Asymptotic directions in the moduli space of curves

1 Introduction

In this paper, we study the local geometry of the Torelli locus in A g . Following the philosophy of Griffiths, the local geometry of the period map often contains information about the global geometry (see [6, 21, 22, 34]). We consider A g endowed with the Siegel metric, that is, the orbifold metric induced by the symmetric metric on the Siegel space H g = Sp ⁡ ( 2 ⁢ g , R ) / U ⁢ ( g ) of which A g is a quotient by the action of Sp ⁡ ( 2 ⁢ g , Z ) . Denote by j : M g → A g the Torelli map. The Torelli locus is the closure of the image of 𝑗. The local geometry of the map 𝑗 is governed by the second fundamental form, which at a non-hyperelliptic curve 𝐶 of genus 𝑔 is a linear map II : I 2 → Sym 2 ⁡ H 0 ⁢ ( K C ⊗ 2 ) , where I 2 is the vector space of quadrics containing the canonical curve.

One of the leading problems in the area is to study totally geodesic subvarieties of A g generically contained in the Torelli locus.

This problem is related to the Coleman–Oort conjecture according to which, for 𝑔 sufficiently high, there should not exist special (or Shimura) subvarieties of A g generically contained in the Torelli locus. We recall that special subvarieties of A g are totally geodesic; hence, in the last years, the study of the second fundamental form has been used to attack this problem. In particular, estimates on the maximal dimension of a totally geodesic subvariety of A g generically contained in the Torelli locus have been given in [9, 17, 16].

In this paper, we take a different point of view. The image of II in Sym 2 ⁡ H 1 ⁢ ( T C ) ∨ is a linear system of quadrics in P ⁢ H 1 ⁢ ( T C ) ≅ P 3 ⁢ g − 4 . This paper is devoted to the study of the base locus of this linear system of quadrics. Following the terminology of differential geometry, we call asymptotic direction a nonzero tangent direction ζ ∈ H 1 ⁢ ( T C ) such that II ⁢ ( Q ) ⁢ ( ζ ⊙ ζ ) = 0 for all Q ∈ I 2 . So asymptotic directions correspond to points in the base locus.

Clearly, a tangent direction to a totally geodesic subvariety is asymptotic. But the locus of asymptotic directions in the projective tangent bundle of the moduli space of curves M g is a natural locus, that is worthwhile investigating. For example, we believe that asymptotic directions could also be useful in the study of fibred surfaces in relation with the Xiao conjecture [35, 18] (see Remark 4.3).

Since II is injective [8, Corollary 3.4], its image in Sym 2 ⁡ H 1 ⁢ ( T C ) ∨ is a linear system of quadrics in P ⁢ H 1 ⁢ ( T C ) ≅ P 3 ⁢ g − 4 of dimension 1 2 ⁢ ( g − 2 ) ⁢ ( g − 3 ) . Hence, for every curve 𝐶 of genus g ≤ 9 , dim ⁡ ( II ⁢ ( I 2 ) ) < 3 ⁢ g − 4 , so the intersections of the quadrics in II ⁢ ( I 2 ) is non-empty; thus there exist asymptotic directions. In fact, for g ≤ 7 , there are examples of special subvarieties of A g generically contained in the Torelli locus (see [12, 13, 14, 15, 27, 28, 31, 32, 33]).

On the other hand, for high values of 𝑔, one would expect that the intersection of a space of quadrics of dimension 1 2 ⁢ ( g − 2 ) ⁢ ( g − 3 ) in P 3 ⁢ g − 4 would be empty.

One main result of this paper is to show that this is not always the case. Indeed, for all 𝑔, there are examples of asymptotic directions given by the Schiffer variations at the ramifications points of the g 3 1 for trigonal curves and by linear combinations of two Schiffer variations on bielliptic curves (see Lemma 8.2 and Theorem 9.2).

This is rather unexpected and intriguing and indicates that understanding the geometry of the locus of asymptotic directions is important. Especially, finding new examples of curves admitting asymptotic directions would be very interesting.

