Jets Separation Thresholds, Seshadri Constants, and Higher Gauss–Wahl Maps on Abelian Varieties

Abstract

1 Introduction

In this article we introduce the vanishing threshold of a closed subscheme of a polarized abelian variety defined over an algebraically closed field. In the case of a closed point (with its reduced structure) this number is no other than the basepoint-freeness threshold introduced by Jiang–Pareschi in [12], which has also been studied by Caucci [3], Ito [10], and Jiang [11], using methods from birational geometry, and by Rojas [27], by means of stability conditions in the 2D case. When we consider the |$p$|-infinitesimal neighborhood of a closed point (i.e., the scheme defined by the |$(p+1)$|-power of the ideal of the point) a natural notion of |$p$|-jets separation threshold arises.

In the first part of this work, we introduce the aforementioned thresholds and show some examples. Afterwards, we prove that if |$Z$| is a closed subscheme and |$Z^{(p)}$| is its |$p$|-infinitesimal neighborhood then the sequence, suitably normalized, of vanishing thresholds of |$Z^{(p)}$| with respect to a polarization |$\ell $| converges to the Seshadri constant with respect to |$\ell $| of the ideal defining |$Z.$| As a particular case, when |$Z$| is a point (with its reduced structure) we get that the sequence of |$p$|-jets separation thresholds (suitably normalized) converges to the classical Seshadri constant of |$\ell .$| This is no surprise since, by a theorem of Demailly [6, Theorem 6.4], the Seshadri constant of a line bundle at a point reflects the asymptotic behaviour of jet separation. However, unlike the invariants |$s(kL,x)$| considered by Demailly (see [13, Proposition 5.1.6] for the definition of these numbers), which are integers and depend on |$x$|⁠, the |$p$|-jets separation thresholds are real numbers and do not depend on an specific closed point |$x.$| In this context, our asymptotic description can be seen as a finer version, valid for abelian varieties, of Demailly’s result.

Subsequently, we show that |$p$|-jets separation thresholds are intimately related, via the Fourier–Mukai transform, to certain higher Gauss–Wahl maps. In fact, we prove that they carry essentially the same information of certain (suitably defined) |$p$|-Gauss–Wahl surjectivity threshold. This parallels the fact, shown by Jiang–Pareschi in [12, Proposition 8.1], that basepoint freeness threshold carries the same information as the surjectivity threshold for multiplication maps of global sections (and therefore, is intimately related to projective normality). As a byproduct of the aforementioned relation we obtain a description of the Seshadri constant in terms of the asymptotic behaviour of the surjectivity of certain Gauss–Wahl maps. Finally, by means of Nadel’s vanishing theorem, we bound the |$p$|-jets separation thresholds in terms of the Seshadri constant, when we work in characteristic zero. As a consequence, we establish precise lower bounds for the Seshadri constant guaranteeing the surjectivity of higher Gauss–Wahl maps, asymptotically improving the results present in the literature.

Turning to a more detailed presentation, we now proceed to define the afore mentioned thresholds. First, we recall that a coherent sheaf |$\mathcal{F}$| on an abelian variety |$A$| is said to satisfy the index theorem with index zero (IT(0) for short) if

where |$P_{\alpha }$| is the line bundle on |$A$| parametrized by |$\alpha $| via a normalized Poincaré bundle. Now, let |$\ell $| be a polarization on |$A$| (i.e., an ample class in |$\textrm{NS}(A)$|⁠). As shown by Jiang–Pareschi, the IT(0) condition can be extended for |${\mathbb{Q}}$|-twisted sheaves |$\mathcal{F}\left <t\ell \right>$| (we recall this notion in Section 2.1). In this context, we define the vanishing threshold of |$\mathcal{F}$| with respect to |$\ell $| as the real number

In this context, we define the vanishing threshold of a closed subscheme |$Z$| of |$A$| with respect to |$\ell $| to be the vanishing threshold of its ideal. In symbols:

