it is geometrically 1-normal. The-
orem 3.5 ensures the existence of g.l.n. irreducible plane curves of degree 10 with
k ≤ 6 cusps and nodes as singularities. But, by using the same ideas as we used
in theorem 3.5, one can prove the existence of g.l.n. plane curves of degree 10 with
nodes and k ≤ 9 cusps. It is enough to consider a sextic Γ6 with six cusps not on a
conic and a rational quartic Γ4 with three cusps intersecting Γ6 transversally. We
choose five points P1, . . . , P5 of Γ4 ∩ Γ6. If Γ
6 and Γ
4 are the normalization curves
of Γ6 and Γ4 respectively and C
′ is the partial normalization of Γ6 ∪Γ4 obtained by
normalizing all its singular points except P1, . . . , P5, by considering the following
exact sequence
0 → OΓ′4(1)(−P1 − · · · − P5) → OC′(1) → OΓ′6(1) → 0
10 CONCETTINA GALATI
we find that h0(C′,OC′(1)) = 3. By using terminology of section 2.1, the statement
follows by smoothing the singular points P1, . . . , P5 of Γ6 ∪Γ4, and by semicontinu-
ity, as in the proof of theorem 3.5. The bound on the number of cusps of theorem
3.5 can be improved also for t = 2 or t = 3. For example, theorem 3.5 ensures
the existence of geometrically 3-normal curves of degree 12 with k ≤ 6 and nodes
as further singularities. But, by considering a geometrically 3-normal curve of de-
gree 8 with six cusps and a quartic with 3 cusps and arguing as before, we can find
geometrically 3-normal irreducible plane curves of degree 12 with nodes and k ≤ 9
cusps.
4. Families of plane curves with nodes and cusps with finite and
expected number of moduli.
Let Σ ⊂ Σnk,d be an irreducible component of Σ
k,d. We want to give sufficient
conditions for Σ to have the expected number of moduli. Let [Γ] ∈ Σ be a general
element, corresponding to a plane curve Γ with normalization map φ : C → Γ. We
shall denote by ωC the canonical sheaf of C and by OC(1) the sheaf associated to
the pullback to C of the divisor cut out on Γ from the general line of P2.
Proposition 4.1. Let Σ ⊂ Σnk,d be an irreducible component of Σ
k,d and let [Γ] ∈ Σ
be a general element, corresponding to a plane curve Γ with normalization map
φ : C → Γ. Suppose that Σ is smooth of the expected dimension equal to 3n+g−1−k
at [Γ]. Moreover, suppose that:
(1) Γ is geometrically linearly normal, i.e. h0(C,OC(1)) = 3,
(2) the Brill-Noether map
µo,C : H
0(C,OC(1))⊗H
0(C, ωC(−1)) → H
0(C, ωC)
is surjective.
Then Σ has the expected number of moduli equal to 3g − 3 + ρ− k.
Proof. The case k = 0 has been proved by Sernesi in [18], section 4. We shall
assume k > 0. Let Γ be a plane curve verifying the hypotheses of the proposition.
By lemma 1.5.(b) of [22], the hypothesis that Σ is smooth of the expected dimension
at [Γ] implies the vanishing H1(C,Nφ) = 0, where Nφ if the normal sheaf of φ. We
recall that, denoting by ΘC and ΘP2 the tangent sheaf of C and P
2 respectively,
then the normal sheaf of φ is defined as the cokernel of the differential map φ∗ of φ
(10) 0 → ΘC
→ φ∗ΘP2 → Nφ → 0
By theorem 3.1 of [13], the vanishing H1(C,Nφ) = 0 is a sufficient condition for the
existence of a universal deformation family
of the normalization map φ, whose parameter space B is smooth at the point 0
corresponding to φ, with tangent space at 0 equal to H0(C, Nφ). On the contrary,
by [3], p. 487, the Severi variety Vn,g = Σ
0,k+d of irreducible plane curves of genus
− d − k is singular at the point [Γ] and the universal deformation space
B of φ is a desingularization of Vn,g at [Γ]. Moreover, by corollary 6.11 of [2],
if Bk = F
−1(Σ) is the locus of points of B corresponding to a morphism with k
ramification points, then the tangent space to Bk at 0 is a subspaceW of H
0(C,Nφ)
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 11
of codimension k such that W ∩H0(C,Kφ) = 0, where Kφ is the torsion subsheaf
of Nφ. By [3], p.487, it follows that, if
F : B → Vn,g
is the natural (1 : 1)-map from B to Vn,g, then the differential map
dF : H0(C, Nφ) → T[Γ]Vn,g
restricts to an isomorphism between W and the tangent space T[Γ]Σ to Σ at [Γ].
We can now go back to the number of moduli of Σ. From the exact sequence
(10), by using that H1(C, Nφ) = 0, we get the following long exact sequence
0 → H0(C,ΘC) → H
0(C, φ∗ΘP2) → H
0(C,Nφ)
→ H1(C,ΘC) → H
1(C, φ∗ΘP2) → 0
Recalling that the space H1(C,ΘC) is canonically identified with the tangent space
T[C]Mg to Mg at the point associated to the normalization C of Γ, the coboundary
map δC : H
0(C,Nφ) → H
1(C,ΘC) sends the Horikawa class of an infinitesimal
deformation of φ to the Kodaira-Spencer class of the corresponding infinitesimal
deformation of C. So, δC |W is the differential map at the point 0 ∈ B of the
moduli map ΠΣ ◦ F : Bk = F
−1(Σ) 99K Mg. Since the point [Γ] is general in Σ,
and recalling the isomorphism dF :W
→ T[Γ]Σ, we have that
the number of moduli of Σ = dim(δC(W )).
Now, from the exact sequence (10), we have that
dim(δC(H
0(C,Nφ)) = 3g − 3− h
1(C, φ∗ΘP2).