In this paper, using the Hodge Gaussian maps introduced in [10], we develop a new technique to calculate the second fundamental form II ⁢ ( Q ) ⁢ ( ζ ⊙ ζ ) on certain tangent directions 𝜁 different from Schiffer variations, computing some residues of meromorphic forms (see Proposition 4.4).

This technique works for those tangent directions 𝜁 whose rank is less than 𝑔, where the rank of an infinitesimal deformation 𝜁 is the rank of the linear map ∪ ζ : H 0 ⁢ ( K C ) → H 1 ⁢ ( O C ) given by the cup product.

One of the main results we obtain by application of these ideas is the following

For a definition of 𝑛-th Schiffer variations, see Section 2.

Notice that the first part of the above result can be seen as a generalisation of the generic Torelli theorem of Griffiths. In fact, denoting by C d the symmetric product of 𝐶, under the assumption Cliff ⁡ ( C ) > d , we characterise the image of the natural map P ⁢ T C d → P ⁢ ( H 1 ⁢ ( T C ) ) as the locus of deformations of rank at most 𝑑. When d = 1 , it is the bicanonical curve.

In particular, if a curve 𝐶 of genus 𝑔 has maximal Clifford index, equal to ⌊ g − 1 2 ⌋ , this gives a characterisation of the image of the natural map P ⁢ T C d → P ⁢ ( H 1 ⁢ ( T C ) ) as the locus of deformations of rank at most 𝑑, for all d < ⌊ g − 1 2 ⌋ . We recall that this applies to the general curve in moduli space, which has maximal Clifford index. Moreover, for such values of 𝑑, Theorem 1.1 (2), for curves of maximal Clifford index, says that the base locus of the linear system of quadrics II ⁢ ( I 2 ) in P ⁢ H 1 ⁢ ( T C ) does not contain any point [ ζ ] with rank ⁡ ( ζ ) ≤ d .

In the case where the rank of 𝜁 is equal to the Clifford index of the curve, we give sufficient conditions ensuring that the infinitesimal deformation 𝜁 is not asymptotic (see Theorem 7.2).

This result allows us to determine all asymptotic directions of rank 1. Notice that, by Theorem 1.1, there can be asymptotic directions of rank 1 only for curves with Clifford index 1, namely trigonal curves or plane quintics. We have the following.

We recall that the general trigonal curve of genus g ≥ 6 has Maroni degree 2. For trigonal curves of genus g = 5 or g = 6 , 7 and Maroni degree 1, we show that there can exist asymptotic directions that are not Schiffer variations in the ramification points of the g 3 1 . We describe these asymptotic directions and we give the explicit equations of the trigonal curves admitting such asymptotic directions (see Section 10).

Finally, we consider infinitesimal deformations of rank 2 and we prove the following.

For bielliptic curves, we show the following.

More precisely, the linear combinations of Schiffer variations which are asymptotic in the above theorem are ξ p ± i ⁢ ξ σ ⁢ ( p ) , where 𝜎 is the bielliptic involution and ( p , σ ⁢ ( p ) ) ∈ C × C is in the zero locus of the meromorphic form η ̂ ∈ H 0 ⁢ ( K C × C ⁢ ( 2 ⁢ Δ ) ) which determines the second fundamental form II (see [9, Theorem 3.7]).

The structure of the paper is as follows.

In Section 2, we describe in Dolbeault cohomology those infinitesimal deformations 𝜁 whose kernel contains a given nonzero form ω ∈ H 0 ⁢ ( K C ) . We define a split deformation 𝜁 as an infinitesimal deformation such that the rank 2 vector bundle of the extension corresponding to 𝜁 splits as the sum of two line bundles, and we give a Dolbeault cohomology description of it. We also recall the definition of (higher order) Schiffer variations.

Section 3 contains some technical results that will be useful to make explicit computations on the second fundamental form of the Torelli map.