As an example, fix a closed point |$x\in A$| with ideal sheaf |$I_{x}$| and consider a positive integer |$p.$| Let |$L$| be a line bundle representing |$\ell .$| As |$L$| is ample, all the line bundles |$L\otimes P_{\alpha }$| are isomorphic to |$t_{y}^{\ast }L$| for some closed point |$y\in A,$| where |$t_{y}$| is the translation map, and hence the vanishing

means that for every |$y\in A$| the line bundle |$t_{y}^{\ast }L$| separates |$p$|-jets at |$x$| and thus, that |$L$| separates |$p$|-jets at every point of |$A.$| As the higher cohomology groups of |$I_{x}^{p+1}\otimes L\otimes P_{\alpha }$| automatically vanish, the condition (2) turns to be equivalent to IT(0); therefore, it is natural to define the |$p$|-jets separation threshold of |$\ell $| using the vanishing threshold with respect to |$\ell $| of the |$p$|-infinitesimal neighborhood of |$x$|⁠, where |$x$| is a (any) closed point of |$A,$| that is:

Our first result relates the vanishing threshold of infinitesimal neighborhoods of a closed subscheme |$Z$| with the Seshadri constant |$\varepsilon (I_{Z},L)$| of the ideal defining |$Z$| (see Section 4 for its definition). More precisely, we prove the following:

Our second result concerns higher Gauss–Wahl maps. They were introduced by Wahl in [29] and they are a hierarchy of linear maps |$\{\gamma _{L,M}^{p}\}_{p\in{\mathbb{Z}}_{\geq 0}}$| associated to a pair of line bundles |$L,M$|⁠, where |$\gamma _{L,M}^{0}$| is no other than the multiplication map of global sections. For the moment, we give an informal introduction to these maps (in Section 5 we recall their precise definition). In brief, given two line bundles |$L,M$| on a projective variety |$A,$| the associated |$p$|-th Gauss–Wahl map is a linear map

where |$R_{L}^{(p-1)}$| is a sheaf that in the case that |$L$| is very ample is no other than the |$p-1$|-order conormal bundle associated to the embedding of |$A$| into |$\mathbb{P}H^{0}(L)$| (for instance, |$R_{L}^{(1)}$| can be identified with |$N_{A/\mathbb{P}H^{0}(L)}^{\vee }\otimes L$|⁠).

When |$L$| separates |$p$|-jets at every point, the surjectivity of |$\gamma _{L,M}^{p}$| is detected by the vanishing of |$H^{1}(R_{L}^{(p)}\otimes M)$| (we refer to section 5 for details). In this context, informally speaking, the vanishing threshold |$\nu _{\ell }(R_{L}^{(p)})$| (see (1)) of the sheaf |$R_{L}^{(p)}$| with respect to |$\ell $| may be seen as the minimal “rational power” of |$L$| such that the “rational” Gauss–Wahl map |$\gamma _{L,\nu L}^{p}$| is surjective. Extending [12, Proposition 8.1], our second result says that the numbers

are related, via the Fourier–Mukai transform (reviewed in Section 2.3), with the jets separation thresholds as follows:

As a consequence, this gives a surprising expression of the Seshadri constant in terms of the asymptotic surjectivity of higher Gauss–Wahl maps. Concretely, we have the following:

An interesting consequence of this relation is the fact that we are able to characterize special kinds of polarized abelian varieties using Gauss-Wahl maps. Concretely, here we highlight two special cases:

1. Decomposable abelian varieties with a curve factor: In [21] Nakamaye proved that if |$(A,\ell )$| is a polarized abelian variety, then |$\varepsilon (\ell )\geq 1$| and, moreover, |$\varepsilon (\ell ) = 1$| if and only if

where |$(E,\theta )$| is a principally polarized elliptic curve and |$(B,m)$| is another polarized abelian variety. In this context, Corollary 1.3 says that varieties of the form (3) are also characterized by the following equivalent conditions:

• a)

  |$\beta _{p}(\ell ) = p+1$| for every |$p\in{\mathbb{Z}}_{\geq 0}$|c–

• b)

  the sequence |$\{\mu _{p}((p+2)\ell )\}_{p\in{\mathbb{Z}}_{\geq 0}}$| is unbounded

