Regular components of moduli spaces of stable maps

GAVRIL FARKAS

1 Introduction

The purpose of this note is to prove the existence of ‘nice’ components of the Hilbert scheme of curves C ⊆ ℙ^{1} × ℙ^{r} of genus g ≥ 2 and bidegree (k, d). We can also phrase our result using the Kontsevich moduli space of stable maps to ℙ^{1} × ℙ^{r}. We work over an algebraically closed field of characteristic zero.

For a smooth projective variety Y and a class β ∈ H_{2}(Y, ℤ), one considers the moduli stack \overline{M}_{g}(Y, β) of stable maps f : C → Y, with C a reduced connected nodal curve of genus g and f_{∗}([C]) = β (see [FP] for the construction of these stacks). The open substack M_{g}(Y, β) of \overline{M}_{g}(Y, β) parametrizes maps from smooth curves to Y. By \overline{M}_{g}(Y, β) we denote the coarse moduli space corresponding to the stack \overline{M}_{g}(Y, β) and similarly \overline{M}_{g} is the moduli space corresponding to the stack \overline{M}_{g} of stable curves of genus g. We denote by π : \overline{M}_{g}(Y, β) → \overline{M}_{g} the natural projection. The expected dimension of the stack \overline{M}_{g}(Y, β) is

χ(g, Y, β) = dim(Y) (1 − g) + 3g − 3 − β · K_{Y} .

Since in general the geometry of \overline{M}_{g}(Y, β) is quite messy (e.g. existence of many components, some nonreduced and/or not of expected dimension), it is not obvious what the definition of a nice component of \overline{M}_{g}(Y, β) should be. Following Sernesi [Se] we introduce the following terminology: Definition. A component V of \overline{M}_{g}(Y, β) is said to be regular if it is generically smooth and of dimension χ(g, Y, β). We say that V has the expected number of moduli if

dim π(V) = min(3g − 3, χ(g, Y, β) − dim Aut(Y)).

In this paper we only construct regular components of moduli spaces of stable maps. We study the stacks \overline{M}_{g}(Y, β) when Y = ℙ^{1}×ℙ^{r}, r ≥ 3 and β = (k, d) ∈ H_{2}(ℙ^{1}×ℙ^{r}, ℤ). We denote by ρ(g, r, d) = g − (r + 1)(g − d + r) the Brill-Noether number governing the existence of 𝔤_{d}^{r}’s on curves of genus g. Our main result is the following:

Theorem 1 Let g, r, d and k be positive integers with r ≥ 3, ρ(g, r, d) < 0 and

(2 − ρ(g, r, d))r + 2 ≤ k ≤ (g + 2)/2.

Then there exists a regular component of the stack of maps M_{g}(ℙ^{1} × ℙ^{r}, (k, d)).

We introduce the Brill-Noether locus M_{g,d}^{r} = {[C] ∈ M_{g} : C has a 𝔤_{d}^{r}}, in the case ρ(g, r, d) < 0. The expected codimension of M_{g,d}^{r} inside M_{g} is −ρ(g, r, d). We view Theorem 1 as a tool in the study of the relative position of the loci M_{g,k}^{1} and M_{g,d}^{r} when r ≥ 3, ρ(g, 1, k) < 0 ⇔ k < (g + 2)/2 and ρ(g, r, d) < 0. The stack M_{g}(ℙ^{1} × ℙ^{r}, (k, d)) comes naturally into play when looking at the intersection in M_{g} of the loci M_{g,k}^{1} and M_{g,d}^{r}. In such a setting, if V is a regular component of M_{g}(ℙ^{1}×ℙ^{r}, (k, d)), then M_{g,k}^{1} and M_{g,d}^{r} intersect properly along π(V). It is very plausible that one has a similar statement to Theorem 1 when ρ(g, r, d) ≥ 0 and/or ρ(g, 1, k) ≥ 0, but from our perspective that seems of less interest because it would be essentially a statement about linear series on the general curve of genus g with no implications on the problem of understanding the geography of the Brill-Noether loci inside M_{g}.

Regarding the problem of existence of regular components of M_{g}(Y, β), so far the spaces M_{g}(ℙ^{r}, d) have received the bulk of attention. When r = 1, 2 the problem boils down to the study of the Hurwitz scheme and of the Severi variety of plane curves which are known to be irreducible and regular. For r ≥ 3 we have the following result of Sernesi (cf. [Se, p. 26]):

Proposition 1.1 For all g, r, d such that d ≥ r + 1 and

−\frac{g}{r} + \frac{r + 1}{r} ≤ ρ(g, r, d) < 0,

there exists a regular component V of M_{g}(ℙ^{r}, d) which has the expected number of moduli. A general point of V corresponds to an embedding C ↪ ℙ^{r} by a complete linear system (i.e. h^{0}(C, O_{C}(1)) = r + 1), the normal bundle N_{C} satisfies H^{1}(C, N_{C}) = 0 and the Petri map

μ_{0}(C) : H^{0}(C, O_{C}(1)) ⊗ H^{0}(C, K_{C}(−1)) → H^{0}(C, K_{C})

is surjective.