Moreover, from the pull-back to C of the Euler exact sequence, we deduce the well
known isomorphism
H1(C, φ∗ΘP2) ≃ coker(µ
0,C) ≃ (ker(µ0,C))
and we conclude that
(11) dim(δC(H
0(C,Nφ))) = 3g − 3− dim(ker(µ0,C)).
Notice that the previous equality is always true, even if Γ doesn’t verify the hypoth-
esis (1) or (2) of the statement. Moreover, if Γ is geometrically linearly normal, i.e.
if h0(C,OC(1)) = 3, we have that
ρ = 3n− 2g − 6 = dim(coker(µo,C))− dim(ker(µo,C)).
When µo,C is surjective, ρ = − dim(ker(µo,C)) and
(12) dim(δC(H
0(C,Nφ)) = 3g − 3 + ρ = dim(B)− 8 = dim(Vn,g)− 8.
Since the dimension of the fibre of the moduli map
ΠVn,g ◦ F : B 99K Mg
has dimension at least equal to 8 = dim(Aut(P2)), from (12) we deduce that the
differential map of ΠVn,g ◦F has maximal rank at 0 and, in particular, we have that
dim((ΠVn,g ◦ F )
−1([C])) = 8. Equivalently, there exist only finitely many g2n on C.
It follows that there are only finitely many g2n on C mapping C to the plane as a
curve with k cusps and d nodes. Then,
dim(δc(W )) = dim(ΠΣ(Σ)) = 3g − 3 + ρ− k.
Remark 4.2. Arguing as in the proof of the previous proposition, it has been proved
in [18] that, if Γ is a geometrically linearly normal plane curve with only d nodes
as singularities and the Brill-Noether map µo,C of the normalization morphism of
Γ is injective, then Σ = Σn0,d has general moduli. If Σ ⊂ Σ
k,d and [Γ] ∈ Σ verify
the hypotheses of proposition 4.1 but we assume that µo,C is injective, we may
only conclude that ΠVn,g ◦ F is dominant with surjective differential map at [Γ].
12 CONCETTINA GALATI
So dim(Π−1Vn,g ([C])) = ρ + 8. But this is not useful to compute the dimension of
δC(W ) = δC(T[Γ]Σ). However, in this case we get that
δC(T[Γ]Σ) + δC(H
0(C,Kφ)) = δC(H
0(C,Nφ)) = H
1(C,ΘC).
Then, by using that dim(δC(H
0(C,Kφ))) ≤ k and by recalling that if Σ has the
expected dimension then the number of moduli of Σnk,d is at most the expected one
(see lemma 2.2 and remark 2.3), we find that
3g − 3− k ≤ number of moduli of Σ ≤ 3g − 3 + ρ− k.
Remark 4.3. Notice that, if a plane curve Γ of genus g verifies the hypotheses
(1) and (2) of the previous proposition, then the Brill-Noether number ρ(2, g, n)
is not positive and, in particular, g ≥ 3. We don’t know examples of complete
irreducible families Σ ⊂ Σnk,d with the expected number of moduli whose general
element [Γ] corresponds to a curve Γ of genus g, with ρ(2, g, n) ≤ 0, which doesn’t
verify properties (1) and (2).
Lemma 4.4 ([5], Corollary 3.4). Let Γ be an irreducible plane curve of degree n
with only nodes and cusps as singularities and let φ : C → Γ be the normalization
morphism of Γ. Suppose that Γ is geometrically 2-normal, i.e. h0(C,OC(2)) = 6.
Then the Brill-Noether map
µo,C : H
0(C,OC(1))⊗H
0(C, ωC(−1)) → H
0(C, ωC)
is surjective.
Proof. By lemma 3.2, the curve Γ is geometrically 2-normal if and only if the scheme
N of the singular points of Γ imposes independent linear conditions to the linear
systemH0(P2,OP2(n−5)) of plane curves of degree n−5. SinceH
0(P2,OP2(n−5)) ⊂
H0(P2,OP2(n−4)), N imposes independent linear conditions plane curves of degree
n−4, and, by using lemma 3.2, we get that h0(C,OC(1)) = 3, i.e. Γ is geometrically
linearly normal. Now, denote by IN |P2 the ideal sheaf of N . Notice that the curve
Γ is geometrically 2-normal if and only if the ideal sheaf IN |P2(n− 4) is 0-regular,
(in the sense of Castelnuovo-Mumford). Indeed, since h2(P2, IN |P2(n− 6)) = 0, the
ideal sheaf IN |P2(n− 4) is 0-regular if and only if h
1(P2, IN |P2(n− 5)) = 0. Because
of the 0-regularity of IN |P2(n− 4), we have the surjectivity of the natural map
H0(P2, IN |P2(n− 4))⊗H
0(P2,OP2(1)) → H
0(P2, IN |P2(n− 3)),
(see [17]). Finally, by the geometric linear normality of Γ, the vertical maps of the
following commutative diagram
H0(P2,OP2(1))⊗H
0(P2, IN |P2(n− 4)) //
H0(P2, IN |P2(n− 3))
H0(C,OC(1))⊗H
0(C, ωC(−1))
// H0(C, ωC)
are surjective and, hence, the Brill-Noether map µo,C is surjective too. �
Corollary 4.5. Let Σ ⊂ Σnk,d be an irreducible component of Σ
k,d of dimension
equal to 3n + g − 1 − k, such that the general point [Γ] ∈ Σ corresponds to a
geometrically 2-normal plane curve. Then Σ has the expected number of moduli
equal to 3g − 3 + ρ− k.
