On Severi varieties and Moduli spaces of curves in arbitrary characteristic

Ilya Tyomkin^{∗} Weizmann Institute of Science, Department of Mathematical and Computer Sciences, Rehovot, Israel, and Max-Planck-Institut für Mathematik Bonn, Germany. Email: ilya.tyomkin@weizmann.ac.il

Abstract In the current paper we prove the irreducibility of Severi varieties on Hirzebruch surfaces in arbitrary characteristic. Our approach is of purely algebro-geometric nature, and it works in any characteristic. As a result, we obtain a deformation-theoretic proof of the irreducibility of moduli spaces M_{g} in positive characteristic, which does not involve reduction to characteristic zero.

1 Introduction

The study of Severi varieties is one of the classical problems in algebraic geometry. Given a smooth projective surface Σ, a line bundle L ∈ P ic(Σ), and an integer g, one defines Severi variety V (Σ, L, g) ⊂ |L| to be the closure of the locus of irreducible nodal curves of genus g. Originally, these varieties were introduced by Severi (in the plane case) in order to prove the irreducibility of the moduli spaces of curves M_{g} in characteristic zero. In Anhang F of his famous book, Vorlesungen über algebraische Geometrie, F. Severi gave a false proof of the irreducibility of V (ℙ^{2}, O_{ℙ^{2}}(d), g). And it took more than sixty years, till, in 1986, Harris proved this result [6] (see also [9] for another proof due to Ran). However, the case of positive characteristic is not covered by these proofs, but it is a particular case of our result presented in this paper.

∗The author was supported by the Postdoctoral fellowship provided by the Clore Foundation.

The study of various properties of Severi varieties, in particular their degrees continued, and in 1989 Ran [10] obtained the first recursive formula for the degree of varieties V_{d,g} = V (ℙ^{2}, O_{ℙ^{2}}(d), g). More formulas were obtained by Caporaso, Harris, Kontsevich, and Mikhalkin for other surfaces. We should mention that recent Mikhalkin’s work on tropical geometry [8] provides us with a formula for the degree of V_{Δ,g} = V (T or(Δ), L(Δ), g), where T or(Δ) is the projective toric surface corresponding to a convex polygon Δ, and L(Δ) is the tautological line bundle on T or(Δ). Mikhalkin’s formula is not recursive, and is of combinatorial nature: namely the degree of V_{Δ,g} is computed in terms of lattice passes inside Δ.

Although the degrees of Severi varieties (in characteristic zero) have been computed for any projective toric surface, the irreducibility problem is still open in most of the cases in characteristic zero, and it is open in all the cases in positive characteristic. In the recent preprint by Shevchishin he claims the irreducibility of Severi varieties on Hirzebruch surfaces in characteristic zero [11]. Shevchishin’s approach is based on the theory of Banach manifolds and involves many analytic and topological arguments. Hence in the case of positive characteristic his approach seems inapplicable.

The goal of the current paper is to give an algebro-geometric characteristic independent proof of the irreducibility of Severi varieties on Hirzebruch surfaces. As a result, we obtain the first direct proof of the irreducibility of moduli spaces M_{g,n} in positive characteristic^{1}. All the previously known proofs of this fact are based on reduction of the problem to the case of characteristic zero (cf. [3, 5]). Moreover, originally these proofs involved analytic methods in characteristic zero, until in 1982 Fulton succeeded in replacing the analytic arguments by algebraic, using the compactification of Hurwitz schemes introduced by Mumford and Harris [7]. To conclude, we shall mention that our proof of the irreducibility of M_{g} is completely different from the previously known, and we think that the methods and techniques developed in this paper can be used in various problems in algebraic geometry.

The idea of the proof of the irreducibility of Severi varieties on Hirzebruch surfaces is as follows. Let Σ_{n} be a Hirzebruch surface, and let L = O_{Σ_{n}}(dL_{0}+ kF) be a line bundle, where L_{0} and F denote the classes generating P ic(Σ_{n}), satisfying L_{0}^{2} = n, F^{2} = 0, and L_{0}.F = 1. We use the notation V_{g,d,k} for the Severi variety V (Σ_{n}, L, g). It turns out that instead of V_{g,d,k} one should consider (for technical reasons) the decorated Severi varieties U_{d,k,δ}, which parameterize nodal curves with δ marked nodes in the linear system |L|. It is important to say that the decorated Severi variety U_{d,k,δ} admits a dominant “forgetful” map to the Severi variety V_{g,d,k}, where g = \frac{(d−1)(nd+2k−2)}{2} − δ. In these notations the first part of the proof is given by Proposition 3.2, which states the following: any irreducible component U ⊆ U_{d,k,δ} contains a pointed curve of a special type, which is denoted by (Γ; p_{1}, ..., p_{δ}), and, as a result, any irreducible component of V_{g,d,k} contains a curve of type Γ (see Section 3 for detailed definitions and precise formulations). Now, the irreducibility follows from Proposition 3.3, claiming that if components U, U′ ⊆ U_{d,k,δ} contain curves of special type (Γ; p_{1}, ..., p_{δ}) and (Γ; p′_{1}, ..., p′_{δ}) respectively, then they have the same image in the appropriate Severi variety.

1It is clear that there exist Severi varieties on Hirzebruch surfaces that dominate M_{g}.

For example V_{g,d,0} on the blown up plane Σ_{1} has this property for large d. The irreducibility of M_{g,n} follows immediately from the irreducibility of M_{g}.

The proof of Proposition 3.2 is done as follows: first, we degenerate the surface Σ_{n} into a fan (in Ran’s terminology [9]) X^{0} ∪...∪X^{r}, where X^{j} ≅ Σ_{n} for all j, and X^{j} ∩ X^{j+1} is L_{0} on X^{j} and L_{∞} on X^{j+1} (L_{∞} ⊂ Σ_{n} denotes the unique irreducible curve with negative self-intersection). Next we show that the degeneration of U contains the degeneration of the curve (Γ; p_{1}, ..., p_{δ}), which allows us to prove that U itself contains the required pointed curve, due to some transversallity and smoothness conditions.

At last, we say a few words about the proof of Proposition 3.3. In the case of characteristic zero it can be reduced to a combinatorial statement using very geometric monodromy-type arguments. However, in order to make the proof characteristic independent, we replaced these arguments by a description of explicit smooth coverings, which are shown to be connected, and hence irreducible.

To finish the introduction we should mention that there are examples of (non-rational) surfaces admitting reducible Severi varieties. Moreover, these Severi varieties can have components of different dimensions. So it is unclear how to characterize the surfaces admitting only irreducible Severi varieties. Nevertheless, we would like to state the following conjecture motivated by our result and Mikhalkin’s work:

Conjecture 1.1 The Severi varieties on any projective toric surface are irreducible in arbitrary characteristic.

We hope that this conjecture can be approached using the rapidly developing methods of tropical geometry combined with the approach presented in the current paper.

Acknowledgement

I am very grateful to J. Bernstein, G.-M. Greuel, E. Shustin, and M. Temkin for helpful discussions. This work was finished while the author was visiting the Max-Planck-Institut für Mathematik at Bonn. I would like to thank this institution for its hospitality.

2 Preliminaries

2.1 Deformation theory

In this section we discuss several (basic) facts from the deformation theory of algebraic varieties and algebraic maps. Most of the statements, ideas, and proofs presented here can be found in different sources (see for example [1, 2, 6], and [12] for related topics). However, I was not able to find a characteristic free presentation of this material, and decided to write it down.

2.1.1 Deformations of maps.

Let X and Y be smooth algebraic varieties over an algebraically closed field 𝕂, and let f : X ^{}Y be an algebraic map. In this section we discuss the deformation theory of the pair (X, f) - namely, we fix Y and vary X and f. Denote 𝔻 = Spec𝕂[ϵ]/(ϵ^{2}). We recall that a first-order deformation of (X, f) is a triple

• a flat family π : X̃ ^{}𝔻,

• a map F : X̃ ^{}Y,

• an isomorphism α : (X̃_{0}, F_{0}) ^{}(X, f), where X̃_{0} = X̃/(ϵ), and F_{0} = F/(ϵ).

Notation 2.1 The set of first order deformations of the pair (X, f) modulo isomorphisms is denoted Def^{1}(X, f).

Proposition 2.2 If 0 ^{}T_{X} ^{}f^{∗}T_{Y} is exact, then

Def^{1}(X, f) ≅ H^{0}(X, N_{f}),

where N_{f} denotes the normal sheaf to f, i.e. the cokernel of the map

df : T_{X} ^{}f^{∗}T_{Y} .

Proof: First, we choose affine coverings Y = ∪_{i=1}^{n}Y_{i} and X = ∪_{i=1}^{n}X_{i} such that f(X_{i}) ⊂ Y_{i} for all i. Now, let ξ ∈ Def^{1}(X, f) be a first-order deformation. We shall use the following well-known claim

Claim 2.3 Let Z = SpecA be a smooth affine variety over an algebraically closed field 𝕂, and let Z_{ϵ} = SpecA_{ϵ} be an infinitesimal extension of Z, i.e. a pair consisting of a flat morphism Z_{ϵ} ^{}𝔻 together with an isomorphism Z_{ϵ}/(ϵ) ≃ Z. Then Z_{ϵ} is isomorphic to the trivial extension, namely A_{ϵ} ≃ A ⊕ ϵA.