A. Lopez has obtained significant improvements on the range of g, r, d such that there exists a regular component of M_{g}(ℙ^{r}, d): if h(r) = (4r^{3} + 8r^{2} − 9r + 3)/(r + 3), then for all g, r, d such that −(2 − 6/(r + 3))g + h(r) ≤ ρ(g, r, d) < 0 there exists a regular component of M_{g}(ℙ^{r}, d) with the expected number of moduli (cf. [Lo]).

When Y is a smooth surface, methods from [AC] can be employed to show that if V is a component of M_{g}(Y, β) with dim(V) ≥ g + 1 and which contains a point [f : C → Y] with deg(f) = 1 (i.e. f is generically injective), then V is regular. Here it is crucial that the normal sheaf N_{f} is of rank 1 as then the Clifford Theorem provides an easy criterion for the vanishing of H^{1}(C, N_{f}), which turns out to be a sufficient criterion for regularity (see Section 2). Acknowledgments. This paper is part of my thesis written at the Universiteit van Amsterdam. The help of my adviser Gerard van der Geer is gratefully acknowledged.

2 Deformations of maps and smoothings of space curves

We review some facts about deformations of maps and smoothings of reducible nodal curves in ℙ^{r}. Our references are [Ran] and [Se].

We start by describing the deformation theory of maps between complex algebraic varieties when the source is (possibly) singular and the target is smooth. Let f : X → Y be a morphism between complex projective varieties, with Y being smooth. We denote by Def(X, f, Y) the space of first-order deformations of the map f when X and Y are not considered fixed. The space of first-order deformations of X (resp. Y) is denoted by Def(X) (resp. Def(Y)). We have the standard identification Def(X) = Ext^{1}(Ω_{X}, O_{X}). The deformation space Def(X, f, Y) fits in the following exact sequence:

Hom_{O_{X}}(f^{∗}Ω_{Y} , O_{X}) ⟶ Def(X, f, Y) ⟶ Def(X) ⊕ Def(Y) ⟶ Ext_{f}^{1}(Ω_{Y} , O_{X}). (1)

The second arrow is given by the natural forgetful maps, the space Hom_{O_{X}}(f^{∗}Ω_{Y} , O_{X}) = H^{0}(X, f^{∗}T_{Y}) parametrizes first-order deformations of f : X → Y when both X and Y are fixed, while for A, B, respectively O_{X} and O_{Y} -modules, Ext_{f}^{i}(B, A) denotes the derived functor of Hom_{f}(B, A) = Hom_{O_{X}}(f^{∗}B, A) = Hom_{O_{Y}} (B, f_{∗}A). Under reasonable assumptions (trivially satisfied when f is a finite map between nodal curves) one has that Ext_{f}^{1}(Ω_{Y} , O_{X}) = Ext^{1}(f^{∗}Ω_{Y} , O_{X}). Using (1) it follows that when X is smooth and irreducible and Y is rigid (e.g. a product of projective spaces) Def(X, f, Y) = H^{0}(X, N_{f}), where N_{f} = Coker{T_{X} → f^{∗}T_{Y} } is the normal sheaf of the map f.

For a smooth variety Y, a class β ∈ H_{2}(Y, ℤ) and a point [f : C → Y] ∈ M_{g}(Y, β) we have that T_{[f]}(\overline{M}_{g}(Y, β)) = H^{0}(C, N_{f}). If moreover deg(f) = 1 and H^{1}(C, N_{f}) = 0, then every class in H^{0}(C, N_{f}) is unobstructed, f is an immersion (cf. [AC, Lemma 1.4]) and \overline{M}_{g}(Y, β) is smooth and of the expected dimension at the point [f], that is,

[f] belongs to a regular component of \overline{M}_{g}(Y, β).

Let C ⊆ ℙ^{r} be a stable curve of genus g and degree d. If I_{C} is the ideal sheaf of C we denote by N_{C} := Hom(I_{C}/I_{C}^{2}, O_{C}) the normal sheaf of C in ℙ^{r}. Assume that H^{1}(C, N_{C}) = 0 and that h^{0}(C, O_{C}(1)) = r + 1, that is, C is embedded by a complete linear system. The differential of the map π : \overline{M}_{g}(ℙ^{r}, d) → \overline{M}_{g} at the point [C ↪ ℙ^{r}] is given by the natural map H^{0}(C, N_{C}) → Ext^{1}(Ω_{C}, O_{C}). If ω_{C} denotes the dualizing sheaf of C, then rk(dπ)_{[C↪ℙ^{r}]} = 3g − 3 − dim Kerμ_{0}(C), where

μ_{0}(C) : H^{0}(C, O_{C}(1)) ⊗ H^{0}(C, ω_{C}(−1)) → H^{0}(C, ω_{C})

is the Petri map. In particular (dπ)_{[C↪ℙ^{r}]} has rank 3g − 3 + ρ(g, r, d) if and only if μ_{0}(C) is surjective.

In the same setting, via the standard identification T_{[C]}(\overline{M}_{g})^{∨} = H^{0}(C, ω_{C} ⊗ Ω_{C}), the annihilator (Im(dπ)_{[C↪ℙ^{r}]})^{⊥} ⊆ H^{0}(C, ω_{C} ⊗ Ω_{C}) can be naturally identified with Im(μ_{1}(C)), where

μ_{1}(C) : Kerμ_{0}(C) → H^{0}(C, Ω_{C} ⊗ ω_{C})

is the Gaussian map obtained from taking the ‘derivative’ of μ_{0}(C) (cf. [CGGH, p. 163]).