Proof. It follows from proposition 4.1 and lemma 4.4. �
In order to produce examples of families of irreducible plane curves with nodes
and cusps with the expected number of moduli, we study how increases the rank of
the Brill-Noether map by smoothing a node or a cusp of the general curve of the
family, (in the sense of section 2.1).
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 13
Let Σ ⊂ Σnk,d, with n ≥ 5, be an irreducible component of Σ
k,d, let [Γ] ∈ Σ be a
general point of Σ and let φ : C → Γ be the normalization of Γ. Choose a singular
point P ∈ Γ and denote by φ′ : C′ → Γ the partial normalization of Γ obtained by
smoothing all singular points of Γ, except the point P . If ωC′ is the dualizing sheaf
of C′ and
µo,C′ : H
0(C′,OC′(1))⊗H
0(C′, ωC′(−1)) → H
0(C′, ωC′),
is the natural multiplication map, we have the following result.
Lemma 4.6. If h0(C,OC(1)) = 3 and the geometric genus g of C is such that g >
n−2, with n ≥ 5, then rk(µo,C′) ≥ rk(µo,C)+1. In particular, if h
0(C,OC(1)) = 3,
n ≥ 5 and µo,C is surjective, then µo,C′ is also surjective.
Proof. Let ψ : C → C′ be the normalization map.
We recall that, if we set φ∗(P ) := p1 + p2 when P is a node and φ
∗(P ) = 2φ−1(P )
when P is a cusp, then the dualizing sheaf of C′ is a subsheaf of ψ∗(ωC(φ
∗(P ))),
(see for example [10], p.80). In particular we have the following exact sequence
(13) 0 → ωC′ → ψ∗ωC(φ
∗(P )) → CP → 0
where CP is the skyscraper sheaf on C with support at P . From this exact sequence,
we deduce that
H0(C′, ωC′) ≃ H
0(C, ωC(φ
∗(P ))).
Moreover, tensoring (13) by OC′(−1), we find the exact sequence
(14) 0 → ωC′(−1) → ψ∗ωC(φ
∗(P ))(−1) → CP → 0
from which we get an injective map H0(C′, ωC′(−1)) → H
0(C, ωC(φ
∗(P ))(−1)).
On the other hand
(15) h0(C′, ωC′(−1)) = h
0(C, ωC(φ
∗(P ))(−1)) = g − n+ 3
and so
H0(C′, ωC′(−1)) ≃ H
0(C, ωC(φ
∗(P ))(−1)).
Moreover, from the hypothesis h0(C,OC(1)) = 3, we have that H
0(C,OC(1)) ≃
H0(C′,OC′(1)) ≃ H
0(P2,OP2(1)). Therefore, in the following commutative dia-
H0(C′,OC′(1))⊗H
0(C′, ωC′(−1))
µo,C′
H0(C′, ωC′)
H0(C,OC(1))⊗H
0(C, ωC(−1)(φ
∗(P )))
// H0(C, ωC(φ
∗(P )))
where we denoted by µ′o,C the natural multiplication map, the vertical maps are
isomorphisms. In particular,
rk(µo,C′) = rk(µ
o,C).
In order to compute the rank of µ′o,C , we consider the following commutative dia-
14 CONCETTINA GALATI
H0(C,OC(1))⊗H
0(C, ωC(−1))
H0(C, ωC)
H0(C,OC(1))⊗H
0(C, ωC(−1)(φ
∗(P )))
// H0(C, ωC(φ
∗(P )))
where the vertical maps are injections. Notice that, since we supposed n ≥ 5,
h0(C,OC(1)) = 3 and g > n−2 ≥ 3, the sheaf OC(1) is special. We deduce that C is
not hyperelliptic and, chosen a basis of H0(C, ωC), the associated map C → P
g−1 is
an embedding. On the contrary, the sheaf ωC(φ
∗(P )) does not define an embedding
on C. Choosing a basis of H0(C, ωC(φ
∗(P )) and denoting by Φ : C → Pg the
associated map, this will be an embedding outside φ∗(P ). If P is a node of C
and φ∗(P ) = p1 + p2, the image of C to P
g, with respect to Φ, will have a node
at the image point Q of p1 and p2. If P ∈ Γ is a cusp, then Φ(C) will have a
cusp at the image point Q of φ−1(P ). The hyperplanes of Pg passing through Q
cut out on C the canonical linear series |ωC |. Moreover, if we denote by B ⊂ P
the subspace which is the base locus of the hyperplanes of Pg corresponding to
Im(µ′o,C), then Q /∈ B. Indeed, B intersects the curve C in the image of the base
locus of |OC(1)|+ |ωC(φ
∗(P ))(−1)| := P(Im(µ′0,C)), which coincides with the base
locus of |ωC(φ
∗(P ))(−1)|, since |OC(1)| is base point free. Now, by (15),
h0(ωC(φ
∗(P ))(−1)) = 3 + g − n = h0(C, ωC(−1)) + 1.
Then φ∗(P ) does not belong to the base locus of |ωC(φ
∗(P ))(−1)|, and so
dim(< Q,B >Pg) = dim(B) + 1.
Finally, we find that
rk(µo,C) = rk(Gµo,C) ≤ dim(Im(G) ∩ Im(µ
o,C))
≤ g + 1− dim(< B,Q >Pg)− 1
= g − 1− dim(B)
= rk(µ′o,C)− 1.
Corollary 4.7. Let Σ ⊂ Σnk,d be a non-empty irreducible component of the expected
dimension of Σnk,d, with n ≥ 5. Suppose that Σ has the expected number of moduli
and that the general element [Γ] ∈ Σ corresponds to a g.l.n. plane curve Γ of
geometric genus g such that, if C → Γ is the normalization of Γ, then the map
µo,C is surjective. Then, for every k
′ ≤ k and d′ ≤ d + k − k′, there is at least an
irreducible component Σ′ ⊂ Σnk′,d′ , such that Σ ⊂ Σ
′, the general element [D] ∈ Σ′
corresponds to a g.l.n. plane curve D of geometric genus g′ with normalization
Dν → D and the Brill-Noether map µ0,Dν surjective. In particular, Σ
′ has the
expected number of moduli.