Due to the claim we can fix trivializations

OX̃(X_{i}) ≃ O_{X_{i}}(X_{i}) ⊕ ϵO_{X_{i}}(X_{i}).  (2.1)

Then we obtain the automorphisms

β_{ij} : O_{X_{ij}}(X_{ij}) ⊕ ϵO_{X_{ij}}(X_{ij}) ^{}O_{X_{ij}}(X_{ij}) ⊕ ϵO_{X_{ij}}(X_{ij}),

equal to identity modulo ϵ. Hence

β_{ij}(x + ϵy) = x + ϵ(y + D_{ij}(x)),

where D_{ij} : O_{X_{ij}}(X_{ij}) ^{}O_{X_{ij}}(X_{ij}) are derivations. The maps

F^{∗} : O_{Y} (Y_{i}) ^{}O_{X_{i}}(X_{i}) ⊕ ϵO_{X_{i}}(X_{i})

are given by F^{∗}(x) = f^{∗}(x) + ϵD_{i}(x), where

D_{i} : O_{Y} (Y_{i}) ^{}O_{X_{i}}(X_{i})

are derivations, and the following equality holds:

D_{i} − D_{j} = D_{ij} ∘ f^{∗}.

Hence the set D^{ξ} = (D_{1}, ..., D_{n}) defines a global section of the sheaf N_{f}. It is clear that D^{ξ} does not depend on the choice of trivializations in (2.1). Now, one can easily check that the constructed correspondence provides us with the bijection

Def^{1}(X, f) ∋ ξ ^{}^{}D^{ξ} ∈ H^{0}(X, N_{f}),

which in fact is an isomorphism of vector spaces. Moreover, this bijection does not depend on the choice of coverings Y = ∪_{i=1}^{n}Y_{i} and X = ∪_{i=1}^{n}X_{i}. ■

2.1.2 Families of curves on algebraic surfaces.

Let Σ be a smooth projective algebraic surface, and let L be a line bundle on Σ. Consider an irreducible variety V ⊆ |L| whose generic element is a reduced curve. The goal of this section is to give a natural upper bound on the dimension of V. Let

C ^{⊂} V × Σ
[figure]
V

be the tautological family of curves over V, and let C̃ ^{}C be its normalization. Then for a generic p ∈ V the fiber C̃_{p} is the normalization of the fiber C_{p}.

Let us choose a generic point 0 ∈ V. Due to the Generic Flatness theorem and Proposition 2.2, we then have a natural map

μ : T_{0}V ^{}Def^{1}(C, f) ≅ H^{0}(C, N_{f}),

where C = C̃_{0} and f is the composition of maps

C = C̃_{0} ^{}C ^{}Σ × {0} = Σ.

Proposition 2.4 (1) The map T_{0}V ^{}H^{0}(C, N_{f}) is an embedding.

(2) Im(T_{0}V) ∩ H^{0}(C, N_{f}^{tor}) = 0.

Proof: First, we choose a smooth projective irreducible curve D ⊂ Σ intersecting f(C) transversally, such that h^{0}(Σ, L(−D)) = 0. Let

[figure: X; F; Σ; 𝔻]

be a first-order deformation of the pair (C, f). We define the new pair X_{D} = X ×_{Σ} D and F_{D} : X_{D} ^{}D.

Claim 2.5 The natural map X_{D} ^{}𝔻 is flat.

So we constructed a map

ρ : Def^{1}(C, f) ^{}Def^{1}(f(C) ∩ D, f_{D}),

where f_{D} = F_{D}/(ϵ).

Claim 2.6 H^{0}(C, N_{f}^{tor}) ⊂ Ker(ρ).

To finish the proof it is enough to show that dim(ρ(μ(T_{0}V))) = dim(V). Consider the exact sequence

H^{0}(Σ, L(−D)) ^{}H^{0}(Σ, L) ^{}H^{0}(D, L ⊗ O_{D}).

The first term is zero, hence the map α : V ^{⊂} ^{}|L ⊗ O_{D}| is an embedding. Now

\begin{aligned}T_{α(0)}|L & ⊗ O_{D}| = \\ & H^{0} (f(C) ∩ D, O_{f(C)∩D}(f(C) ∩ D)) = Def^{1}(f(C) ∩ D, f_{D}) \end{aligned}

and ρ ∘ μ = dα(0), which implies dim(ρ(μ(T_{0}V))) = dim(V). ■ Proof of Claim 2.5: First of all we can assume that D, Σ, and C are affine. Thus X and X_{D} are affine as well. Next we shall use the following lemma:

Lemma 2.7 (Proposition 6.1 [4]) Let R be a commutative ring with identity, and let M be an R−module. If I is an ideal of R, then T or_{R}^{1}(R/I, M) = 0 if and only if the map I ⊗_{R} M ^{}M is an injection. The module M is flat if and only if this condition is satisfied for every finitely generated ideal I ⊂ R.

In our case R = 𝕂[ϵ]/(ϵ^{2}) and the only non-trivial ideal we have is I = 𝕂ϵ. By the lemma it is enough to show that the map

O_{X_{D}} ⊗_{O_{𝔻}} 𝕂ϵ ^{γ}^{}O_{X_{D}}

is an embedding. Since O_{X} is flat over O_{𝔻}, we have an exact sequence

0 ^{}O_{X} ⊗_{O_{𝔻}} 𝕂ϵ ^{}O_{X}  ^{}O_{C}  ^{}0.

Tensoring this sequence with O_{D} over O_{Σ} we obtain

0 ^{}T or_{O_{Σ}}^{1}(O_{D}, O_{C}) ^{}O_{X_{D}} ⊗_{O_{𝔻}} 𝕂ϵ ^{γ}^{}O_{X_{D}}.

It remains to prove that T or_{O_{Σ}}^{1}(O_{D}, O_{C}) = 0. Let

0 ^{}O_{Σ} ^{·j}^{}O_{Σ}  ^{}O_{D}  ^{}0

be a free resolution of O_{D}, where j ∈ O_{Σ} is a function defining D. Thus

T or_{O_{Σ}}^{1}(O_{D}, O_{C}) = Ker (O_{C} ^{f^{∗}(j)}^{}O_{C}) = 0,

since f(C) intersects D transversally. ■ Proof of Claim 2.6: To prove this claim we have to construct the map ρ explicitly. Consider the fibered product diagram

[figure: f(C) ∩ D; f_{D}; D; C; i; f; Σ; i_{D}]

The intersection f(C) ∩ D is transversal and its points are smooth on both f(C) and D. Hence we have the following exact sequences on f(C) ∩ D:

[figure: 0; i^{∗}T_{C}; i^{∗}(df); i^{∗}f^{∗}T_{Σ}; i^{∗}N_{f}; 0]
∥
[figure: 0; f_{D}^{∗}T_{D}; df_{D}; f_{D}^{∗}i_{D}^{∗}T_{Σ}; f_{D}^{∗}N_{i_{D}}; 0]
∥
N_{f_{D}}

Moreover,

i^{∗}(df)(i^{∗}T_{C}) ⊕ df_{D}(f_{D}^{∗}T_{D}) = i^{∗}f^{∗}T_{Σ}.

Hence the map γ : N_{f_{D}} ^{}i^{∗}N_{f} is an isomorphism, and ρ is given by the composition

H^{0}(C, N_{f}) ^{i^{∗}}^{}H^{0}(f(C) ∩ D, i^{∗}N_{f}) ^{γ^{−1}}^{}H^{0}(f(C) ∩ D, N_{f_{D}}).

To finish the proof we note that df(p) ≠ 0 for any p ∈ i(f(C) ∩ D). Thus i^{∗}N_{f}^{tor} = 0, which implies H^{0}(C, N_{f}^{tor}) ⊂ Ker(ρ). ■

Theorem 2.8 Let Σ be a smooth projective algebraic surface, C′ ⊂ Σ be a smooth curve, and let L be a line bundle on Σ. Consider an irreducible variety V ⊆ |L| whose generic element C_{0} is a reduced curve. Assume that C_{0}^{i}.K_{Σ} < −1 for any irreducible component C_{0}^{i} ⊆ C_{0}. Then

dim(V) ≤ −C_{0}.K_{Σ} + g − 1.  (2.2)

Furthermore, if the equality holds, and

C_{0}^{i}.K_{Σ} < −3  (2.3)

for any singular component C_{0}^{i} of C_{0}, then C_{0} has only nodes as its singularities, and C_{0} intersects C′ transversally.

Proof: Let C be the normalization of C_{0} and let f : C ^{}Σ be the natural map. Then dim(h^{0}(C, N_{f}/N_{f}^{tor})) ≤ −C_{0}.K_{Σ} +g −1 implies (2.2), by Proposition 2.4. So it is enough to show the analogous inequality for every irreducible component of C. Thus we can assume that C is irreducible.