In Section 3 we will smooth curves X ⊆ ℙ^{r} which are unions of two smooth curves C and E meeting quasi-transversally (i.e. having distinct tangent lines) at a finite set Δ. For such a curve one has the exact sequences (cf. [Se, p. 35])

0 ⟶ O_{E}(−Δ) ⟶ O_{X} ⟶ O_{C} ⟶ 0,  (2)

and

0 ⟶ Ω_{E} ⟶ ω_{X} ⟶ Ω_{C}(Δ) ⟶ 0.  (3)

Also in Section 3 we will use an inductive procedure to construct curves C ⊆ ℙ^{1}×ℙ^{r} with H^{1}(C, N_{C/ℙ^{1}×ℙ^{r}}) = 0. The induction step uses the following result (cf. [BE, Lemma 2.3]):

Proposition 2.1 Let C ⊆ ℙ^{r} be a smooth curve with H^{1}(C, N_{C}) = 0. We take r + 2 points p_{1}, …, p_{r+2} ∈ C in general linear position and a smooth rational curve E ⊆ ℙ^{r} of degree r which meets C quasi-transversally at p_{1}, …, p_{r+2}. Then X = C ∪ E is smoothable in ℙ^{r} and H^{1}(X, N_{X}) = 0.

3 Existence of regular components of M_{g}(ℙ^{1}×ℙ^{r}, (k, d))

In this section we prove the existence of regular components of \overline{M}_{g}(ℙ^{1} × ℙ^{r}, (k, d)) in the case k ≥ r + 2, d ≥ r ≥ 3, and ρ(g, r, d) < 0. We achieve this by constructing smooth curves C ⊆ ℙ^{1} × ℙ^{r} of bidegree (k, d) satisfying H^{1}(C, N_{C/ℙ^{1}×ℙ^{r}}) = 0.

Let us fix integers g ≥ 2, d ≥ r ≥ 3 and k ≥ 2, as well as a smooth curve C of genus g with maps f_{1} : C → ℙ^{1}, f_{2} : C → ℙ^{r}, such that deg(f_{1}) = k, deg(f_{2}(C)) = d and f_{2} is generically injective. Let us denote by f : C → ℙ^{1} × ℙ^{r} the product map. As usual we denote by G_{d}^{r}(C) the scheme parametrizing 𝔤_{d}^{r}’s on C.

There is a commutative diagram of exact sequences

0 ↓ T_{C} ↓

0 ⟶  T_{C}  ⟶  f^{∗}(T_{ℙ^{1}×ℙ^{r}})  ⟶  N_{f}  ⟶ 0
↓  ↓^{=}  ↓
0 ⟶ T_{C} ⊕ T_{C} ⟶ f_{1}^{∗}(T_{ℙ^{1}}) ⊕ f_{2}^{∗}(T_{ℙ^{r}}) ⟶ N_{f_{1}} ⊕ N_{f_{2}} → 0

↓

0  .

By taking cohomology in the last column, we see that the condition H^{1}(C, N_{f}) = 0 is equivalent to H^{1}(C, N_{f_{1}}) = 0 (automatic), H^{1}(C, N_{f_{2}}) = 0, and

Im{δ_{1} : H^{0}(C, N_{f_{1}}) → H^{1}(C, T_{C})} + Im{δ_{2} : H^{0}(C, N_{f_{2}}) → H^{1}(C, T_{C})} = H^{1}(C, T_{C}), (4) where δ_{1} and δ_{2} are coboundary maps. Condition (4) is equivalent (cf. Section 2) to

(dπ_{1})_{[f_{1}]} (T_{[f_{1}]}(M_{g}(ℙ^{1}, k))) + (dπ_{2})_{[f_{2}]} (T_{[f_{2}]}(M_{g}(ℙ^{r}, d))) = T_{[C]}(M_{g}),  (5)

where the projections π_{1} : M_{g}(ℙ^{1}, k) → M_{g} and π_{2} : M_{g}(ℙ^{r}, d) → M_{g} are the natural forgetful maps. Slightly abusing terminology, if C is a smooth curve and (l_{1}, l_{2}) ∈ G_{k}^{1}(C) × G_{d}^{r}(C) is a pair of base point free linear series on C, we say that (C, l_{1}, l_{2}) satisfies (5), if (C, f_{1}, f_{2}) satifies (5), where f_{1} and f_{2} are maps associated to l_{1} and l_{2}.

Recall that a base point free pencil 𝔤_{k}^{1} is said to be simple if the induced covering f : C → ℙ^{1} has a single ramification point x over each branch point and moreover e_{x}(f) = 2.

We prove the existence of regular components of M_{g}(ℙ^{1} × ℙ^{r}, (k, d)) using the following inductive procedure:

Proposition 3.1 Fix positive integers g, r, d and k with d ≥ r ≥ 3, k ≥ r + 2 and ρ(g, r, d) < 0. Let us assume that C ⊆ ℙ^{r} is a smooth nondegenerate curve of degree d and genus g, such that h^{1}(C, N_{C}) = 0, h^{0}(C, O_{C}(1)) = r + 1 and the Petri map

μ_{0}(C) = μ_{0}(C, O_{C}(1)) : H^{0}(C, O_{C}(1)) ⊗ H^{0}(C, K_{C}(−1)) → H^{0}(C, K_{C})

is surjective. Assume furthermore that C possesses a simple base point free pencil 𝔤_{k}^{1} say l, such that |O_{C}(1)|(−l) = ∅ and (C, l, |O_{C}(1)|) satisfies (5).