2. Hyperelliptic jacobians: In [5, Theorem 7] it is shown that if |$C$| is a curve of genus |$g\geq 5$| and |$C$| is not hyperelliptic then |$\varepsilon (\operatorname{Jac}C,\theta )>2$| while |$\varepsilon (\operatorname{Jac}C,\theta ) = 2g/(g+1)<2$| in the hyperelliptic case. Moreover, in the cited reference it is conjectured that the condition |$\varepsilon (\theta )<2$| should characterize jacobians of hyperelliptic curves among indecomposable principally polarized abelian varieties (i.p.p.a.v) of dimension |$g\geq 5.$|

In this context, Corollary 1.3 says that jacobians of hyperelliptic curves are characterized among jacobians (and conjecturally among all i.p.p.a.v) by the fact that |$\mu _{p}((p+2)\theta )>1$| for |$p\gg 0,$| that is, for the failure of surjectivity of the Gauss–Wahl map |$\gamma _{(p+2)\theta ,(p+2)\theta }^{p}$| for |$p\gg 0.$|

Finally, as an application, we use Theorem 1.2 above to study the surjectivity of Gauss-Wahl maps |$\gamma _{cL,dM}^{p},$| where |$c,d,p\in{\mathbb{Z}}_{\geq 0}$| and |$L,M$| are ample and algebraically equivalent line bundles. Our result is the following:

For instance, the above result says that if |$\beta _{p}(\ell )<1,$| that is, if |$\ell $| separates |$p$|-jets at every point, then |$\gamma _{2L,2M}^{p}$| is surjective, while |$\gamma _{L,M}^{p}$| is already surjective whenever |$\beta _{p}(\ell )<1/2.$| On the other hand, using Theorem 1.1 3) we obtain a condition ensuring the surjectivity of certain Gauss–Wahl maps in terms of the Seshadri constant, which (at least asymptotically) improves the results present in the literature (e.g., [22, Theorem 2.2]). More precisely, we get the following:

This article structures as follows: in Section 2 we survey the notions of generic vanishing for |${\mathbb{Q}}$|-twisted objects and their relation with the Fourier–Mukai functor, whose definition we also recall. In Section 3 we formally introduce the notion of vanishing threshold of a closed subscheme of an abelian variety and, in particular, the |$p$|-jets separation thresholds. In this context, we establish Theorem 1.1 2.a. In Section 4 we discuss Seshadri constants of ideal sheaves, and relate them with the vanishing threshold of the corresponding closed subscheme, proving Theorem 1.1.1 and 2.b. In Section 5 we discuss Gauss–Wahl maps, introduce their surjectivity thresholds and prove a slightly more general version of Theorem 1.2 above by computing the Fourier transform of |$I_{0}^{p+1}\otimes L.$| In Section 6, we prove Theorem 1.1.3 and show its consequences. In particular, in this section we prove Theorem 1.4 and Corollary 1.5. Finally, in Section 7 we state a couple of questions for future research: the first one regards the possibility to render our asymptotic results more effective and the second one regards the possibility to extend Theorem 1.1.3 to abelian varieties defined over fields of positive characteristic.

2 Preliminaries

In this section we recall the preliminary materials that we will need. We start by fixing some notation. Let |$A$| be an abelian variety defined over an algebraically closed field |$k$|⁠. By a polarization on |$A$| we mean an ample class |$\ell \in \textrm{NS}(A).$| For |$A$| as before, we write |$\hat{A}$| for the dual of |$A,$| that is, |$\hat{A} = \textrm{Pic}^{0}(A)$| is the abelian variety parametrizing translation-invariant line bundles on |$A.$| For a polarization |$\ell $| we write |$\varphi _{\ell }$| (or |$\varphi _{L}$|⁠) for the isogeny |$A\rightarrow \hat{A}$| which sends a point |$x\in A$| to the point of |$\hat{A}$| corresponding to the line bundle |$t_{x}^{\ast }L\otimes L^{-1},$| where |$t_{x}:A\rightarrow A$| is the translation map given by |$z\mapsto z+x$| and |$L$| is a (any) line bundle representing |$\ell $|⁠. For a positive integer |$n$| we write |$\lambda _{n}:A\rightarrow A$| for the isogeny |$x\mapsto nx.$| Finally, we fix |$\mathcal{P}$| a normalized Poincaré bundle on |$A\times \hat{A}$| and for |$\alpha \in \hat{A},$| write |$P_{\alpha }$| for the line bundle on |$A$| parametrized by |$\alpha $| (equivalently, |$P_{\alpha }$| is the fiber of |$\mathcal{P}$| over |$\alpha $|⁠).