Choose an invertible sheaf F on C such that the sequence

0 ^{}N_{f}/N_{f}^{tor}  ^{}F ^{}N_{f}^{tor}  ^{}0

is exact (the existence of such F is completely obvious). Then

c_{1}(F) = c_{1}(N_{f}) = c_{1}(f^{∗}T_{Σ}) − c_{1}(T_{C}) = 2g − 2 + c_{1}(f^{∗}T_{Σ}) > 2g − 1. (2.4)

Hence

h^{0}(C, N_{f}/N_{f}^{tor}) ≤ h^{0}(C, F) = c_{1}(F) + 1 − g = −C_{0}.K_{Σ} + g − 1

by the Riemann-Roch theorem, and the equality holds if and only if N_{f}^{tor} = 0.

For the second part we note that if the dimension of V equals −C_{0}.K_{Σ} + g − 1, then N_{f}^{tor} = 0, and hence df ≠ 0 everywhere. So it remains to prove that C_{0} has no triple points, and that all its double points have two different tangent directions. If p ∈ C_{0} is a triple point and q_{1}, q_{2}, q_{3} ∈ C are three points mapped to p, then any ξ ∈ T_{0}V vanishing at q_{1} and q_{2}, must vanish at q_{3} as well. However, due to the Riemann-Roch theorem, inequality (2.4), and condition (2.3), there exists η ∈ H^{0}(C, N_{f}/N_{f}^{tor}) such that η(q_{1}) = η(q_{2}) = 0 ≠ η(q_{3}). Thus

dim T_{0}V < h^{0}(C, N_{f}/N_{f}^{tor}) = −C_{0}.K_{Σ} + g − 1,

which contradicts the equality in (2.2). If p ∈ C_{0} is a double point with a unique tangent direction and q_{1}, q_{2} ∈ C are the two pre-images of p, then any ξ ∈ T_{0}V vanishing at q_{1}, must also vanish at q_{2}. However, applying Riemann-Roch theorem, inequality (2.4), and condition (2.3), we can find η ∈ H^{0}(C, N_{f}/N_{f}^{tor}), such that η(q_{1}) = 0 ≠ η(q_{2}). Hence

dim T_{0}V < h^{0}(C, N_{f}/N_{f}^{tor}) = −C_{0}.K_{Σ} + g − 1,

which is a contradiction.

It remains to prove that if dim(V) = −C_{0}.K_{Σ} + g − 1, then C_{0} intersects C′ transversally. Assume that dim(V) = −C_{0}.K_{Σ} + g − 1. Then the linear system V has no fixed components; hence C_{0} does not contain C′. If the intersection C_{0} ∩ C′ is not transversal at some point p, then either p has at least two pre-images q_{1}, q_{2} ∈ C or any ξ ∈ T_{0}V vanishes at q, where q ∈ C is the unique pre-image of p. In the first case any ξ ∈ T_{0}V vanishing at q_{1} must also vanish at q_{2}. So both cases contradict the Riemann-Roch theorem, due to (2.4) and (2.3). ■

2.2 Severi varieties on Hirzebruch surfaces

Let Σ_{n} = ℙroj(O_{ℙ^{1}}⊕O_{ℙ^{1}}(n)) be the Hirzebruch surface and let π : Σ_{n} ^{}ℙ^{1} be the natural projection π : Σ_{n} ^{}ℙ^{1}. Consider two sections (1, 0), (0, σ) ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}⊕O_{ℙ^{1}}(n)). They define the maps (ℙ^{1}\Z(σ)) ^{}Σ_{n}. We denote the closures of the images of these maps by L_{0} and L_{∞}, respectively. It is clear that L_{∞} is independent of the choice of σ. The following facts will be useful

• The Picard group P ic(Σ_{n}) is a free abelian group generated by the classes F and L_{∞}, where F denotes the fiber of the projection π. It is important to mention that L_{0} ≡ nF + L_{∞}.

• The intersection form on NS(Σ_{n}) = P ic(Σ_{n}) is given by F^{2} = 0, L_{∞}^{2} = −n, and F.L_{∞} = 1.

• Any effective divisor M ∈ Div(Σ_{n}) is linearly equivalent to a linear combination of F and L_{∞} with non-negative coefficients. Moreover, if M does not contain L_{∞}, then it is linearly equivalent to a combination of F and L_{0} with non-negative coefficients.

• The canonical class is K_{Σ_{n}} ≡ −(2L_{∞} + (2 + n)F).

• Any smooth curve C ≡ dL_{0} + kF has genus g(C) = \frac{(d−1)(dn+2k−2)}{2}.

Now let us define the Severi varieties on Σ_{n}.

Definition 2.9 (1) Let

Σ̆_{n}^{δ} = {(p_{1}, ..., p_{δ}) ∈ Σ_{n}^{δ} | p_{i} ≠ p_{j} for any i ≠ j}

be the configuration space of δ points in Σ_{n}. For non-negative integers d, k, δ, we define the decorated Severi variety

U_{d,k,δ} ⊂ |O_{Σ_{n}}(dL_{0} + kF)| × Σ̆_{n}^{δ}

to be

{(C; p_{1}, ..., p_{δ}) | C is reduced, L_{∞} ⊈ C, and p_{1}, ..., p_{δ} ∈ C are nodes } .

(2) Let g, d, k be non-negative integers. We define the Severi variety V_{g,d,k} ⊆ |O_{Σ_{n}}(dL_{0} + kF)| to be the closure of the locus of reduced nodal curves of genus g which do not contain L_{∞}.

Next, we establish the basic properties of (decorated) Severi varieties:

Claim 2.10 (1) Let d, k, δ be non-negative integers. Then either U_{d,k,δ} = ∅ or U_{d,k,δ} is a smooth equidimensional variety of dimension

\frac{(d + 1)(nd + 2k + 2)}{2} − 1 − δ = dim |O_{Σ_{n}}(dL_{0} + kF)| − δ.

(2) Let ψ : U_{d,k,δ} ^{}|O_{Σ_{n}}(dL_{0} + kF)| be the projection to the first factor. Then for g = \frac{(d−1)(nd+2k−2)}{2} − δ

V_{g,d,k} = \{ \begin{matrix}\overline{ψ (U_{d,k,δ})} & if δ ≥ 0; \\ ∅ & otherwise. \end{matrix}

Proof: (1) Let us choose arbitrary (C^{0}; p_{1}^{0}, ..., p_{δ}^{0}) ∈ U_{d,k,δ}. We can find an open subset U ⊂ Σ_{n} isomorphic to 𝔸^{2} and containing all the points p_{1}^{0}, ..., p_{δ}^{0}. Fix a trivialization of O_{Σ_{n}}(dL_{0} + kF)(U) ≃ 𝕂[x, y]. Then in a neighborhood of (C^{0}; p_{1}^{0}, ..., p_{δ}^{0}) U_{d,k,δ} is given by the (homogeneous in the coefficients) equations f(p_{l}) = f′_{x}(p_{l}) = f′_{y}(p_{l}) = 0, 1 ≤ l ≤ δ, where f ∈ H^{0}(Σ_{n}, O_{Σ_{n}}(dL_{0} + kF)). We denote (x_{l}^{0}, y_{l}^{0}) = p_{l}^{0} and define a_{ij}^{l} to be the coefficients of f(x + x_{l}^{0}, y + y_{l}^{0}), f = ∑ a_{ij}x^{i}y^{j} ∈ 𝕂[x, y]. Let f_{0}(x, y) = ∑ β_{ij}x^{i}y^{j} be an equation defining C^{0}. Then

d(f(p_{l}))_{|_{(f_{0},p^{0}_{1},...,p^{0}_{δ})}} = da_{00}^{l},
d(f′_{x}(p_{l}))_{|_{(f_{0},p^{0}_{1},...,p^{0}_{δ})}} = da_{10}^{l} + 2β_{20}^{l}dx_{l} + β_{11}^{l}dy_{l},

and

d(f′_{y}(p_{l}))_{|_{(f_{0},p^{0}_{1},...,p^{0}_{δ})}} = da_{01}^{l} + β_{11}^{l}dx_{l} + 2β_{02}^{l}dy_{l}.

The points p_{1}^{0}, ..., p_{δ}^{0} are nodes of C^{0}, hence the matrices

(\begin{matrix}2β_{20}^{l} & β_{11}^{l} \\ β_{11}^{l} & 2β_{02}^{l} \end{matrix})

are invertible (even in characteristic 2). So, it remains to prove that da_{00}^{1} = ... = da_{00}^{δ} = 0 defines a subspace of codimension δ in the tangent space to |O_{Σ_{n}}(dL_{0} + kF)| at C^{0}. In other words we have to prove that

h^{0}(C^{0}, J (dL_{0} + kF)) = h^{0}(C^{0}, O_{C^{0}}(dL_{0} + kF)) − δ,  (2.5)

where J denotes the ideal of the zero-dimensional reduced subscheme

X = ∪_{i=1}^{δ}p_{i} ⊂ C^{0}.

H^{1}(C^{0}, J (dL_{0} + kF)) = 0 by Claim 2.12, which implies (2.5).