Then there exists a smooth nondegenerate curve Y ⊆ ℙ^{r} with g(Y) = g + r + 1, deg(Y) = d + r and a simple base point free pencil l′ ∈ G_{k}^{1}(Y), so that Y enjoys exactly the same properties: h^{1}(Y, N_{Y}) = 0, h^{0}(Y, O_{Y} (1)) = r + 1, the Petri map μ_{0}(Y) is surjective, |O_{Y} (1)|(−l′) = ∅ and (Y, l′, |O_{Y} (1)|) satifies (5).

Proof. We first construct a reducible k-gonal nodal curve X ⊆ ℙ^{r}, with p_{a}(X) = g + r + 1, deg(X) = d + r, having all the required properties, then we prove that X can be smoothed in ℙ^{r} preserving all properties we want.

Let f_{1} : C → ℙ^{1} be the degree k map corresponding to the pencil l. The covering f_{1} is simple hence the monodromy of f_{1} is the full symmetric group. Then since |O_{C}(1)|(−l) = ∅, we have that for a general λ ∈ ℙ^{1} the fibre f_{1}^{−1}(λ) = p_{1} + ⋯ + p_{k} consists of k distinct points in general linear position. Let Δ = {p_{1}, …, p_{r+2}} be a subset of f_{1}^{−1}(λ) and let E ⊆ ℙ^{r} be a rational normal curve (deg(E) = r) passing through p_{1}, …, p_{r+2}. (Through any r + 3 points in general linear position in ℙ^{r}, there passes a unique rational normal curve). We set X := C ∪ E, with C and E meeting quasi-transversally at Δ. Of course p_{a}(X) = g + r + 1 and deg(X) = d + r. Note that ρ(g, r, d) = ρ(g + r + 1, r, d + r).

We first prove that [X] ∈ \overline{M}_{g+r+1,k}^{1} (that is, X is a limit of smooth k-gonal curves), by constructing an admissible covering of degree k having as domain a curve X′, stably equivalent to X. Let X′ := X ∪ D_{r+3} ∪ … ∪ D_{k}, where D_{i} ≃ ℙ^{1} and D_{i} ∩ X = {p_{i}}, for i = r + 3, …, k. Take Y := (ℙ^{1})_{1} ∪_{λ} (ℙ^{1})_{2} a union of two lines identified at λ. We construct a degree k admissible covering f′ : X′ → Y as follows: take f′_{|C} = f_{1} : C → (ℙ^{1})_{1}, f′_{|E} = f_{2} : E → (ℙ^{1})_{2} a map of degree r + 2 sending the points p_{1}, …, p_{r+2} to λ, and finally f′_{|D_{i}} : D_{i} ≃ (ℙ^{1})_{2} isomorphisms sending p_{i} to λ. Clearly f′ is an admissible covering, so X which is stably equivalent to X′ is a k-gonal curve.

Let us consider now the space \overline{H}_{g+r+1,k} of Harris-Mumford admissible coverings of degree k (cf. [HM]) and denote by π_{1} : \overline{H}_{g+r+1,k} → \overline{M}_{g+r+1} the natural projection which sends a covering to the stable model of its source. We assume for simplicity that Aut(C) = {Id_{C}} which implies that Aut(f′) = {Id_{X′}}, so [f′] is a smooth point of \overline{H}_{g+r+1,k}. In the case when C has nontrivial automorphisms the argument carries through without change if we replace the space of admisible coverings with the space of twisted covers of Abramovich, Corti and Vistoli (cf. [ACV]).

We compute the differential of the map π_{1} at [f′]. We have T_{[f′]}(\overline{H}_{g+r+1,k}) = Def(X′, f′, Y) = Def(X, f, Y), where f = f′_{|_{X}} : X → Y. The differential (dπ_{1})_{[f′]} is the forgetful map Def(X, f, Y) → Def(X) and from the sequence (2.1) we get that Im(dπ_{1})_{[f′]} = u_{1}^{−1}(Im u_{2}), where u_{1} : Def(X) → Ext^{1}(f^{∗}Ω_{Y} , O_{X}) and u_{2} : Def(Y) → Ext^{1}(f^{∗}Ω_{Y} , O_{X}) are the dual maps of u_{1}^{∨} : H^{0}(X, ω_{X} ⊗ f^{∗}Ω_{Y}) → H^{0}(X, ω_{X} ⊗ Ω_{X}) and u_{2}^{∨} : H^{0}(X, ω_{X} ⊗ f^{∗}Ω_{Y}) → H^{0}(Y, ω_{Y} ⊗ Ω_{Y}). Here u_{2}^{∨} is induced by the trace map tr : f_{∗}ω_{X} → ω_{Y} . Starting with the exact sequence on X,