Proof. Let Γ be the curve corresponding to the general element [Γ] of Σ ⊂ Σnk,d.
Since by hypothesis Σ is smooth of the expected dimension at [Γ], by section 2.1,
for every k′ ≤ k and for every d′ ≤ d+ k− k′ there exists an irreducible component
Σ′ of Σnk′,d′ containing Σ. In order to prove the statement, it is enough to show it
under the hypotheses k′ = k− 1 and d′ = d+1, k = k′ and d′ = d− 1 or d = d′ and
k′ = k − 1. If k′ = k − 1 and d′ = d + 1, then the statement follows by standard
semiconinuity arguments. If k = k′ and d′ = d − 1 or d = d′ and k′ = k − 1, the
statement follows by lemma 4.6 and by standard semicontinuity arguments. �
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 15
The following lemma has been stated and proved by Sernesi in [18]. Actually,
Sernesi supposes that Γ has only nodes as singularities. But, since his proof works
for plane curves Γ with any type of singularities and, since we need it for curves
with nodes and cusps, we state the lemma in a more general form.
Lemma 4.8. ([18], lemma 2.3) Let Γ be an irreducible and reduced plane curve of
degree n ≥ 5 with any type of singularities. Denote by C the normalization of Γ.
Suppose that h0(C,OC(1)) = 3 and the Brill-Noether map
µo,C : H
0(C,OC(1))⊗H
0(C, ωC(−1)) → H
0(C, ωC),
has maximal rank. Let R be a general line and let P1, P2 and P3 be three fixed points
of Γ ∩ R. We denote by C′ the partial normalization of Γ′ = Γ ∪ R, obtained by
smoothing all the singular points, except P1, P2 and P3. Then h
0(C′,OC′(1)) = 3
and, denoting by ωC′ the dualizing sheaf of C
′, the multiplication map
µo,C′ : H
0(C′,OC′(1))⊗H
0(C′, ωC′(−1)) → H
0(C′, ωC′),
has maximal rank.
Theorem 4.9. Let Σnk,d be the algebraic system of irreducible plane curves of degree
n ≥ 4 with k cusps, d nodes and geometric genus g =
− k − d. Suppose that:
(16) n− 2 ≤ g equivalently k + d ≤ h0(P2,OP2(n− 4))
(17) k ≤ 6 +
if 3n− 9 ≤ g and n ≥ 6,
(18) k ≤ 6 otherwise.
Then Σnk,d has at least an irreducible component Σ which is not empty and such
that, if Γ ⊂ P2 is the curve corresponding to the general element of Σ and C is the
normalization curve of Γ, then h0(C,OC(1)) = 3 and the map µo,C has maximal
rank. In particular, when ρ ≤ 0, the algebraic system Σ has the expected number of
moduli equal to 3g − 3 + ρ− k.
Proof. Suppose that (17) holds. Then, by observing that
g ≥ 3n− 9 if and only if k + d ≤ h0(P2,OP2(n− 6))
and by using theorem 3.5 for t = 3, we have that there exists an irreducible com-
ponent Σ of Σnk,d whose general element is a geometrically 3-normal plane curve Γ.
By remark 3.4, it follows that also the linear systems cut out on C by the conics
and the lines are complete. The statement follows from corollary 4.5.
In order to prove the theorem under the hypothesis (18), we consider the following
subcases:
(1) 2n− 5 ≤ g ≤ 3n− 9, i.e. h0(OP2(n− 6)) ≤ k+ d ≤ h
0(OP2(n− 5)) and
n ≥ 5,
(2) n− 2 ≤ g ≤ 2n− 7 and n ≥ 5,
(3) g = 2n− 6 and n ≥ 4.
Suppose that (1) holds. By theorem 3.5 for t = 2, we know that, under this
hypothesis, there exists a nonempty component Σ ⊂ Σnk,d, whose general element is
geometrically 2-normal. We conclude as in the previous case, by corollary 4.5.
Now, suppose that g and n verify (2). We shall prove the theorem by induction
on n and g. Set g = 2n − 7 − a, with a ≥ 0 fixed. Suppose that the theorem is
true for the pair (n, g), with n ≥ 7. We shall prove the theorem for (n + 1, g + 2),
observing that g + 2 = 2(n + 1) − 7 − a. Let Γ be a g.l.n. irreducible plane curve
16 CONCETTINA GALATI
of degree n and genus g = 2n − 7 − a with k ≤ 6 cusps, d nodes and no more
singularities. Let C be the normalization of Γ. Suppose that the Brill-Noether map
µo,C has maximal rank. Let R ⊂ P
2 be a general line and let P1, P2 and P3 be three
fixed points of Γ∩R. By section 2.1, since k ≤ 6 < 3n, one can smooth the singular
points P1, P2, P3 and preserve the other singularities of Γ ∪ R ⊂ P
2, obtaining a
family of plane curves C → ∆ whose general fibre is irreducible, has degree n+1 and
genus g+2. We conclude by lemma 4.8 and by standard semicontinuity arguments.
Now we prove the first step of the induction for n ≥ 7. If n = 7, we get 0 ≤ a ≤ 2.