(2) The inclusion

V_{g,d,k} ⊆ \{ \begin{matrix}\overline{ψ (U_{d,k,δ})} & if δ ≥ 0; \\ ∅ & otherwise. \end{matrix}

is obvious. Let us prove the opposite direction. If g ≥ \frac{(d−1)(nd+2k−2)}{2}, then there is nothing to prove. Otherwise let U ⊆ U_{d,k,δ} be an irreducible component, and let (C; p_{1}, ..., p_{δ}) ∈ U be a generic point. Then by part (1) and Theorem 2.8

−C.K_{Σ_{n}} + g(C) − 1 ≥ dim ψ (U_{d,k,δ}) = dim |O_{Σ_{n}}(dL_{0} + kF)| − δ. (2.6)

Hence g(C) ≥ \frac{(d−1)(nd+2k−2)}{2} − δ = g_{ar}(C) − δ, which is possible only if the equality holds. Thus the equality holds in (2.6) as well, and by Theorem 2.8 this implies the nodality of the curve C. So

ψ(C; p_{1}, ..., p_{δ}) ∈ V_{g,d,k}

and we are done. ■

It is easy to see that if (C; p_{1}, ..., p_{δ}) ∈ U_{d,k,δ} and C has exactly δ nodes, then the map ψ : U_{d,k,δ} ^{}V_{g,d,k} for g = \frac{(d−1)(nd+2k−2)}{2} − δ, is étale in a neighborhood of (C; p_{1}, ..., p_{δ}).

Corollary 2.11 If V_{g,d,k} ≠ ∅ then it has pure dimension nd+2k +2d+g −1, and for any C ∈ V_{g,d,k} having exactly δ = \frac{(d−1)(nd+2k−2)}{2} − g nodes, V_{g,d,k} is smooth at C, and T_{C}(V_{g,d,k}) ≃ H^{0}(C, I(C)), where I ⊂ O_{C} is the conductor ideal.

Claim 2.12 Let C ∈ |O_{Σ_{n}}(dL_{0} + kF)| be a reduced curve. Assume that the restriction of the canonical class to any irreducible component of C is negative^{2}. Let q_{1}, ..., q_{r} ∈ C be some of the nodes of C, and let J ⊂ O_{C} be the ideal of ∪_{i=1}^{r}q_{i} ⊂ C. Then H^{1}(C, J (C)) = 0.

Proof: We prove this claim by induction on the number of irreducible components of C. Assume that C is irreducible. Let p : C̃ ^{}C be its normalization. Then

H^{1}(C, J (dL_{0} + kF)) = H^{1}(C̃, p^{∗}J (dL_{0} + kF)) = 0

by the Riemann-Roch theorem, since

deg(J (dL_{0} + kF)) = C^{2} − 2r = C.(C + K_{Σ_{n}}) − 2r − C.K_{Σ_{n}} > 2g(C̃) − 2.

We assume now that C has l > 1 irreducible components and that the statement is correct for any curve with the number of components less than l. Let C_{1} be an irreducible component. Then C = C_{1}∪C_{2} and without loss of generality assume q_{1}, ..., q_{r_{1}} ∈ C_{1}\C_{2}, q_{r_{1}+1}, ..., q_{r_{2}} ∈ C_{1} ∩ C_{2}, q_{r_{2}+1}, ..., q_{r} ∈ C_{2}\C_{1}. Let J_{2} ⊂ O_{C_{2}} be the ideal of ∪_{i=r_{2}+1}^{r}q_{i}, and let J_{1} ⊂ O_{C_{1}} be the ideal of ∪_{i=1}^{r_{2}}q_{i}. Then the following sequence

0 ^{}J_{2}(C_{2}) ^{×C_{1}}^{}J (C) ^{}J_{1}(C) ^{}0

is exact. Hence, by the induction hypothesis it is enough to prove that H^{1}(C_{1}, J_{1}(C)) = 0. Let p : C̃_{1} ^{}C_{1} be the normalization of C_{1}. Then

H^{1}(C_{1}, J_{1}(C)) = H^{1}(C̃_{1}, p^{∗}J_{1}(C)) = 0

by the Riemann-Roch theorem, since

\begin{aligned}deg(p^{∗}J_{1}(C)) & = & C_{1}^{2} − 2r_{1} + C_{1}.C_{2} − (r_{2} − r_{1}) \\ & ≥ & C_{1}.(C_{1} + K_{Σ_{n}}) − 2r_{1} − C_{1}.K_{Σ_{n}} > 2g(C̃_{1}) − 2. ■ \end{aligned}

3 The Result

Theorem 3.1 Let g, k, d be non-negative integers. If V_{g,d,k} ≠ ∅, then it has a unique component, whose generic point corresponds to an irreducible curve.

Proof: Let L_{i}, 1 ≤ i ≤ d, be generic curves in the linear system |O_{Σ_{n}}(L_{0})|, and let F_{i}, 1 ≤ i ≤ k, be generic curves in the linear system |O_{Σ_{n}}(F)|. Define

Γ = L_{1} ∪ ... ∪ L_{d} ∪ F_{1} ∪ ... ∪ F_{k} ∈ |O_{Σ_{n}}(dL_{0} + kF)| .

2If n < 2 then this condition is empty, otherwise it is equivalent to C not containing L_{∞} as its component.

Proposition 3.2 Consider an arbitrary component V of the Severi variety V_{g,d,k}. Then Γ ∈ V.

Proposition 3.3 There exists a unique component V ⊂ V_{g,d,k} containing Γ, whose generic point corresponds to an irreducible curve.

The theorem now follows. ■

3.1 Proof of Proposition 3.2

To prove the proposition we will need the following lemma

Lemma 3.4 Let (Σ, L) be a smooth rational surface equipped with a line bundle, and let L ⊂ Σ be a smooth curve. Let p_{1}, ..., p_{r} ∈ L be arbitrary points, k_{1}, ..., k_{r} be non-negative integers, and let

R ⊂ {C ∈ |L| : L ∩ C = ∑_{i=1}^{r} k_{i}p_{i}}

be a non-empty subvariety. Choose a generic curve C ∈ R. If C is reduced and C_{i}.(K_{Σ} + L) < −1 for any irreducible component C_{i} ⊂ C, then

dim(R) ≤ −C.K_{Σ} + g(C) − 1 − L.C.

Moreover, if the equality holds, and for any singular component C_{i} of C we have C_{i}.(K_{Σ} + L) < −3, then C has only nodes as its singularities outside of L, and for any smooth irreducible curve C′ that does not contain any of the p_{i}, a generic curve D ∈ R intersects C′ transversally.

Proof: The proof is by induction on L.C. If L.C = 0, then the lemma follows from Theorem 2.8, since L.C_{i} = 0 for any irreducible component C_{i} ⊆ C.

Assume now that k_{i} > 0 for some i. Without loss of generality k_{1} > 0. Consider the blow up Σ̃ = Bl_{p_{1}}(Σ) with its natural projection π : Σ̃ ^{}Σ. We denote the proper transform of L by L̃. Then OΣ̃(L̃) ≃ π^{∗}O_{Σ}(L) ⊗ OΣ̃(−E), where E denotes the exceptional divisor π^{−1}(p_{1}). Let C̃ ⊂ Σ̃ be the proper transform of C. Then OΣ̃(C̃) ≃ π^{∗}O_{Σ}(C) ⊗ OΣ̃(−m_{1}E), where m_{1} = mult_{p_{1}}(C). Define p′_{1} = E ∩ L̃ and p′_{i} = p_{i} for any i > 1, and consider the family

R_{1} = {X ∈ |OΣ̃(C̃)| : g(X) = g(C) and X ∩ L̃ = ∑_{i=1}^{r} k′_{i}p′_{i}} ,

where k_{1} = m_{p′_{1}}(C̃, L̃) and k′_{i} = k_{i} for all i > 1. It is clear that dim(R_{1}) ≥ dim(R), since C ∈ R is generic. Since KΣ̃ ≡ π^{∗}K_{Σ} + E,

C̃_{i}.(KΣ̃ + L̃) = C_{i}.(K_{Σ} + L) < −1

for any irreducible component C̃_{i} ⊂ C̃. Since L̃.C̃ = L.C − m_{1} < L.C,

dim(R) ≤ dim(R_{1})
≤ −C̃.KΣ̃ + g(C̃) − 1 − L̃.C̃ = −C.K_{Σ} + g(C) − 1 − L.C,

by the induction hypothesis. Moreover, if the equalities hold and for any singular component C_{i} of C we have C_{i}.(K_{Σ} + L) < −3, which implies C̃_{i}.(KΣ̃ + L̃) < −3, then C has only nodes as its singularities outside of L, and for any irreducible curve C′ satisfying C′ ∩ {p_{1}, ..., p_{r}} = ∅, a generic curve D ∈ R intersects C′ transversally. ■

Corollary 3.5 Let n, d, k be non-negative integers and let p_{1}, ..., p_{r} ∈ L_{∞} ⊂ Σ_{n} be arbitrary points. Choose k_{1}, ..., k_{r} positive integers such that ∑_{i=1}^{r} k_{i} = k. Let

R ⊂ {C ∈ |O_{Σ_{n}}(dL_{0} + kF)| : L∞ ∩ C = ∑_{i=1}^{r} k_{i}p_{i}} ,

be a non-empty subvariety, and let C ∈ R be generic element. If C is reduced, then

dim(R) ≤ −C.K_{Σ_{n}} + g(C) − 1 − k.  (3.1)

Moreover, if the equality holds, then C has only nodes as its singularities outside of L_{∞}, and C intersects L_{0} transversely.