2.1 |${\mathbb{Q}}$|-twisted objects

For |$A$| as before, we write |$D^{b}(A)$| for the bounded derived category of coherent sheaves on |$A.$| We consider pairs |$(\mathcal{F},t\ell ),$| where |$\mathcal{F}$| is an object in |$D^{b}(A),$|  |$t$| is a rational number, and |$\ell \in \textrm{NS}(A)$| is a polarization. On these pairs we impose the relation given by identifying |$(\mathcal{F}\otimes L^{\otimes m},t\ell )$| with the pair |$(\mathcal{F},(t+m)\ell )$| for every integer |$m$| and for every |$L$| representing |$\ell .$|

For a |${\mathbb{Q}}$|-twisted object |$\mathcal{F}\left <t\ell \right>$| we would like to define its cohomological support loci |$V^{i}(\mathcal{F}\left <t\ell \right>).$| To start, let |$L$| be a line bundle representing |$\ell $| and consider |$a,b\in{\mathbb{Z}}$| with |$b>0$| and |$t=a/b.$| Write

where |$H^{i}$| stands for the sheaf (hyper)cohomology. These subsets depend on the choices of |$L,a$| and |$b.$| For instance, if |$N$| is another representant of |$\ell $| then |$V^{i}(\mathcal{F},N,a,b)$| is just a translation of |$V^{i}(\mathcal{F},L,a,b).$| On the other hand, if we change |$a,b$| for |$ac,bc$| with |$c$| a positive integer not divisible by |$\operatorname{char} k$| and we consider a symmetric |$L$|⁠, then:

where, to get the direct sum we used [17, p.72] and the projection formula. However, we can make the following:

In particular, although |$V^{i}(\mathcal{F}\left <t\ell \right>)$| is not a set, its dimension and codimension are well defined and they are denoted by |$\dim V^{i}(\mathcal{F}\left <t\ell \right>)$| and |$\mathrm{codim}_{\hat{A}}V^{i}(\mathcal{F}\left <t\ell \right>),$| respectively. In [12, Section 5] the cited authors use these loci in order to define generic vanishing notions for |${\mathbb{Q}}$|-twisted objects:

For example, if |$\ell $| is a polarization, then from the index theorem (see [17, III.16], or Kodaira vanishing, in the characteristic zero case) it follows that |$\mathcal{O}_{A}\left <t\ell \right>$| is IT(0) for |$t>0$|⁠. Another important example of these generic vanishing conditions regards the notion of separation of jets that we now recall:

In particular, |$L$| separates |$0$|-jets at |$x$| if and only if |$x$| is not a basepoint of |$L.$| Furthermore, if |$x$| is not a basepoint of |$L,$| then |$L$| separates 1-jets at |$x$| if and only if |$L$| separates tangent vectors at |$x$| (i.e., the map between tangent spaces induced by the rational morphism |$\phi _{L}:X \mathbb{P}H^{0}(L)$| is injective).

We have the following characterization of the behaviour of the generic vanishing conditions of |$\mathcal{F}\left <t\ell \right>$| for varying |$t$|⁠:

There is also the following important result of preservation of vanishing:

The conditions defined in this subsection can also be reformulated in terms of the Fourier–Mukai transform with kernel |$\mathcal{P},$| which we briefly survey in the following subsection.

2.2 Fourier–Mukai transform

If we write |$p_{A},p_{\hat{A}}$| for the product projections from |$A\times \hat{A},$| then we can consider the Fourier–Mukai functor |$\Phi _{\mathcal{P}}: D^{b}(A)\rightarrow D^{b}(\hat{A})$| given by

which by [16] is an equivalence of categories. For an object |$\mathcal{F}\in D^{b}(A)$| we will write |$R^{i}\Phi _{\mathcal{P}}(\mathcal{F})$| for the |$i$|-th cohomology sheaf of |$\Phi _{\mathcal{P}}(\mathcal{F}).$|