0 ⟶ Tors(ω_{X} ⊗ Ω_{X}) ⟶ ω_{X} ⊗ Ω_{X} ⟶ Ω_{C}^{⊗2}(Δ) ⊕ Ω_{E}^{⊗2}(Δ) ⟶ 0,

we can write the following commutative diagram of sequences

0  0  0
↓  ↓  ↓
H^{0}(Tors(ω_{X} ⊗ f^{∗}Ω_{Y})) ↪ H^{0}(ω_{X} ⊗ f^{∗}Ω_{Y}) ↠ H^{0}(2K_{C} − R_{1} + Δ) ⊕ H^{0}(2K_{E} − R_{2} + Δ)
↓^{(u_{1}^{∨})_{tors}} ↓^{u_{1}^{∨}} ↓
H^{0}(Tors(ω_{X} ⊗ Ω_{X})) ↪ H^{0}(ω_{X} ⊗ Ω_{X}) ↠  H^{0}(2K_{C} + Δ) ⊕ H^{0}(2K_{E} + Δ)

where R_{1} (resp. R_{2}) is the ramification divisor of the map f_{1} (resp. f_{2}). Taking into account that H^{0}(E, 2K_{E} −R_{2}+Δ) = 0 and that H^{0}(Y, ω_{Y} ⊗Ω_{Y}) = H^{0}(Tors(ω_{Y} ⊗Ω_{Y})), we obtain that

Im(dπ_{1})_{[f′]} = (H^{0}(C, 2K_{C} − R_{1} + Δ) ⊕ Ker(u_{2}^{∨})_{tors})^{⊥},  (6)

where (u_{2}^{∨})_{tors} : H^{0}(Tors(ω_{X} ⊗ f^{∗}Ω_{Y})) → H^{0}(Tors(ω_{Y} ⊗ Ω_{Y})) is the restriction of u_{2}^{∨}. The space Ker(u_{2}^{∨})_{tors} is just a hyperplane in H^{0}(Tors(ω_{X} ⊗ f^{∗}Ω_{Y})) ≃ ℂ^{r+2}.

Intermezzo. If we also assume that ρ(g, 1, k) < 0 and that [C] is a smooth point of M_{g,k}^{1} (which happens precisely when Aut(C) = {Id_{C}}, C has exactly one 𝔤_{k}^{1} and dim|2𝔤_{k}^{1}| = 2), then we can prove that the locus \overline{M}_{g+r+1,k}^{1} is smooth at [X] as well. Indeed, since Δ ∈ C_{r+2} was chosen generically in a fibre of the 𝔤_{k}^{1} on C, from Riemann-Roch we have that h^{0}(C, 2K_{C} − R_{1} + Δ) = g − 2k + 3 + r = codim(\overline{M}_{g+r+1,k}^{1}, \overline{M}_{g+r+1}). The fibre over [X] of the map π_{1} : \overline{H}_{g+r+1,k} → \overline{M}_{g+r+1} is identified with the space of degree r + 1 maps f_{2} : E → ℙ^{1} such that f_{2}(p_{1}) = … = f_{2}(p_{r+2}) = λ, hence it is r + 1 dimensional. We compute the tangent cone

T C_{[X]}(\overline{M}_{g+r+1,k}^{1}) = ⋃\{Im(dπ_{1})_{z} : z ∈ π_{1}^{−1}([X])\} = H^{0}(C, 2K_{C} − R_{1} + Δ)^{⊥},

which shows that [X] is a smooth point of the locus \overline{M}_{g+r+1,k}^{1}.

We compute now the differential

(dπ_{2})_{[X]} : T_{[X]}(Hilb_{d+r,g+r+1,r}) → T_{[X]}(\overline{M}_{g+r+1}),

which is the same thing as the differential at the point [X ↪ ℙ^{r}] of the projection π_{2} : \overline{M}_{g+r+1}(ℙ^{r}, d + r) → \overline{M}_{g+r+1}. We start by noticing that X is smoothable in ℙ^{r} and that H^{1}(X, N_{X}) = 0 (apply Proposition 2.1). We also have that X is embedded in ℙ^{r} by a complete linear system, that is, h^{0}(X, O_{X}(1)) = r + 1. Indeed, on one hand, since X is nondegenerate, h^{0}(X, O_{X}(1)) ≥ h^{0}(ℙ^{r}, O_{ℙ^{r}(1)}) = r + 1; on the other hand from the sequence (2) we have that h^{0}(X, O_{X}(1)) ≤ h^{0}(C, O_{C}(1)) = r + 1.

If X is embedded in ℙ^{r} by a complete linear system, we know (cf. Section 2) that

Im(dπ_{2})_{[X]} = (Imμ_{1}(X))^{⊥},

where μ_{1}(X) : Kerμ_{0}(X) → H^{0}(X, ω_{X} ⊗Ω_{X}) is the ‘derivative’ of the Petri map μ_{0}(X) : H^{0}(X, O_{X}(1)) ⊗ H^{0}(X, ω_{X}(−1)) → H^{0}(X, ω_{X}). We compute the kernel of μ_{0}(X) and show that μ_{0}(X) is surjective in a way that resembles the proof of Proposition 2.3 in [Se].