Proof: By definition k = C.L_{∞}; hence it is enough to prove (3.1) only for irreducible C. If C = F, then there is nothing to prove, otherwise d > 0. Thus C.(K_{Σ_{n}} + L_{∞}) = −k − (2 + n)d < −1 and if C is singular, then d ≥ 2, and C.(K_{Σ_{n}} + L_{∞}) = −k − (2 + n)d < −3. Now, (3.1) follows from the lemma.

Assume that the equality holds in (3.1). Let C_{1}, ..., C_{s} be the irreducible components of type F, and let C_{s+1}, ..., C_{r} be the irreducible components of other types. Then C_{1}, ..., C_{s} are generic elements of |O_{Σ_{n}}(F)|, and C_{s+1} ∪ ...∪C_{r} is a nodal curve intersecting L_{0} transversally, due to the lemma. This implies the last part of the corollary. ■

Claim 3.6 Let g, k, n, d ≥ 0 be natural numbers such that V_{g,d,k} ≠ ∅, and let V ⊂ V_{g,d,k} be any component. Then given g + nd + 2k + 2d − 1 points p_{1}, ..., p_{g+nd+2k+2d−1} ∈ Σ_{n} in general position, there exists a finite positive number of curves C ∈ V passing through these points.

Proof: The claim follows from Corollary 2.11 and Riemann-Roch Theorem. ■ Proof of Proposition 3.2:

Step 0: Prelude. First, we would like to explain the idea of the proof. The proof goes by induction on d. For d = 0, 1 it is easy. To prove the induction step, we degenerate the surface Σ_{n} into a fan (in Ran’s terminology [9]) X^{0} ∪...∪X^{r}, where X^{j} ≅ Σ_{n} for all j, and X^{j} ∩X^{j+1} is L_{0} on X^{j} and L_{∞} on X^{j+1}. Next we show that the degeneration of V contains the degeneration of the curve Γ, which allows us to prove that V itself contains the required curve.

The following remark is important: it is enough to prove the statement for V whose generic element is irreducible, since if the generic curve C ∈ V is reducible, then C = C_{1} ∪ ... ∪ C_{l}, where C_{i} ∈ V_{i} ⊆ V_{g(C_{i}),d_{i},k_{i}} for some d_{i}, k_{i}. So we have a natural map ∏_{i=1}^{l} V_{i} ^{}V and if the proposition is true for all V_{i}, then it is true for V as well.

Step 1: The base of induction. If d = 0, then k = 1, g = δ = 0 and the proposition is obvious. If d = 1, then any nodal curve C ∈ V contains fibers F_{1}, ..., F_{δ} passing through the nodes of C, since C.F = 1. Thus δ = 0, since the generic C ∈ V is irreducible. So

V = V_{g,1,k} = |O_{Σ_{n}}(L_{0} + kF)|

and Γ ∈ V. Next we show the induction step.

Step 2: The construction of the fan. Our goal is to construct a flat family of surfaces ρ : Z ^{}𝔸^{1} and a flat family of curves C ⊂ Z ^{}𝔸^{1} such that

1. Z_{t} = Σ_{n} for all t ≠ 0

2. Z_{0} = X^{0} ∪ ... ∪ X^{r} where X^{j} ≅ Σ_{n} for all j, and X^{j} ∩ X^{j+1} is L_{0} = L_{0}^{j} on X^{j} and L_{∞} = L_{∞}^{j+1} on X^{j+1}

3. C_{t} ∈ V for almost all t ≠ 0

4. C_{0} does not contain L_{∞}^{j} for all j

5. There are at least two indices j_{1} ≠ j_{2}, such that C_{0} ∩ X^{j_{i}} is not a collection of fibers of X^{j_{i}} ≅ Σ_{n}.

Consider the trivial family Z^{0} = Σ_{n} × 𝔸^{1} ^{}𝔸^{1}. Let

σ : 𝔸^{1} ^{}Σ_{n}^{g+nd+2k+2d−1}

be a generic algebraic map for which σ(0) ∈ Σ_{n}^{g+dn+k+2d−2} × L_{∞}^{k+1} is in general position. Then σ takes the generic point of 𝔸^{1} to the generic point of Σ_{n}^{g+nd+2k+2d−1}; hence we can find a flat family of curves C^{0} ⊂ Σ_{n} × 𝔸^{1} such that C_{t}^{0} ∈ V for almost all t. The curve C_{0}^{0} contains L_{∞} as its component, so C_{0}^{0} = lL_{∞} ∪ C′ and l > 0. We claim that l < d, since otherwise C_{0}^{0} would be a union of dL_{∞} with dn + k fibers, but such a curve can not pass through g + dn + k + 2d − 2 points in general position, since d > 1 and g ≥ 0 due to Step 0. Thus

0 < l < d.  (3.2)

Next we shall modify the pair (Z^{0}, C^{0}) in order to obtain the required family (Z, C). Namely, we shall blow up Z^{0} along L_{∞} × {0}, replace C^{0} by its strict transform, and continue this process till we get (Z, C) satisfying conditions 1-5.

Let F ⊂ Σ_{n} be a generic fiber and let L ⊂ Σ_{n} be a generic curve in the system |O_{Σ_{n}}(L_{0})|. Then W = Σ_{n}\(F ∪L) ≃ 𝔸^{2}. We choose coordinates (x, y) on W such that x = 0 defines L_{∞} ∩W and for any c ∈ 𝕂 y = c defines F_{c} ∩W for some fiber F_{c} ⊂ Σ_{n}. Let f(x, y, t) ∈ 𝕂[x, y, t] be a polynomial defining C^{0} ∩ (W × 𝔸^{1}). Consider f_{y}(x, t) = f(x, y, t) ∈ 𝔽[x, t], where 𝔽 = \overline{𝕂(y)}.

Claim 3.7 Let 𝔽 be an algebraically closed field, and let

S_{0} = Spec 𝔽[x, t].

We define

S_{r} = Bl_{p_{r}}(Bl_{p_{r−2}}(...(Bl_{p_{1}})..),

where p_{1} = (0, 0) and p_{i+1} is the intersection of the exceptional divisor E_{i} with the strict transform of the t-axis. Let g(x, t) ∈ 𝔽[x, t] be a polynomial not divisible by t and x, but vanishing at (0, 0). Then there exist N and r such that

Ĉ ∩ {p_{r+1}} = Ĉ ∩ E_{i} ∩ E_{i−1} = ∅

for any 1 ≤ i ≤ r, where E_{0} is the strict transform of the x-axis and Ĉ denotes the strict transform on S_{r} of the curve given by g(x, t^{N}) = 0 if N is not divisible by cahr(𝔽), and given by g(x, t^{N} + t^{N+1}) = 0 if N is a multiple of cahr(𝔽).

Since f_{y}(x, t) is not divisible by t and x, there exist numbers N and r as in the claim. Without loss of generality we can assume that N = 1, since this will be the case after replacing σ by σ′ defined by either σ′(t) = σ(t^{N}) or σ′(t) = σ(t^{N} + t^{N+1}), respectively.

Now we can construct the pair (Z, C) satisfying conditions 1-5. To do this, we first blow up Z^{0} along L_{∞} × {0}. Then the exceptional divisor X^{1} is isomorphic to Σ_{n}. Next, we blow up the resulting family along L_{∞} ⊂ X^{1}, and we proceed this way r times till, finally, we get the required family of surfaces ρ : Z ^{}𝔸^{1}. The family of curves C is defined to be the strict transform of C^{0}. It is clear that conditions 1, 2, 3 are satisfied for the pair (Z, C). Condition 4 follows immediately from Claim 3.7, and finally (3.2) implies condition 5.