Let a = 0, i.e. g = 2n−7−a= 7. Let Γ be a g.l.n. irreducible plane curve of degree
n = 7, of genus g = n = 7 with k ≤ 6 cusps and nodes as singularities, such that
no seven singular points of Γ lie on an irreducible conic. To prove that there exists
such a plane curve, notice that, by applying theorem 3.5 for t = 1, we get that, for
any fixed k ≤ 6, there exists a g.l.n. irreducible sextic D of genus four with k cusps
and d = 6 − k nodes. Let R1, . . . , R6 be the singular points of D. Since the points
R1, . . . , R6 of D impose independent linear conditions to the conics, however we
choose five singular points Ri1 , . . . , Ri5 of D, with I = (i1, . . . , i5) ⊂ (1, . . . , 6), there
exists only one conic CI , passing through these points. Let us set S =
CI ∩D
and let R be a line intersecting D transversally at six points out of S. By Bezout
theorem, no seven singular points of Γ′ = D ∪ R belong to an irreducible conic.
Moreover, if D̃ is the normalization of D, if Q1, . . . , Q4 are four fixed points of
D∩R and D′ is the partial normalization of Γ′ obtained by smoothing the singular
points except Q1, . . . , Q4, then, by the following exact sequence
(19) 0 → OR(1)(−Q1 − · · · −Q4) → OD′(1) → OD̃(1) → 0
we find that h0(D′, OD′(1)) = 3. By section 2.1, one can smooth the singulari-
ties Q1, . . . , Q4 and preserve the other singularities of D ∪ R, getting a family of
irreducible septics G → ∆ whose general fibre Γ is a geometrically linearly normal
irreducible septic with k cusps and 8 − k nodes such that no seven singular point
of Γ belong to an irreducible conic. Let, now, C be the normalization of Γ and let
∆ ⊂ C be the adjoint divisor of the normalization map φ : C → Γ. We shall prove
that ker(µo,C) = 0. Since Γ is geometrically linearly normal, we have that
h0(C, ωC(−1)) = h
0(C,OC(3)(−∆))) = g − n+ 2 = 2.
Then, by the base point free pencil trick, we find that
ker(µo,C) = H
0(C, ω∗C(B)⊗OC(2)),
where B is the base locus of |ωC(−1) = OC(3)(−∆)|. Let F be the pencil of plane
cubics passing through the eight double points P1, . . . , P8 of Γ and let BF be the
base locus of the pencil F . Let Γ3 be the general element of F . Suppose that BF
has dimension one. If BF contains a line l, then, by Bezout theorem, at most three
points among P1, . . . , P8, say P1, . . . , P3 can lie on l and the other points have to be
contained in the base locus of a pencil of conics F ′. Using again Bezout theorem,
we find that the curves of F ′ are reducible and the base locus of F ′ contains a line
l′. But also l′ contains at most three points of P4, . . . , P6. It follows that there is
only one cubic through P1, . . . , P8. This is not possible by construction. Suppose
that BF contains an irreducible conic Γ2. By Bezout theorem, at most seven points
among P1, . . . , P8 may lie on Γ2. On the other hand, since dim(F) = 1, there
are exactly seven points of P1, . . . , P8, say P1, . . . , P7, on Γ2 and the general cubic
Γ3 of F is union of Γ2 and a line passing through P8. Since, by construction, no
seven singular points of Γ lie on an irreducible conic, also in this case we get a
contradiction. So the general element Γ3 of F is irreducible. Using again Bezout
theorem, we find that Γ3 is smooth and F has only one more base point Q. We
consider the following cases:
a) Q doesn’t lie on Γ;
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 17
b) Q lies on Γ, but Q 6= P1, . . . , P8;
c) Q is infinitely near to one of the points P1, . . . , P8, say Pî, i.e. the cubics of F
have at P
the same tangent line l, but l is not contained in the tangent cone to Γ
d) Q is like in the case c), but l is contained in the tangent cone to Γ at P
Suppose that the case a) or c) holds. Thus B = 0 and
ker(µo,C) = H
0(C, ω∗C ⊗OC(2)) = H
0(C,OC(−2)(∆)).
By Riemann-Roch theorem, h0(C,OC(−2)(∆)) = h
0(C,OC(6)(−2∆)) − 4. One
sees that h0(C,OC(6)(−2∆)) = 4, by blowing-up the plane at P1, . . . , P8 and by
using some standard exact sequences. Suppose now that the case b) holds. Thus
B = Q and
dim(ker(µo,C)) = h
0(C,OC(−2)(∆ +Q)) = h
0(C,OC(6)(−2∆−Q))− 3.
Also in this case one sees that h0(C,OC(6)(−2∆ − Q)) = 3 by blowing-up at
P1, . . . , P8 and Q and by using standard exact sequences. Finally, we analyze the
case d). Let Φ : S → P2 be the blow-up of the plane at P1, . . . , P8 with exceptional
divisors E1, . . . , E8. Let Q ∈ Eî be the intersection point of Eî and the strict
transform C3 of the general cubic Γ3 of the pencil F . We denote by Φ̃ : S̃ → S the
blow-up of S at Q and by Ψ : S̃ → P2 the composition map of the maps Φ and Φ̃.
We still denote by E1, . . . , E8 their strict transforms on S̃, by C and C3 the strict
transforms of Γ and Γ3 and by EQ the new exceptional divisor of S̃. In this case we
have that Ψ∗(Γ) = C+2
Ei+3EQ, Ψ
∗(Γ3) = C3+
Ei+2EQ. Moreover, the
divisor ∆ is cut out on C by
iEi + EQ and the base locus B of the linear series
|ωC(−1)| coincides with the intersection point of EQ and C. So, we have that
dim(ker(µo,C)) = h
0(C,OC(−2)(
Ei+2EQ)) = h
0(C,OC(6)(−2
Ei−3EQ))−3.