In Section 4, we recall the definition of the second fundamental form, we define asymptotic directions and we use the results in Section 3 to give a formula that computes the second fundamental form II ⁢ ( Q ) ⁢ ( ζ ⊙ ζ ) on some 𝜁 with nontrivial kernel, in terms of residues of meromorphic forms (see Proposition 4.4).

In Section 5, we consider infinitesimal deformations of rank 𝑑 less than the Clifford index of the curve, and we analyse the extension corresponding to 𝜁 and the rank 2 vector bundle 𝐸 of the extension. First we show that if either d < Cliff ⁡ ( C ) , or d = Cliff ⁡ ( C ) < g − 1 2 and 𝐸 is not globally generated, then 𝜁 is a linear combination of Schiffer variations supported on a divisor 𝐷 of degree 𝑑 (see Theorem 5.1). One of the main technical tools is a theorem of Segre–Nagata and Ghione (see [24, p. 84]) on the existence of a subline bundle 𝐴 of 𝐸 such that deg ⁡ ( A ) ≥ g − 1 2 . Then we show that if d < Cliff ⁡ ( C ) , no linear combinations of Schiffer variations of rank 𝑑 is asymptotic (see Theorem 5.4). Here we use a result of Green and Lazarsfeld [20, Theorem 1] that allows us to find some quadrics where our technique to compute the second fundamental form works well. Finally, we prove Theorem 5.5.

In Section 6, we define some special split deformations that we call double split, where we can compute explicitly the second fundamental form, showing that they are not asymptotic.

In Section 7, we treat the case where the rank of 𝜁 is equal to the Clifford index of the curve and we give sufficient conditions ensuring that 𝜁 is not asymptotic (see Theorem 7.2).

In Section 8, we determine all asymptotic directions of rank 1 and we show Theorem 1.2 (Theorem 8.4 and Theorem 8.5).

In Section 9, we concentrate on deformations of rank 2. We show Theorem 1.3 (Theorem 9.1, Theorem 9.6) and we give the example of asymptotic directions on bielliptic curves proving Theorem 1.4 (Theorem 9.2).

In Section 10, we consider trigonal curves of genus g = 6 , 7 and Maroni degree 1 and trigonal curves of genus g = 5 , showing that there can exist asymptotic directions that are not Schiffer variations in the ramification points of the g 3 1 . We also describe these asymptotic directions and we give the equation of the trigonal curves admitting such asymptotic directions.

2 Preliminaries on deformations

Recall that an infinitesimal deformation ζ ∈ H 1 ⁢ ( T C ) corresponds to a class of an extension

Taking global sections, we have

We shall now describe in Dolbeault cohomology those deformations 𝜁 having a given form ω ∈ H 0 ⁢ ( K C ) in the kernel of ∪ ζ . Let ω ∈ H 0 ⁢ ( K C ) , ω ≠ 0 , be a holomorphic 1-form, Z = ∑ n i ⁢ p i its divisor; consider the sequence

and the corresponding exact sequence in cohomology

By the above exact sequence, the elements in ζ ∈ H 1 ⁢ ( T C ) such that ω ∈ ker ⁡ ( ∪ ζ ) are exactly those belonging to the image of 𝛿.

We want now to give an explicit description in Dolbeault cohomology of the image of 𝛿. Suppose that { U i , z i } are open pairwise disjoint coordinate neighbourhoods centred on p i and p i ∈ D i ⊂ Δ i ⊂ U i and D i and Δ i are two closed discs, where D i is in the interior of Δ i .

Let s ∈ H 0 ⁢ ( O Z ) be a holomorphic section and assume that

is its polynomial expression in U i . Let ρ ̃ be a C ∞ function on 𝐶 which is equal to 1 on ⋃ i D i and equal to 0 on C ∖ ⋃ i Δ i .

Denote by 𝜌 the C ∞ function on 𝐶 given by ∑ i ρ ̃ ⁢ f i . Let us introduce the following notation: writing in a local coordinate 𝑧, ω : = g ( z ) d z , we set 1 ω = 1 g ⁢ ( z ) ⁢ ∂ ∂ z . Hence 1 ω defines a meromorphic section of T C .