Step 3: The construction of the degeneration. First we shall prove that ρ_{∗}O_{Z}(C) is locally free. By the semi-continuity theorems, it is enough to show that H^{1}(Z_{t}, O_{Z_{t}}(C_{t})) = 0 for all t ∈ 𝔸^{1}. For t ≠ 0 this is obvious. Consider the short exact sequence

0 ^{}O_{Z_{0}}(C_{0}) ^{}⊕_{i=0}^{r} O_{X^{i}}(C_{0}) ^{}⊕_{i=0}^{r−1} O_{X^{i}∩X^{i+1}}(C_{0}) ^{}0

and define C_{0}^{i} = C_{0} ∩ X_{i}. Then C_{0}^{i} ∈ |O_{X^{i}}(d_{i}L_{0}^{i} + k_{i}F^{i})| for some d_{i}, k_{i} ≥ 0, satisfying ∑_{i=0}^{r} d_{i} = d, k_{i} = nd_{i+1} + k_{i+1}, and k_{r} = k. Thus we have: ⊕_{i=1}^{r}H^{1}(X^{i}, O_{X^{i}}(C_{0})) = 0, and in order to prove h^{1}(Z_{0}, O_{Z_{0}}(C_{0})) = 0, it is enough to show that the map

⊕_{i=0}^{r}H^{0}(X^{i}, O_{X^{i}}(C_{0})) ^{}⊕_{i=0}^{r−1} H^{0}(X^{i} ∩ X^{i+1}, O_{X^{i}∩X^{i+1}}(C_{0})) (3.3)

is surjective. It is clear that the map

H^{0}(Σ_{n}, O_{Σ_{n}}(aL_{0} + bF)) ^{}H^{0}(L_{∞}, O_{L_{∞}}(aL_{0} + bF))

is surjective for any a, b ≥ 0. This implies the surjectivity in (3.3).

Now we can construct the degeneration of V: ρ_{∗}O_{Z}(C) is locally free, and therefore the natural map

ρ_{∗}O_{Z}(C) ⊗ k(t) ^{}H^{0}(Z_{t}, O_{Z_{t}}(C_{t}))

is an isomorphism for any t ∈ 𝔸^{1}. Thus V = V_{t} ⊂ |O_{Z_{t}}(C_{t})| for t ≠ 0, degenerates into a variety V_{0} ⊂ |O_{Z_{0}}(C_{0})|, which is defined to be the zero fiber of

\overline{V × 𝕂^{∗}} ⊂ ℙroj(ρ_{∗}O_{Z}(C)) ^{}𝔸^{1}.

Step 4: The count of dimensions. The goal of this step is to prove that V_{0} contains a nodal curve. Let W ⊆ V_{0} be the irreducible component containing C_{0}. We can assume that C_{0} is a generic element of W. Denote C_{0}^{i} = (C_{0}^{i})^{red}. Then by Corollary 3.5 and by the genus formula^{3}

dim(W) ≤ ∑_{i=0}^{r} (−C_{0}^{i}.K_{X^{i}} + g(C_{0}^{i}) − 1) − ∑_{i=0}^{r−1} C_{0}^{i}.L_{∞}^{i}
≤ −(dL_{0} + kF).K_{Σ_{n}} + ∑_{i=0}^{r} (g(C_{0}^{i}) − 1) + ∑_{i=0}^{r−1} C_{0}^{i}.L_{∞}^{i}
≤ −(dL_{0} + kF).K_{Σ_{n}} + g − 1 = dim(V).

3We leave it as an exercise to the reader to check that ∑_{i=0}^{r} (g(C_{0}^{i}) − 1) + ∑_{i=0}^{r−1} C_{0}^{i}.L_{∞}^{i} ≤ g − 1 (see [9], proof of Lemma 2.1, for similar calculation).

Moreover, the second inequality becomes an equality if and only if C_{0}^{i} = C_{0}^{i}. Thus if the first and second inequalities are equalities, then C_{0}^{i} is nodal and intersects L_{0}^{i} transversally for any i, due to Corollary 3.5.

Since V_{0} is a flat limit of V_{t}, all its components have the same dimension as V_{t}, in particular dim(W) = dim(V). Thus a generic element of W is a nodal curve, and for any i the image of W under the natural projection |O_{Z_{0}}(C_{0})| ^{}|O_{X^{i}}(C_{0}^{i})| is a component of some Severi variety.

Step 5: Final accord. By the induction hypothesis we can find a curve Γ_{0} ∈ V_{0} such that all the curves Γ_{0}^{i} = Γ_{0} ∩ X^{i} are of the form Γ. Then Γ_{0} has δ′ = ∑_{i=0}^{r}(d_{i}k_{i} + n\frac{d_{i}(d_{i}−1)}{2}) nodes outside of ∪_{i=1}^{r}L_{0}^{i}. We denote these nodes by p_{0}^{1}, ..., p_{0}^{δ′}.

Next, we shall construct a deformation Γ_{t} ∈ V_{t} of Γ_{0} ∈ W preserving the δ′ nodes. To do this, consider the relative decorated Severi varieties U_{d,k,δ} ^{}𝔸^{1} and U_{d,k,δ′} ^{}𝔸^{1}, where δ = \frac{(d−1)(nd+2k−2)}{2} −g. Let U ⊆ U_{d,k,δ} be the irreducible component whose projection to ℙroj(ρ_{∗}O_{Z}(C)) is contained in \overline{V × 𝕂^{∗}}. Then (Γ_{0}; p_{0}^{i_{1}}, ..., p_{0}^{i_{δ}}) ∈ U ⊗ k(0) for some i_{1}, ..., i_{δ}.

Similarly to Claim 2.10, one can easily check that U_{d,k,δ′} and U_{d,k,δ} are smooth at the points (Γ_{0}; p_{0}^{1}, ..., p_{0}^{δ′}) and (Γ_{0}; p_{0}^{i_{1}}, ..., p_{0}^{i_{δ}}) respectively, and U_{d,k,δ′} is irreducible. Consider the natural projection ϕ : U_{d,k,δ′} ^{}U_{d,k,δ}, sending (C; p^{1}, ...., p^{δ′}) to (C; p^{i_{1}}, ...., p^{i_{δ}}). Then ϕ_{0}(U_{d,k,δ′} ⊗ k(0)) ⊂ U ⊗ k(0), which implies

ϕ_{t}(U_{d,k,δ′} ⊗ k(t)) ⊂ U ⊗ k(t)

for almost all t ∈ 𝔸^{1}. This allows us to construct a deformation Γ_{t} over a (formal) neighborhood of 0 ∈ 𝔸^{1}.

Using the relations ∑_{i=0}^{r} d_{i} = d, k_{i} = nd_{i+1} + k_{i+1}, and k_{r} = k, it is easy to see that

\begin{aligned}g(Γ_{t}) & = & \frac{(d−1)(nd+2k−2)}{2} − ∑_{i=0}^{r} (d_{i}k_{i} + n\frac{d_{i}(d_{i}−1)}{2}) \\ & = & \frac{(d−1)(nd+2k−2)}{2} − dk + n\frac{d(d−1)}{2} = 1 − k − d. \end{aligned}

Thus Γ_{t} has at least k + d components. Since L_{∞}^{r} ⊈ Γ_{0}, we know that L_{∞} ⊈ Γ_{t} for almost all t, and hence Γ_{t} = Γ for generic t. So we are done. ■ Proof of Claim 3.7: Let 𝕂 = \overline{𝔽((t))} and let ξ_{1}, ..., ξ_{l} ∈ 𝕂 be the roots of g_{t}(x) = g(x, t) = 0. Let ν be the natural valuation on the field 𝕂. We define N to be the common denominator of ν(ξ_{1}), ..., ν(ξ_{l}). Then the valuation has integral values on the roots of h_{t}(x) = g(t^{N}, x) (h_{t}(x) = g(t^{N} + t^{N+1}, x) respectively). Since g(x, t) is not divisible by t and x, there exist m_{1}, ..., m_{l} ∈ ℤ_{≥0} and α ∈ 𝔽(t) ⊂ 𝕂 such that

h_{t}(x) = α ∏_{i=1}^{l}(t^{m_{i}}x − ξ′_{i}),

ν(α) = 0, ν(ξ′_{i}) ≥ 0, and if m_{i} > 0 then ν(ξ′_{i}) = 0. Thus without loss of generality, we can assume that m_{1} = ... = m_{l} = 0 and ν(ξ′_{i}) > 0 for all i. Now we define r = max{ν(ξ′_{i})}, and one can easily check that it satisfies the required property. ■

3.2 Proof of Proposition 3.3

We start with some combinatorics.

Definition 3.8 (1) An ordered subset μ ⊆ Γ^{sing} consisting of r nodes is called an r−marking on the curve Γ.

(2) An r−marking μ is called irreducible if and only if the curve Γ\μ is connected.

(3) We define D-moves on the set of r−markings as follows: let μ = {p_{1}, ..., p_{r}} be an r−marking, let D, D′ ⊂ Γ be two different irreducible components, and let q, q′ ∈ D ∩ D′ be two nodes, such that q = p_{i} for some i. Then

D_{q,q′}(μ) = \begin{cases}\{p_{1}, ..., p_{i−1}, q′, p_{i+1}, ..., p_{r}\} & if q′ ∉ μ, \\ \{p_{τ_{ij}(1)}, ..., p_{τ_{ij}(r)}\} & if q′ = p_{j} for some j, \end{cases}

where τ_{ij} ∈ 𝔖_{r} denotes the elementary transposition τ_{ij} = (i j).