From the sequence (3) we obtain H^{0}(X, ω_{X}) = H^{0}(C, K_{C} + Δ), while from (2) we have that H^{0}(X, O_{X}(1)) = H^{0}(E, O_{E}(1)) (keeping in mind that H^{0}(C, O_{C}(1)(−Δ)) = 0, as p_{1}, …, p_{r+2} are in general linear position). Finally, using (3) again, we have that H^{0}(X, ω_{X}(−1)) = H^{0}(C, K_{C}(−1) + Δ). Therefore we can write the following commutative diagram:

H^{0}(C, O_{C}(1)) ⊗ H^{0}(C, K_{C}(−1)) ⟶^{μ_{0}(C)} H^{0}(C, K_{C})
↓ ↓
H^{0}(C, O_{C}(1)) ⊗ H^{0}(C, K_{C}(−1) + Δ) ⟶ H^{0}(C, K_{C} + Δ)
↓^{=}  ↓^{=}
H^{0}(X, O_{X}(1)) ⊗ H^{0}(X, ω_{X}(−1)) ⟶^{μ_{0}(X)} H^{0}(X, ω_{X}) .

It follows that Kerμ_{0}(C) ⊆ Kerμ_{0}(X). By Corollary 1.6 from [CR], our assumptions (μ_{0}(C) surjective and card(Δ) ≥ 4) enable us to conclude that μ_{0}(X) is surjective too. Then Kerμ_{0}(C) = Kerμ_{0}(X) for dimension reasons, hence also Imμ_{1}(X) = Imμ_{1}(C) ⊆ H^{0}(C, 2K_{C}) ⊆ H^{0}(X, ω_{X} ⊗ Ω_{X}). We thus get that Im(dπ_{2})_{[X]} = (Imμ_{1}(X))^{⊥} = (Imμ_{1}(C))^{⊥}.

The assumption that (C, f_{1}, f_{2}) satisfies (5) can be rewritten by passing to duals as

H^{0}(C, 2K_{C} − R_{1})^{⊥} + (Imμ_{1}(C))^{⊥} = H^{1}(C, T_{C}) ⟺ H^{0}(C, 2K_{C} − R_{1}) ∩ Imμ_{1}(C) = 0.

Then it follows that Imμ_{1}(X) ∩ (H^{0}(C, 2K_{C} − R_{1} + Δ) ⊕ Ker((u_{2}^{∨})_{tors})) = 0, which is the same thing as

(dπ_{1})_{[f′]} (T_{[f′]}(\overline{H}_{g+r+1,k})) + (dπ_{2})_{[X↪ℙ^{r}]} (T_{[X↪ℙ^{r}]}(\overline{M}_{g+r+1}(ℙ^{r}, d + r))) = Ext^{1}(Ω_{X}, O_{X}). (7)

This means that the images of \overline{H}_{g+r+1,k} under the map π_{1} and of \overline{M}_{g+r+1}(ℙ^{r}, d + r) under the map π_{2}, meet transversally at the point [X] ∈ \overline{M}_{g+r+1}. Claim. The curve X can be smoothed in such a way that the 𝔤_{k}^{1} and the very ample 𝔤_{d+r}^{r} are preserved (while (7) is an open condition on \overline{H}_{g+r+1,k} × \overline{M}_{g+r+1}(ℙ^{r}, d + r)).

Indeed, the tangent directions that fail to smooth at least one node of X are those in ⋃_{i=1}^{r+2} H^{0}(Tors_{p_{i}}(ω_{X} ⊗ Ω_{X}))^{⊥}, whereas the tangent directions that preserve both the 𝔤_{k}^{1} and the 𝔤_{d+r}^{r} are those in

((Imμ_{1}(C) + H^{0}(C, 2K_{C} − R_{1} + Δ)) ⊕ Ker(u_{2}^{∨})_{tors})^{⊥}.

Since H^{0}(Tors_{p_{i}}(ω_{X} ⊗ Ω_{X})) ⊈ Ker(u_{2}^{∨})_{tors} for i = 1, …, r + 2, by moving in a suitable direction in the tangent space at [f′] of π_{1}^{−1}π_{2}(\overline{M}_{g+r+1}(ℙ^{r}, d + r)), we finally obtain a smooth curve Y ⊆ ℙ^{r} with g(Y) = g + r + 1, deg(Y) = d + r and satisfying all the required properties. □

Using the previous result together with Proposition 1.1 we construct now regular components of M_{g}(ℙ^{1} × ℙ^{r}, (k, d)). Theorem 1 Let g, r, d and k be positive integers such that r ≥ 3, ρ(g, r, d) < 0 and

(2 − ρ(g, r, d))r + 2 ≤ k ≤ (g + 2)/2.

Then there exists a regular component of the stack of maps M_{g}(ℙ^{1} × ℙ^{r}, (k, d)).

Proof. All integer solutions (g_{0}, d_{0}) of the equation ρ(g_{0}, r, d_{0}) = ρ(g, r, d) with g_{0} ≤ g and d_{0} ≤ d, are of the form g_{0} = g − a(r + 1) and d_{0} = d − ar with a ≥ 0. Using our numerical assumptions, by a routine check we find that there exists a > 0 such that g_{0} = g − a(r + 1) > 0, d_{0} = d − ar ≥ r + 1, k ≥ g_{0} + 1 and

−\frac{g_{0}}{r} + \frac{r + 1}{r} ≤ ρ(g_{0}, r, d_{0}) < 0.