Then, in Dolbeault cohomology, we have δ ⁢ ( s ) = [ ∂ ̄ ⁢ ( ρ ω ) ] . Setting ζ : = [ ∂ ̄ ( ρ ω ) ] , we clearly have ω ∪ ζ = 0 . Conversely, by the exact sequence (2.2), any 𝜁 such that ω ∪ ζ = 0 is in the image of 𝛿; hence it has a Dolbeault representative as above.

We will now give a description in Dolbeault cohomology of split deformations. Take a form ω ∈ H 0 ⁢ ( K C ) and choose a decomposition of its zero divisor Z = D + F , where the supports of 𝐷 and 𝐹 are disjoint. Set [ σ D ] ∈ H 0 ⁢ ( O Z ) , where σ D ≡ 1 on ⋃ p i ∈ Supp ⁡ ( D ) Δ i and σ D ≡ 0 on ⋃ p i ∈ Supp ⁡ ( F ) Δ i .

With the above notation, set ρ D : = ρ ̃ σ D . Setting Θ = ∂ ̄ ⁢ ( ρ D ω ) , we have in the exact sequence (2.2), in Dolbeault cohomology,

One has Θ ⁢ ω = ∂ ̄ ⁢ ρ D , which is ∂ ̄ -exact; hence ω ∪ ζ = 0 in H 1 ⁢ ( C , O C ) .

Let 𝐶 be a smooth complex projective curve of genus g ≥ 2 . Let 𝑝 be a point in 𝐶 and 𝑧 a local coordinate centred in 𝑝. We define the 𝑛-th Schiffer variation at 𝑝 to be the element ξ p n ∈ H 1 ⁢ ( C , T C ) ≅ H ∂ ̄ 0 , 1 ⁢ ( T C ) whose Dolbeault representative is ∂ ̄ ⁢ ( ρ p z n ) ⁢ ∂ ∂ z , where ρ p is a bump function in 𝑝 which is equal to one in a small neighbourhood 𝑈 containing 𝑝, ξ p n = [ ∂ ̄ ⁢ ( ρ p z n ) ⁢ ∂ ∂ z ] . Clearly, ξ p n depends on the choice of the local coordinate 𝑧. Take the exact sequence

and the induced exact sequence in cohomology

By Riemann–Roch, if n < 2 ⁢ g − 2 , we have h 0 ⁢ ( T C ⁢ ( n ⁢ p ) ) = 0 ; hence we have an inclusion

and the image of δ p n in H 1 ⁢ ( C , T C ) is the 𝑛-dimensional subspace ⟨ ξ p 1 , … , ξ p n ⟩ .

Clearly, H 0 ⁢ ( K C ⁢ ( − n ⁢ p ) ) ⊂ ker ⁡ ( ∪ ξ p n ) for n ≤ g ; hence ξ p n has rank at most 𝑛. For n = 1 , the first Schiffer variations ξ p 1 are the usual Schiffer variations, that we denote simply by ξ p , and they have rank 1 (see Section 7).

More generally, any linear combination

has rank at most ∑ i = 1 k m i , since H 0 ⁢ ( K C ⁢ ( − ∑ i = 1 k m i ⁢ p i ) ) ⊂ ker ⁡ ( ∪ ζ ) .

3 Some computations

We will now show some technical results that will be useful in the sequel to make explicit computations on the second fundamental form of the Torelli map.

Consider two holomorphic one forms ω 1 and ω 2 . With the notation introduced in Section 2, take an infinitesimal deformation 𝜁 having a form ω ≠ 0 in its kernel and denote by Z = ∑ n i ⁢ p i its zero divisor. Then ζ = [ Θ ] ∈ H 1 ⁢ ( T C ) with Θ : = ∂ ̄ ( ρ ω ) , and ρ = ∑ i ρ ̃ ⁢ f i . We have Θ ⁢ ω i = γ i + ∂ ̄ ⁢ h i , where γ i is harmonic and h i is a C ∞ function on 𝐶, so

Consider the two meromorphic functions g 1 = ω 1 / ω and g 2 = ω 2 / ω . Then, clearly, by the above construction, we have

Note that if ρ ⁢ g 1 is C ∞ , then ω 1 ∈ ker ⁡ ( ∪ ζ ) and γ 1 = 0 . Moreover, up to a constant, we have h 1 = ρ ⁢ g 1 .