Moreover, from the following exact sequence
0 → O
(−1) → O
(6)(−2
Ei − 3EQ) → OC(6)(−2
Ei − 3EQ) → 0
we find that H0(C,OC(6)(−2
Ei− 3EQ)) = H
0(S̃,O
(6)(−2
Ei− 3EQ)). In
order to show that h0(S̃,O
(6)(−2
Ei − 3EQ)) = 3, we consider the following
exact sequence
0 → O
(3)(−
Ei − EQ) → OS̃(6)(−2
Ei − 3EQ) →(20)
→ OC3(6)(−2
Ei − 3EQ) → 0
By Riemann-Roch theorem, we have that
h0(C3,OC3(6)(−2
Ei − 3EQ)) = 1 and h
1(C3,OC3(6)(−2
Ei − 3EQ) = 0.
Moreover, by Serre duality we have that
H1(S̃,O
(3)(−
Ei − EQ))) = H
1(S̃,O
(−6)(2
Ei + 3EQ))).
From the exact sequence
(21) 0 → O
(−6)(+2
Ei + 3EQ)) → OS̃(1) → OC(1) → 0
18 CONCETTINA GALATI
by using that the map H0(S̃,O
(1)) → H0(C,OC(1)) is surjective and that
h1(S̃,O
(1)) = 0, we find that
H1(S̃,O
(−6)(+2
Ei + 3EQ))) = H
1(S̃,O
(3)(−
Ei − EQ))) = 0.
Then, by (20), h0(S̃,O
(6)(−2
Ei − 3EQ)) = h
0(S̃,O
(3)(−
Ei − EQ)) +
h0(C3,OC3(6)(−2
Ei−3EQ)) = 3 and ker(µo,C) = 0. The first step of induction
for g = n = 7 and k ≤ 6 is proved.
We complete the proof of the first step of the induction, for n and g verifying
(2). When n = 7 and 1 ≤ a ≤ 2, the existence of a g.l.n. plane curve Γ follows
from theorem 3.5. Using the above notation, h0(C, ωC(−1)) = 1 if a = 1 and
h0(C, ωC(−1)) = 0 if a = 2. In any case µo,C is injective. When n ≥ 8 and
a ≤ n − 6 the theorem follows by induction from the case n = 7. For n ≥ 8 and
a = n− 5, we find that g = n− 2, or, equivalently, k + d = h0(P2,OP2(n− 4)). In
theorem 3.5, we proved the existence of geometrically linearly normal plane curves
of degree n ≥ 8 and genus g = n− 2, with nodes and k ≤ 6 cusps. For every such
plane curve Γ, using the above notation, the Brill-Noether map µo,C is injective
since h0(C, ωC(−1)) = 0. The cases n = 5 and n = 6 are similar.
Suppose now that n and g verify (3). First of all we prove the theorem for
(n, g) = (4, 2), (5, 4), (6, 6). For n = 4 and g = 2, we find n = g + 2 and we argue
as in the case n ≥ 8 and g = n − 2. Similarly, for (n, g) = (5, 4). For n = 6
and g = 6 in theorem 3.5 we proved the existence of geometrically linearly normal
plane curves Γ with k ≤ 4 cusps and nodes as singularities. For every such a plane
curve Γ, denoting by C its normalization, we get that h0(C, ωC(−1)) = 2, i.e. the
linear system F of conics passing through the four singular points P1, . . . , P4 of Γ
is a pencil which cuts out on C the complete linear series |ωC(−1)|. We have two
possibilities: either the general element of this pencil is irreducible or it consists of
a line containing exactly three singular points P1, P2, P3 of Γ and a line passing
through P4. In any case the base locus of F intersects Γ only at P1, . . . , P4 and
the linear series |ωC(−1)| has no base points. Then, by the base point free pencil
trick , we find that ker(µo,C) = H
0(C, ω∗C ⊗ O(2)) = H
0(C,OC(−1)(∆)), where
∆ ⊂ C is the adjoint divisor of the normalization map C → Γ. By Riemann-Roch
theorem, we have that h0(C,OC(−1)(∆)) = h
0(C,OC(4)(−2∆))−3. By blowing-up
at P1, . . . , P4, one can see that h
0(C,OC(4)(−2∆)) = 3, as we wanted.
Finally, we show the theorem under the hypothesis (3) for n ≥ 7, by using
induction on n. In order to prove the inductive step we may use lemma 4.8, exactly
as we did in the case (2). We prove the first step of induction. If n = 7 we have
that g = 8. On pages 15 and 16 we proved the existence of geometrically linearly
normal plane curves Γ of degree 7 and genus 7 with k ≤ 6, such that, if P1, . . . , P8
are the singular points of Γ, then no seven points among P1, . . . , P8 lie on a conic.
In particular, we proved that, for every such a plane curve Γ, the general element
of the pencil of cubics passing through P1, . . . , P8 is irreducible and, if φ : C → Γ
is the normalization of Γ, then the Brill-Noether map µo,C is injective. Let C
the partial normalization of Γ which we get by smoothing all the singular points of
Γ except a node, say P8. By using the same notation and by arguing exactly as in
the proof of lemma 4.6, we get the following commutative diagram
H0(C′,OC′(1))⊗H
0(C′, ωC′(−1))
µo,C′
H0(C′, ωC′)
H0(C,OC(1))⊗H
0(C, ωC(−1)(φ
∗(P8)))
// H0(C, ωC(φ
∗(P8)))
where µ′o,C is the multiplication map and the vertical maps are isomorphisms. We
want to prove that the map µo,C′ is surjective. By the previous diagram it is enough
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 19
to prove that µ′o,C is surjective. Since h
0(C, ωC(φ
∗(P8))) = 8 and
h0(C,OC(1))h
0(C, ωC(−1)(φ
∗(P8))) = 3(7− 7 + 3) = 9,
we have that dim(ker(µo,C′)) ≥ 1 and µo,C′ is surjective if dim(ker(µo,C′)) = 1. By
recalling that Γ is geometrically linearly normal, we have that, if Z is the scheme of
the points P1, . . . , P7 and IZ|P2 is the ideal sheaf of Z in P
2, then in the following
commutative diagram
H0(C,OC(1))⊗H
0(C, ωC(−1)(φ
∗(P8)))
H0(C, ωC(φ
∗(P8)))
H0(P2,OP2(1))⊗H
0(P2, IZ|P2(3))
// H0(P2, IZ|P2(4))
the vertical maps are isomorphisms. Hence, it is enough to prove that the kernel
of the multiplication map µ has dimension one. Let {f0, f1, f2} be a basis of the
vector space H0(P2, IZ|P2(3)). Since the general cubic passing through P1, . . . , P8
is irreducible, we may assume that f0, f1 and f2 are irreducible. Suppose, by
contradiction, that there exist at least two linearly independent vectors in the kernel
of µ. Then, there exist sections u0, u1, u2 and v0, v1, v2 of H
0(P2,OP2(1)) such that
the sections
ui⊗ fi and
vi⊗ fi are linearly independent in H
0(P2,OP2(1))⊗
H0(P2, IZ|P2(3)) and
i=0 uifi = 0
i=0 vifi = 0.