By Proposition 1.1 there exists a smooth curve C_{0} ⊆ ℙ^{r} of genus g_{0} and degree d_{0}, with H^{1}(C_{0}, N_{C_{0}/ℙ^{r}}) = 0, h^{0}(C_{0}, O_{C_{0}}(1)) = r + 1 and μ_{0}(C_{0}) surjective. Moreover, since k ≥ g_{0} + 1, there exists an open dense subset U ⊆ Pic^{k}(C_{0}) such that for each L_{1} ∈ U there exists a pencil l_{1} = (L_{1}, V_{1}) ∈ G_{k}^{1}(C_{0}) with V_{1} ∈ Gr(2, H^{0}(C_{0}, L)), such that l_{1} is simple and base point free (cf. [Fu, Proposition 8.1]).

We denote by π_{1} : M_{g_{0}}(ℙ^{1}, k_{0}) → M_{g_{0}} the natural projection and by f_{1} : C → ℙ^{1} the map corresponding to l_{1}. By Riemann-Roch we have H^{1}(C_{0}, L_{1}^{⊗2}) = 0, hence using Section 2 (dπ_{1})_{[f_{1}]} : T_{[f_{1}]}(M_{g_{0}}(ℙ^{1}, k)) → T_{[C_{0}]}(M_{g_{0}}) is surjective since Coker(dπ_{1})_{[f_{1}]} = H^{1}(C_{0}, f_{1}^{∗}T_{ℙ^{1}}) = 0. It follows that (C_{0}, |O_{C_{0}}(1)|, l_{1}) satisfies (5).

We claim that if L ∈ U is general then |O_{C_{0}}(1) ⊗ L^{∨}| = ∅. Suppose not, that is O_{C_{0}}(1) ⊗ L^{∨} ∈ W_{d_{0}−k}(C_{0}) for a general L ∈ Pic^{k}(C_{0}). This is possible only for d_{0} − k ≥ g_{0}, hence

r + 2 ≤ k ≤ d_{0} − g_{0} < r, (because ρ(g_{0}, r, d_{0}) = ρ(g, r, d) < 0),

a contradiction. Thus (C_{0}, |O_{C_{0}}(1)|, l_{1}) satisfies all conditions required by Proposition 3.1 which we can now apply a times to get a smooth curve C ⊆ ℙ^{1} × ℙ^{r} of genus g and bidegree (k, d) such that H^{1}(C, N_{C/ℙ^{1}×ℙ^{r}}) = 0. The conclusion of Theorem 1 now follows. □

In the special case ρ(g, r, d) = −1 we can extend the range of possible g, r, d and k for which there is a regular component:

Theorem 2 Let g, r, d, k be positive integers such that r ≥ 3, ρ(g, r, d) = −1 and

\frac{2r^{2} + r + 1}{r − 1} ≤ k ≤ \frac{g + 2}{2} .

Then there exists a regular component of the stack of maps M_{g}(ℙ^{1} × ℙ^{r}, (k, d)).

Proof. We find a solution (g_{0} = g −a(r +1), d_{0} = d−ar) of the equation ρ(g_{0}, r, d_{0}) = ρ(g, r, d) with a ∈ ℤ_{≥0} such that d_{0} ≥ k + r and ρ(g_{0}, 1, k) ≥ r − 1. Our numerical assumptions ensure that such an a ≥ 0 exists. Note that in this case k ≤ g_{0} + 1, so we are not in the situation covered by Theorem 1.

It also follows that −\frac{g_{0}}{r} + \frac{r+1}{r} ≤ −1 = ρ(g_{0}, r, d_{0}) and d_{0} ≥ r + 1, hence by Proposition 1.1 there exists an irreducible smooth open subset U of M_{g_{0}}(ℙ^{r}, d_{0}) of the expected dimension, such that all points of U correspond to embeddings of smooth curves C ↪ ℙ^{r}, with h^{1}(C, N_{C}) = 0, h^{0}(C, O_{C}(1)) = r + 1 and μ_{0}(C) surjective.

Since we are in the case ρ(g_{0}, r, d_{0}) = −1, a combination of results by Eisenbud, Harris and Steffen gives that the Brill-Noether locus M_{g_{0},d_{0}}^{r} is an irreducible divisor in M_{g_{0}} (see [St, Theorem 0.2]). It follows that the natural projection π_{2} : U → M_{g_{0},d_{0}}^{r} is dominant.

To apply Proposition 3.1 we now find a curve [C_{0}] ∈ M_{g_{0},d_{0}}^{r_{0}} having a complete base point free 𝔤_{k}^{1} such that 2𝔤_{k}^{1} is non-special. Then by semicontinuity we get that the general [C] ∈ U also possesses a pencil 𝔤_{k}^{1} with these properties. To find one particular such curve we proceed as follows: take C_{0} a general (r + 1)-gonal curve of genus g_{0}. These curves will have rather few moduli (r + 1 < [(g + 3)/2]) but we still have that [C_{0}] ∈ M_{g_{0},d_{0}}^{r}. Indeed, according to [CM] we can construct a 𝔤_{d_{0}}^{r} = |𝔤_{r+1}^{1} + F | on C_{0}, where F is an effective divisor on C_{0} with h^{0}(C_{0}, F) = 1. Since k ≤ g_{0}, using Corollary 2.2.3 from [CKM] we find that C_{0} also carries a complete base point free 𝔤_{k}^{1}, not composed with the 𝔤_{r+1}^{1} computing gon(C_{0}), and such that 2𝔤_{k}^{1} is non-special. Since these are open conditions, they will hold generically along a component of G_{k}^{1}(C_{0}). Applying semicontinuity, for a general element [C] ∈ M_{g_{0},d_{0}}^{r} (hence also for a general element [C] ∈ U), the variety G_{k}^{1}(C) will contain a component A with general point l ∈ A being complete, base point free and with 2l non-special.