(4) Assume that n > 0. We define T-moves on the set of r−markings as follows: let μ = {p_{1}, ..., p_{r}} be an r−marking, let D, D′, D′′ ⊂ Γ be three different irreducible components, and let q ∈ D′∩D′′, q′ ∈ D∩D′′, q′′ ∈ D∩D′ be three nodes satisfying q′ ∉ μ and q = p_{i} for some i. Then

T_{q,q′,q′′}(μ) = \begin{cases}\{p_{1}, ..., p_{i−1}, q′′, p_{i+1}, ..., p_{r}\} & if q′′ ∉ μ, \\ \{p_{τ_{ij}(1)}, ..., p_{τ_{ij}(r)}\} & if q′′ = p_{j} for some j. \end{cases}

(5) Assume that n = 0. We define Q^{h}-moves and Q^{v}-moves as follows: let μ = {p_{1}, ..., p_{r}} be an r−marking, let X, X′, Y, Y′ ⊂ Γ be four different irreducible components satisfying X ≡ X′ ≡ L_{0} and Y ≡ Y′ ≡ F, and let q ∈ X ∩ Y, q′ ∈ X ∩ Y′, q′′ ∈ X′ ∩ Y, q′′′ ∈ X′ ∩ Y′ be nodes. Assume that q = p_{i} for some i, and q′′′ ∉ μ. If q′′ ∉ μ, we define

Q_{q,q′,q′′,q′′′}^{h}(μ) = \begin{cases}\{p_{1}, ..., p_{i−1}, q′, p_{i+1}, ..., p_{r}\} & if q′ ∉ μ, \\ \{p_{τ_{ij}(1)}, ..., p_{τ_{ij}(r)}\} & if q′ = p_{j} for some j. \end{cases}

If q′ ∉ μ, we define

Q_{q,q′,q′′,q′′′}^{v}(μ) = \begin{cases}\{p_{1}, .., p_{i−1}, q′′, p_{i+1}, .., p_{r}\} & if q′′ ∉ μ, \\ \{p_{τ_{ij}(1)}, .., p_{τ_{ij}(r)}\} & if q′′ = p_{j} for some j. \end{cases}

(6) Two r−markings, μ and μ′, are called equivalent if and only if they can be joined by a sequence of T-moves, Q-moves, and D-moves.

Notation 3.9 Let μ be any r−marking and let C, C′ ⊂ Γ be two different components. The following notation will be useful:

μ_{C,C′} = |μ ∩ C ∩ C′|, μ_{C} = |μ ∩ C|, and μ_{i} = p_{i}.

Claim 3.10 Let r > 0 be such an integer, that the set of irreducible r−markings on the curve Γ is not empty. Then for any pair of distinct irreducible components C, C′ ⊂ Γ satisfying C.C′ > 0 and for any q ∈ C ∩C′, there exist irreducible r−markings μ and μ′ such that q ∈ μ and q ∉ μ′.

Proof: Obvious. ■

¿From now on we will assume that n > 0. The remaining case, n = 0, is much easier, and the proof in this case can be obtained via the same lines as in the case we consider. Thus, we leave it to the reader.

Lemma 3.11 If n > 0 then any two irreducible r−markings on the curve Γ are equivalent.

Proof: It is enough to prove the lemma for the case r = k(d − 1) + n\frac{d(d−1)}{2} − (d − 1), since any irreducible r−marking can be extended to an irreducible marking of order k(d − 1) + n\frac{d(d−1)}{2} − (d − 1). We prove the lemma by induction on d + k. If either d = 2 = k + 2 or d ≤ 1, then the lemma is obvious. Assume now that the statement is true for all d + k ≤ m and let us prove it for d + k = m + 1. We can assume that m ≥ 2 and d ≥ 2, and let μ be an irreducible r−marking.

Step 1: The goal of this step is to prove that there exists a marking μ′ ∼ μ such that

μ′_{C,C′} = \begin{cases}C.C′ − 1 & if L_{d} ∈ \{C, C′\}, \\ C.C′ & otherwise. \end{cases}  (3.4)

Choose a component D ⊂ Γ in the following way: if k ≠ 0 then D = F_{k}, otherwise D = L_{1}. Then any irreducible component D′ ⊂ Γ different from D and satisfying D.D′ > 0 belongs to the linear system |O_{Σ_{n}}(L_{0})|. Moreover, there are at least two such components. Now, let us choose μ̃ ∼ μ, such that

μ̃_{D} = max_{μ′∼μ}{μ′_{D}}.

Then μ̃_{|_{Γ\D}} is an irreducible marking on Γ\D, since otherwise, due to the choice of D, we would be able to find two distinct irreducible components D′, D′′ ⊂ Γ different from D such that μ̃_{D,D′} < D.D′, μ̃_{D,D′′} < D.D′′, and μ̃_{D′,D′′} = D.D′′ > 0. Hence, there would exist q′′ ∈ D ∩ D′, q′ ∈ D ∩ D′′, and q ∈ D′ ∩ D′′, such that q′, q′′ ∉ μ and q = μ̃_{i} for some i. Thus μ̃_{D} < μ′_{D}, where μ′ = T_{q,q′,q′′}(μ̃) ∼ μ, which would be a contradiction.

Next, we shall prove that

μ̃_{D} = D.(Γ − D) − 1.  (3.5)

Consider two cases: k > 0 and k = 0. Case 1: k > 0. In this case D = F_{k}, and

μ̃_{D} ≥ r − ((k − 1)(d − 1) + n\frac{d(d − 1)}{2} − (d − 1)) ,

since μ̃_{|_{Γ\D}} is an irreducible marking on Γ\D. However,

r − ((k − 1)(d − 1) + n\frac{d(d − 1)}{2} − (d − 1)) = d − 1 = D.(Γ − D) − 1,

and we are done. Case 2: k = 0. In this case D = L_{1}, and

μ̃_{D} ≥ r − (n\frac{(d − 1)(d − 2)}{2} − (d − 2)) ,

since μ̃_{|_{Γ\D}} is irreducible. However,

r − (n\frac{(d − 1)(d − 2)}{2} − (d − 2)) = n(d − 1) − 1 = D.(Γ − D) − 1.

Thus, μ̃_{D} = D.(Γ − D) − 1 in both cases.

Now we can complete the proof of the first step. By (3.5) there exists a unique component D_{1} ⊂ Γ satisfying μ̃_{D,D_{1}} = D.D_{1} − 1, and μ̃_{D,D′} = D.D′ for all D′ ⊂ Γ different from D_{1}. By the induction hypothesis it remains to prove that there exists μ′ ∼ μ̃ such that μ′_{D,L_{d}} = D.L_{d} − 1 and μ′_{D} = μ̃_{D}. If D_{1} = L_{d} then there is nothing to prove. Otherwise, by the induction assumption and Claim 3.10, we can find an irreducible marking μ̂ ∼ μ̃ with the following properties: μ̂_{D,D′} = μ̃_{D,D′} for all D′ ⊂ Γ, and μ̂_{D_{1},L_{d}} < D_{1}.L_{d}. Then there exist q′′ ∈ D ∩ D_{1}, q ∈ D ∩ L_{d}, and q′ ∈ D_{1} ∩ L_{d} such that q′, q′′ ∉ μ̂, and q = μ̂_{i} for some i. Thus μ′ = T_{q,q′,q′′}(μ̂) satisfies the required condition.

Step 2: The goal of this step is to prove that any two markings satisfying (3.4) are equivalent. Let μ, μ′ be two such markings. Applying several Dmoves we can find a marking μ′′ ∼ μ′ such that μ′′ differs from μ only by the order of marked points. Namely, there exists τ ∈ 𝔖_{r} such that μ′′_{i} = μ_{τ(i)} for all i. We use the notation τ(μ) for such μ′′. It remains to prove that μ ∼ τ(μ). Without loss of generality we can assume that τ = τ_{ij} is a simple transposition; moreover we can assume that i = 1 and j = 2. Let D_{1}, D_{2}, D_{3}, D_{4} be components such that μ_{1} ∈ D_{1} ∩ D_{2} and μ_{2} ∈ D_{3} ∩ D_{4}. If {D_{1}, D_{2}} = {D_{3}, D_{4}}, then we apply D_{μ_{1},μ_{2}} to finish the proof. So we can assume that D_{1} ≠ D_{3}, D_{4} and D_{2} ≠ D_{3}.

If μ_{1} ∈ L_{d}∩D′ for some D′, then D′ ≡ L_{0} due to (3.4). Thus there exists a component D′′ ≠ D, L_{d} such that D′′.D′ > 0. Let μ_{i} ∈ D′′ ∩ D′ be any node, and let q ∈ L_{d} ∩ D′′ be the node not belonging to μ. Then μ ∼ T_{μ_{1},q,μ_{i}}(μ), and hence we can reduce the statement to the case when μ_{1} ∉ L_{d}. Similarly, we can assume that μ_{2} ∉ L_{d}, and thus that D_{i} ≠ L_{d} for all i. Now we shall consider several cases: Case 1: D_{2} = D_{4} ≡ L_{0}. Let q_{1} ∈ L_{d} ∩ D_{1}, q_{2} ∈ L_{d} ∩ D_{2}, and q_{3} ∈ L_{d} ∩ D_{3} be the nodes not belonging to μ. Then

τ_{12}(μ) = T_{μ_{2},q_{3},q_{2}}(T_{μ_{1},q_{1},q_{2}}(T_{μ_{2},q_{3},q_{2}}(μ))) ∼ μ.