We can look at (22) as a linear system in the variables f0, f1, f2. The space of
solutions of (22) is generated by the vector
(u1v2 − u2v1, u3v0 − u0v3, u0v1 − u1v0).
In particular, if we set qi = (−1)
1+iuivj − viuj, we find that fjqi = fiqj , for every
i 6= j. But this is not possible since f1, f2 and f3 are irreducible. We deduce that
dim(ker(µ)) = dim(ker(µo,C′)) = 1
and µo,C′ is surjective. The existence of a plane septic of genus 8 with k ≤ 6
cusps and nodes as singularities, with injective Brill-Neother map, follows now by
smoothing the node P8 (in the sense of section 2.1) and by standard semicontinuity
arguments. �
Remark 4.10. Notice that the conditions which we found in theorem 4.9 in order
that Σnk,d has at least an irreducible component with the expected number of moduli,
are not sharp, even if we suppose ρ ≤ 0. To see this, notice that in remark 3.6 we
proved the existence of an irreducible component Σ of Σ129,0 whose general element
corresponds to a 3-normal plane curve. By remark 3.4 and corollary 4.5, we have
that Σ has the expected number of moduli.
Theorem 4.11. Σn1,d has the expected number of moduli, for every d ≤
Proof. First of all, we recall that, by [16], Σn1,d is irreducible for every d ≤
Moreover, from theorem 4.9 and from corollary 2.7, we know that Σn1,d is not empty
and it has the expected number of moduli if either ρ ≤ 0 or ρ ≥ 2. Next we shall
prove that, if ρ = 1, then the algebraic system
Σn1,d = Σ
(n−3)2
has general moduli. Equivalently, we will show that, if [Γ] ∈ Σn1,d is a general point
and g =
− 1 − d = 3n−7
, then, on the normalization curve C of Γ there are
only finitely many linear series g2n with at least a ramification point. Notice that,
20 CONCETTINA GALATI
if g =
− 1 − d = 3n−7
, then n is odd and n ≥ 5. We prove the statement by
induction on n.
If n = 5 then g = 4. Let C ⊂ P3 be the canonical model of a general curve of
genus four and let 2P + Q, with P 6= Q be a divisor in a g13 on C. This divisor is
cut out on C by the tangent line to C at P . The projection of C from Q is a plane
quintic of genus four with a cusp. This proves that Σ51,1 has general moduli.
Now we suppose that the theorem is true for n and we prove the theorem for
n+2. Let Γ ⊂ P2 be the plane curve with a cusp and
(n−3)2
−1 nodes corresponding
to a general point [Γ] ∈ Σn
(n−3)2
and let C2 be an irreducible conic intersecting
Γ transversally. By section 2.1, the point [C2 ∪ Γ] belongs to Σ
(n+2−3)2
particular, however we choose four points P1, . . . , P4 of intersection between Γ and
C2, there exists an analytic branch SP1,..., P4 of Σ
(n−1)2
, passing through [C2∪Γ]
and whose general point corresponds to an irreducible plane curve of degree n + 2
with a cusp in a neighborhood of the cusp of Γ and a node at a neighborhood of
every node of C2 ∪ Γ different from P1, . . . , P4. Moreover, S := SP1,..., P4 is smooth
at the point [C2 ∪ Γ], (see [7], chapter 2). Let
Π : Σn+2
(n−1)2
99K M 3(n+2)−7
be the moduli map of Σn+2
(n−1)2
. In order to prove that Π is dominant it is
sufficient to show that Π(S) = M 3n−1
. By section 2.1, there exist an analytic open
sets Si ⊂ Σn+2
(n−3)2
−1+2n−i
, with i = 1, 2, 3, such that
S0 := S ∩ (P5 × Σn
(n−3)2
) ⊂ S1 ⊂ S2 ⊂ S3 ⊂ S.