We claim that for a general l ∈ A we have that |O_{C}(1)|(−l) = ∅. Suppose not. Then if we denote by V_{d_{0}−k}^{r−1}(|O_{C}(1)|) the variety of effective divisors of degree d_{0} − k on C imposing ≤ r − 1 conditions on |O_{C}(1)|, we obtain

dim V_{d_{0}−k}^{r−1}(|O_{C}(1)|) ≥ dim A ≥ ρ(g_{0}, 1, k) ≥ r − 1.

Therefore C ⊆ ℙ^{r} has at least ∞^{r−1} (d_{0} − k)-secant (r − 2)-planes, hence also at least ∞^{r−1} r-secant (r −2)-planes (because d_{0} −k ≥ r). This last statement contradicts the Uniform Position Theorem (see [ACGH, p. 112]), hence the general point [C] ∈ U enjoys all properties required to make Proposition 3.1 work. □

Remark. From the proof of Theorem 2 the following question appears naturally: let us fix g, k such that g/2 + 1 ≤ k ≤ g. One knows (cf. [ACGH]) that if l ∈ G_{k}^{1}(C) is a complete, base point free pencil then dim T_{l}(G_{k}^{1}(C)) = ρ(g, 1, k) + h^{1}(C, 2l). Therefore if A is a component of G_{k}^{1}(C) such that dim A = ρ(g, 1, k) and the general l ∈ A is base point free such that 2l is special, then A is nonreduced. What is then the dimension of the locus

V_{g,k} := {[C] ∈ M_{g} : every component of G_{k}^{1}(C) is nonreduced }?

A result of Coppens (cf. [Co]) says that for a curve C, if the scheme W_{k}^{1}(C) is reduced and of dimension ρ(g, 1, k), then the scheme W_{k+1}^{1}(C) is reduced too and of dimension ρ(g, 1, k + 1). It would make then sense to determine dim(V_{g,k}) when ρ(g, 1, k) ∈ {0, 1} (depending on the parity of g). We suspect that V_{g,k} depends on very few moduli and if g is suitably large we expect that V_{g,k} = ∅.

References

[ACV] D. Abramovich, A. Corti, A. Vistoli, Twisted bundles and admissible covers, math.AG/0106211 preprint.

[AC] E. Arbarello, M. Cornalba, Footnotes to a paper of Beniamino Segre, Math. Annalen 256(1981), 341–362.

[ACGH] E. Arbarello, M. Cornalba, P. Griffiths, J. Harris, Geometry of algebraic curves, Grundlehren der Mathematischen Wissenschaften, 267, Springer Verlag, 1985

[BE] E. Ballico, Ph. Ellia, On the existence of curves with maximal rank in ℙ^{n}, J. reine angew. Math. 397(1989), 1-22.

[CGGH] J. Carlson, M. Green, P. Griffiths, J. Harris, Infinitesimal variations of Hodge structure (I), Compositio Math. 50(1983), no. 2-3, 109-205.

[CR] M.C. Chang, Z. Ran, Deformations and smoothing of complete linear systems on reducible curves, in: Algebraic geometry-Bowdoin 1985 (S. Bloch ed.), Proc. Symp. Pure Math. 46, Part 1, 1987, 63-75.

[Co] M. Coppens, Some remarks on the schemes W_{d}^{r}, Ann. Mat. Pura Appl. (4) 157(1990), no.1, 183-197.

[CKM] M. Coppens, C. Keem, G. Martens, The primitive length of a general k-gonal curve, Indag. Math., N.S., 5(1994), 145-159.

[CM] M. Coppens, G. Martens, Linear series on a general k-gonal curve, Abh. Math. Sem. Univ. Hamburg 69(1999), 347-361.

[FP] W. Fulton, R. Pandharipande, Notes on stable maps and quantum cohomology, in: Algebraic geometry-Santa Cruz 1995 (J. Kollar, R. Lazarsfeld, D. Morrison eds.), Proc. Symp. Pure Math., 62, Part 2, 1997, 45-96.

[Fu] W. Fulton, Hurwitz schemes and irreducibility of moduli of algebraic curves, Annals of Math. 90(1969), 542-575.

[HM] J. Harris, D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67(1982), no.1, 23–88.

[Lo] A. Lopez, On the existence of components of the Hilbert scheme with the expected number of moduli, Comm. Algebra 27(1999), no. 7, 3485-3493.

[Ran] Z. Ran, Deformations of maps, in: Algebraic Curves and Projective Geometry: Proceedings, Trento 1988, Lecture Notes in Math. 1389, Springer Verlag 1989, 246-253.

[Se] E. Sernesi, On the existence of certain families of curves, Invent. Math. 75(1984), no.1, 25–57.

[St] F. Steffen, A generalized principal ideal theorem with applications to Brill-Noether theory, Invent. Math. 132(1998), no.1, 73-89.

University of Michigan, Department of Mathematics East Hall, 525 East University, Ann Arbor, MI 48109-1109

e-mail: gfarkas@umich.edu