Case 2: D_{2} = D_{4} ≡ F. In this case D_{1} ≡ D_{3} ≡ L_{0}. Let μ_{i} ∈ D_{1} ∩ D_{3} be any node. Then τ_{12} = τ_{1i} ∘ τ_{i2} ∘ τ_{1i}, and thus τ_{12}(μ) ∼ μ. So we reduce to the previous case. Case 3: {D_{1}, D_{2}} ∩ {D_{3}, D_{4}} = ∅. Without loss of generality D_{1} ≡ D_{3} ≡ L_{0}. Let μ_{i} ∈ D_{1} ∩ D_{3} be any node. Then τ_{12} = τ_{1i} ∘ τ_{i2} ∘ τ_{1i}, so τ_{12}(μ) ∼ μ, by the first case, and we are done, since cases 1, 2, 3 cover all the possibilities. ■

Lemma 3.12 Let X = X_{1} ∪ X_{2} ∪ ... ∪ X_{r} be a nodal curve. Assume that X_{2}, ..., X_{r} are generic curves of types L_{0} and F, and assume that X_{1} is a generic nodal rational curve whose type is one of L_{0}, L_{0}+F or 2L_{0}. Consider the variety W (X, L) ⊂ (X^{smooth})^{s} × |L| given by

{(p_{1}, ..., p_{s}; ξ) : p_{i} ≠ p_{j} for all i ≠ j, and ξ(p_{1}) = ... = ξ(p_{s}) = 0} ,

where s = deg(L ⊗ O_{X}) and either L ≃ O_{Σ_{n}}(L_{0}) or L ≃ O_{Σ_{n}}(F). Let (p_{1}, ..., p_{s}; ξ) ∈ W (X, L) be an arbitrary point, and let W ⊆ W (X, L) be the irreducible component containing (p_{1}, ..., p_{s}; ξ). Assume that n > 0 and there exist a ≠ b satisfying p_{a}, p_{b} ∈ X_{1}. Then

(p_{τ_{ab}(1)}, ..., p_{τ_{ab}(s)}; ξ) ∈ W,

where τ_{ab} ∈ 𝔖_{s} denotes the elementary transposition τ_{ab} = (a b).

Proof: If L ≃ O_{Σ_{n}}(F), we can assume that X_{1} ≡ 2L_{0}. Consider the variety W (X_{1}, L), which is irreducible since X_{1} is irreducible and s = 2. The natural forgetful map f : W ^{}W (X_{1}, L) is dominant, and since F.X_{i} ≤ 1 for all i > 1 it is also one-to-one. This implies the statement. Therefore we can assume that L ≃ O_{Σ_{n}}(L_{0}).

It is easy to see that W (X, L) is smooth. Let (x_{1}, ..., x_{s}; η) ∈ \overline{W (X, L)} be any point with the following properties: x_{i} ∈ X^{smooth} for all i, x_{a} = x_{b}, η(x_{a}) = d(η_{|_{X}})(x_{a}) = 0, and x_{i} ≠ x_{j} for all {i, j} ≠ {a, b}. Then (x_{1}, ..., x_{s}; η) is a smooth point of \overline{W (X, L)}. Thus to prove the lemma it suffices to show that \overline{W} contains a point with such properties. Consider the forgetful map f : \overline{W} ^{}X_{1}^{2} given by f(x_{1}, ..., x_{s}; η) = (x_{a}, x_{b}). This map is surjective, since L ≃ O_{Σ_{n}}(L_{0}) and n > 0, and it is clear that a generic α ∈ f^{−1}(Δ_{X_{1}}) satisfies the properties mentioned above. ■ Proof of Proposition 3.3: We start with the following remark: let U ⊂ U_{d,k,δ} be an irreducible component containing (Γ; p_{1}, ..., p_{δ}). We define a δ−marking μ^{Γ;p_{1},...,p_{δ}} on the curve Γ in the following way:

μ^{Γ;p_{1},...,p_{δ}} = {p_{1}, ..., p_{δ}}.

Then the generic curve C ∈ \overline{ψ(U)} is irreducible if and only if the marking μ^{Γ;p_{1},...,p_{δ}} is irreducible. The collection of all δ−markings corresponding to U is denoted by

M(U) = {μ^{Γ;p_{1},...,p_{δ}} | (Γ; p_{1}, ..., p_{δ}) ∈ U}.

Thus Proposition would be proven once we show that M(U) is closed under T-moves and under D-moves. This is what we proceed to proving now under the assumption n > 0.

Step 1: First, we shall prove that M(U) is closed under T-moves. We choose an arbitrary marking μ = μ^{Γ;p_{1},...,p_{δ}} ∈ M(U) and label the rest of the nodes of Γ by p_{δ+1}, ..., p_{δ′}, where δ′ = dk + n\frac{d(d−1)}{2}. Let D, D′, D′′ ⊂ Γ be three different irreducible components, and let q ∈ D′ ∩ D′′, q′ ∈ D ∩ D′′, q′′ ∈ D ∩ D′ be three nodes such that q′ ∉ μ and q ∈ μ. We shall show that T_{q,q′,q′′}(μ) ∈ M(U). Without loss of generality, q = p_{1}, q′′ = p_{δ′−1}, and q′ = p_{δ′}.

Consider the irreducible component U′ ⊂ U_{d,k,δ′−1} containing the pointed curve (Γ; p_{1}, ..., p_{δ′−1}), and let f : U′ ^{}U be the natural forgetful map. It is sufficient to prove that

(Γ; p_{δ′−1}, p_{2}, ..., p_{δ′−2}, p_{1}) ∈ U′,

since T_{q,q′,q′′}(μ) = μ^{f(Γ;p_{δ′−1},p_{2},...,p_{δ′−2},p_{1})}.

Let (C; x_{1}, ..., x_{δ′−1}) ∈ U′ be a generic element. Then C has a unique component C_{2} of type D + D′′ among its d + k − 1 irreducible components. Moreover, there exists another irreducible component C_{1} such that x_{1}, x_{δ′−1} ∈ C_{1}∩C_{2}. We denote C^{fix} = ∪_{i=2}^{d+k−1}C_{i}, where C_{1}, ..., C_{d+k−1} are the irreducible components of C.

Consider the locus U′′ ⊂ U′ of pointed curves (C′; x′_{1}, ..., x′_{δ′−1}) with the following property: C′ = C′_{1} ∪ C^{fix}, where C′_{1} ≡ C_{1} is generic. Let U′′′ ⊂ U′′ be the irreducible component containing the pointed curve (C; x_{1}, ..., x_{δ′−1}), and let

ϕ : U′′′ ^{}W (C^{fix}, O_{C^{fix}}(C_{1}))

be the natural forgetful map (cf. Lemma 3.12). Then ϕ is dominant and one-to-one. Hence

(Γ; p_{δ′−1}, p_{2}, ..., p_{δ′−2}, p_{1}) ∈ U′′′ ⊂ U′

by Lemma 3.12.

Step 2: The goal of this step is to prove that M(U) is closed under Dmoves. We choose an arbitrary marking μ = μ^{Γ;p_{1},...,p_{δ}} ∈ M(U) and label the rest of the nodes of Γ by p_{δ+1}, ..., p_{δ′}, where δ′ = dk + n\frac{d(d−1)}{2}. Let D, D′ ⊂ Γ be two different irreducible components, and let q, q′ ∈ D ∩ D′ be two nodes such that q ∈ μ. We shall show that D_{q,q′}(μ) ∈ M(U).

Consider the irreducible component U′ ⊂ U_{d,k,δ′} containing the pointed curve (Γ; p_{1}, ..., p_{δ′}), and let f : U′ ^{}U be the natural forgetful map. Then

D_{q,q′}(μ) = μ^{f(Γ;p_{τ_{ij} (1)},...,p_{τ_{ij} (δ′)})},

where q = p_{i}, q′ = p_{j}, and τ_{ij} ∈ 𝔖_{r} denotes the elementary transposition τ_{ij} = (i j). Hence it is sufficient to prove that

(Γ; p_{τ_{ij}(1)}, ..., p_{τ_{ij}(δ′)}) ∈ U′.

Let Γ_{1}, ..., Γ_{d+k} be the irreducible components of Γ. We can assume that D = Γ_{1}, and we denote Γ^{fix} = ∪_{i=2}^{d+k}Γ_{i}.

Consider the locus U′′ ⊂ U′ of pointed curves (C; x_{1}, ..., x_{δ′}) with the following property: C = C_{1} ∪ Γ^{fix}, where C_{1} ≡ Γ_{1} is generic. Let U′′′ ⊂ U′′ be the irreducible component containing (Γ; p_{1}, ..., p_{δ′}), and let

ϕ : U′′′ ^{}W (Γ^{fix}, O_{Γ^{fix}}(Γ_{1}))

be the natural forgetful map (cf. Lemma 3.12). Then ϕ is dominant and one-to-one. Hence

(Γ; p_{τ_{ij}(1)}, ..., p_{τ_{ij}(δ′)}) ∈ U′′′ ⊂ U′

by Lemma 3.12. ■

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