Every Si, with i = 1, 2, 3, has
irreducible components, passing through [C2∪Γ]
and intersecting transversally at [C2 ∪Γ], (see [7], chapter 2 or [25]). Moreover, the
general point of every irreducible component of Si, with i = 1, 2, 3, corresponds
to an irreducible plane curve Γi of degree n + 2 with a cusp in a neighborhood of
the cusp of Γ, a node in a neighborhood of every node of C2 ∪ Γ different from
P1, . . . , P4 and 4 − i nodes specializing to 4 − i fixed points among P1, . . . , P4, as
Γi specializes to C2 ∪ Γ. Now, notice that the moduli map Π is not defined at the
point [C2 ∪ Γ], but, if S is sufficiently small, then the restriction of Π to S extends
to a regular function on S. More precisely, let C → ∆ be any family of curves,
parametrized by a projective curve ∆ ⊂ S, passing through the point [C2 ∪ Γ] and
whose general point corresponds to an irreducible plane curve of degree n+2 of genus
3(n+2)−7
with a cusp and nodes as singularities. If we denote by C′ → ∆
the family of curves obtained from C → ∆ by normalizing the total space, we have
that the general fibre of C′ → ∆ is a smooth curve of genus 3n−1
, corresponding
to the normalization of the general fibre of C → ∆, whereas the special fibre C′0 is
the partial normalization of C2∪Γ, obtained by normalizing all the singular points,
except P1, . . . , P4. Then, the map Π|S is defined at [C2 ∪Γ] and it associates to the
point [C2∪Γ] the isomorphism class of C
0. Similarly, if [Γi] is a general point in one
of the irreducible components of Si, with i = 1, 2, 3, then Π|S ([Γi]) is the partial
normalization of Γi obtained by smoothing all the singular points except for the 4−i
nodes of Γi tending to 4−i fixed points among P1, . . . , P4 as Γi specializes to C2∪Γ.
It follows that, if we denote by M
the locus of M 3n−1
parametrizing j-nodal
curves, then ΠS(S
i) ⊆ M4−i3n−1
, for every i = 0, . . . , 4, and ΠS(S
i)  ΠS(S
i+1). In
particular, we find that
dim(Π|S (S)) ≥ dim(Π|S (S
0)) + 4.
NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 21
In order to compute the dimension of Π|S (S
0) we consider the rational map
F : Π|S (S
0) 99K M 3n−7
forgetting the rational tail. By the hypothesis that Σn
(n−3)2
has general moduli
and hence F is dominant. Moreover, if C is the normalization curve of Γ, by the
generality of [Γ] in Σn
(n−3)2
, we may assume that C is general in M 3n−7
want to show that dim(F−1([C])) = 5. In order to see this, we recall that, by the
hypothesis that Σn
(n−3)2
has general moduli, on C there exist only finitely many
linear series of degree n and dimension two, mapping C to the plane as curve with a
cusp and nodes as singularities. Let g2n be one of these linear series, let {s0, s1, s2}
be a basis of g2n and φ
′ : C → Γ′ ⊂ P2 the associated morphism. If Q1, . . . , Q4
are four general points of Γ′, then the linear system of conics through Q1, . . . , Q4
is a pencil F(Q1, . . . , Q4). Let C2 and D2 be two general conics of F(Q1, . . . , Q4).
We claim that, if η : P1 → C2 and β : P
1 → D2 are isomorphisms between P
1 and
C2 and D2 respectively, then the points η
−1(Q1), . . . , η
−1(Q4) are not projectively
equivalent to the points β−1(Q1), . . . , β
−1(Q4). In order to prove this, it is enough to
prove that there are at least two conics in the pencil F(Q1, . . . , Q4) which verify the
claim. LetD ⊂ P2 be a conic. If we choose two sets of points p1, . . . , p4 and q1, . . . , q4
ofD not projectively equivalent onD, we may always find projective automorphisms
A : P2 → P2 and A′ : P2 → P2 such that A(pi) = Qi and A
′(qi) = (Qi), for every
i. By construction, the conics C2 = A(D) and D2 = A
′(D) belong to the pencil
F (Q1, . . . , Q4) and verify the claim. This implies that the partial normalizations C
and D′ of Γ′∪C2 and Γ
′∪D2, obtained by smoothing all the singular points except
Q1, . . . , Q4, are not isomorphic. Now, let C
2 be a general conic of F(Q1, . . . , Q4)
and let R1, . . . , R4 be four general points of Γ
′, different from Q1, . . . , Q4. If D
is a general conic of the pencil F(R1, . . . , R4), then the partial normalization C
and D′ of Γ′ ∪C′2 and Γ
′ ∪D′2 obtained, respectively, by smoothing all the singular
points except Q1, . . . , Q4 and R1, . . . , R4, are not isomorphic. Indeed, since C is a
general curve of genus 3n−7
≥ 7, the only automorphism of C is the identity. This
proves that dim(F−1([C])) = 5. In particular, we deduce that
dim(Π|S (S
0)) = 3
3n− 7
− 3 + 5
dim(Π|S (S) ≥ 3
3n− 7
− 3 + 9 = 3
3(n+ 2)− 7
Remark 4.12. We expect that it is possible to prove that Σnk,d has expected number
of moduli for every ρ also when k = 2 or k = 3. By corollary 2.7 and theorem 4.9,
Σnk,d is not empty, irreducible and it has expected number of moduli for ρ ≤ 0 and
ρ ≥ 2k. In order to extend theorem 4.11 to the case k = 2 and k = 3 one needs to
consider a finite number of cases.
Acknowledgment. The results of this paper are part of my PhD-thesis. I would
like to express my gratitude to my advisor Prof. C. Ciliberto who initiated me into
the subject of algebraic geometry and who provided me many invaluable suggestions.
I have also enjoyed and benefited from conversation with many people including F.
Flamini, E. Sernesi, L. Chiantini, L. Caporaso and G. Pareschi. Finally, I would
like to thank the referee for useful remarks which allowed me to improve the finale
version of this paper.
22 CONCETTINA GALATI
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Dipartimento di Matematica, Università degli Studi della Calabria, via P. Bucci,
cubo 30B, Arcavacata di Rende (CS)
E-mail address: galati@mat.unical.it
	1. Introduction
	2. Preliminaries
	2.1. On Severi-Enriques varieties
	2.2. Known results on the number of moduli of nk,d
	3. On the existence of certain families of plane curves with nodes and cusps in sufficiently general position
	4. Families of plane curves with nodes and cusps with finite and expected number of moduli.
	Acknowledgment
	References