Set

In the next section, we will show that this expression will be very useful in the computation of the second fundamental form of the Torelli map.

4 Second fundamental form and Hodge Gaussian map

Denote by M g the moduli space of smooth complex projective curves of genus 𝑔 and let A g be the moduli space of principally polarised abelian varieties of dimension 𝑔. The space A g is a quotient of the Siegel space H g = Sp ⁡ ( 2 ⁢ g , R ) / U ⁢ ( g ) by the action of the symplectic group Sp ⁡ ( 2 ⁢ g , Z ) . The space H g is a Hermitian symmetric domain and it is endowed with a canonical symmetric metric. We consider the induced orbifold metric (called the Siegel metric) on the quotient A g .

Denote by j : M g → A g , [ C ] ↦ [ j ⁢ ( C ) , Θ C ] the Torelli map, where j ⁢ ( C ) is the Jacobian of 𝐶 and Θ C is the principal polarisation induced by cup product. If g ≥ 3 , it is an orbifold embedding outside the hyperelliptic locus [29].

Consider the complement M g 0 of the hyperelliptic locus in M g and the cotangent exact sequence of the Torelli map,

where q = d ⁢ j ∗ is the dual of the differential of the Torelli map. Call ∇ the Chern connection on Ω A g | M g 0 1 with respect to the Siegel metric and let

be the second fundamental form of the above exact sequence. From now on, we will assume g ≥ 4 , so N M g 0 / A g ∗ is nontrivial.

If we take a point [ C ] ∈ M g 0 , we have

where I 2 : = I 2 ( K C ) is the vector space of quadrics containing the canonical curve and the dual of the differential of the Torelli map 𝑞 is the multiplication map of global sections. Then, at the point [ C ] , the second fundamental form is a linear map II : I 2 → Sym 2 ⁡ H 0 ⁢ ( K C ⊗ 2 ) .

In [10, Theorem 2.1], it is proven that II is equal (up to a constant) to the Hodge Gaussian map 𝜌 of [10, Proposition-Definition 1.3]. We briefly recall its definition.

Let

with a i , j = a j , i , ω i ∈ H 0 ⁢ ( K C ) and Θ ∈ A 0 , 1 ⁢ ( T C ) , [ Θ ] = : ζ . We write Θ ⁢ ω i = γ i + ∂ ̄ ⁢ h i , where γ i is a harmonic ( 0 , 1 ) -form. Now, identifying Sym 2 ⁡ H 0 ⁢ ( K C ⊗ 2 ) with the symmetric homomorphisms H 1 ⁢ ( T C ) → H 0 ⁢ ( K C ⊗ 2 ) , we have (see [10])

In [10, Theorem 3.1] (see also [9, Theorem 2.2]), it is proven that if 𝐶 is a non-hyperelliptic curve of genus g ≥ 4 and p , q are two distinct points in 𝐶, we have

where μ 2 : I 2 → H 0 ⁢ ( K C ⊗ 4 ) is the second Gaussian map of the canonical bundle (see [10], or [9] for more details).

Now consider a quadric Q ∈ I 2 . Notice that we can always assume that 𝑄 has the following expression:

Fix a holomorphic form ω ∈ H 0 ⁢ ( K C ) , ω ≠ 0 , with zero locus 𝑍. With the notation of Section 2, we set ζ = [ ∂ ̄ ⁢ ( ρ ω ) ] . So we have ζ ⁢ ω = 0 and Θ ⁢ ω j = γ j + ∂ ̄ ⁢ h j , γ j harmonic. Let g i be the meromorphic function given by ω i ω . We would like to compute II ⁢ ( Q ) ⁢ ( ζ ⊙ ζ ) .