From the cohomology and base change theorem [8, Théorème (7.7.5) II] we have that

In particular, we see that if |$\mathcal{F}\left <\frac{a}{b}\ell \right>$| is IT(0) then |$R^{i}\Phi _{\mathcal{P}}(\lambda _{b}^{\ast }\mathcal{F}\otimes L^{ab})=0$| for every |$i\neq 0.$| Moreover, we have the following equivalence [25, Lemma 3.6]:

On the other hand, if |$c$| is a positive integer (not necessarily coprime to the characteristic of the base field), then by [16, (3.4)] we have that

In particular, as |$\lambda _{c}$| is finite, we see that the following conditions are equivalent:

• a)

  |$\mathcal{F}\left <t\ell \right>$| is GV;

• b)

  |$\operatorname{codim}_{\hat{A}} V^{i}(\mathcal{F},L,a,b)\geq i$| for every |$i\geq 1$| and a (any) representation |$t=a/b;$| and

• c)

  |$\operatorname{codim}_{\hat{A}} \operatorname{Supp} R^{i}\Phi _{\mathcal{P}}(\lambda _{b}^{\ast }\mathcal{F}\otimes L^{ab}) \geq i$| for every |$i$| and a (any) representation |$t=a/b.$|

2.3 Cohomological rank functions

For varying |$t\in{\mathbb{Q}},$| the fact that whether |$V^{i}(\mathcal{F}\left <t\ell \right>) = \hat{A}$| or not, can be expressed using the cohomological rank functions introduced by Jiang–Pareschi in [12]. In this subsection we recall their definition and fix some notation

One of the main properties of the cohomological rank functions is the following fundamental transformation formula with respect to the Fourier–Mukai functor:

Moreover, in Theorem 3.2 from loc-cit it is also shown that cohomological rank functions can be extended to continuous functions |${\mathbb{R}}\rightarrow{\mathbb{R}}.$|

3 Vanishing Thresholds

In this section we introduce the notion of vanishing thresholds for coherent sheaves. Afterwards, we give some examples and at the end we prove Theorem 1.1.1 from the introduction.

We note that, by definition, we have that |$\nu _{n\ell }(\mathcal{F}) = n^{-1}\nu _{\ell }(\mathcal{F}).$| Also, thanks to Theorem 2.6 we have the following equality:

A couple of formal properties are the following:

We also have the following:

In order to prove the theorem, the following subadditivity lemma is fundamental:

An immediate consequence is the following:

4 Seshadri Constants

In Section 3 above we proved that for a closed subscheme |$Z$| of dimension at most one, the sequence |$\{(p+1)^{-1}\beta _{p}(Z,\ell )\}_{p}$| converges. In this section we establish that the limit is actually the inverse of the Seshadri constant of the ideal |$I_{Z}$| with respect to |$\ell $|⁠.

First, recall that for an ample line bundle |$L$| and an ideal sheaf |$I,$| the Seshadri constant of |$I$| with respect to |$L$| is the real number

where |$\sigma :\mathrm{Bl}_{I}A\rightarrow A$| is the blow-up along |$I$| with exceptional divisor |$E.$|

5 Gauss–Wahl Maps

In this section we recall the definition of the Gauss–Wahl maps and in particular we introduce some thresholds that may be thought as the surjectivity thresholds of such maps. Then we establish Theorem 1.2 from the introduction, which relates these thresholds with the jets separation thresholds.

To start, consider a smooth projective variety |$X.$| We write |$p_{1},p_{2}:X\times X\rightarrow X$| for the projections, |$\Delta \subset X\times X$| for the diagonal and |$I_{\Delta }\subset \mathcal{O}_{X\times X}$| for the corresponding ideal sheaf. For |$n\in{\mathbb{Z}}_{\geq 0}$| we write |$n\Delta $| for the closed subscheme of |$X\times X$| defined by the ideal |$I_{\Delta }^{n}$| (we consider |$I_{\Delta }^{0} = \mathcal{O}_{X\times X}$|⁠), thus

From the definition it follows that |$P^{-1}(L) = 0$| and |$P^{0}(L) = L.$| Now, applying |$p_{1\ast }(p_{2}^{\ast }L\otimes -)$| to the exact sequence |$0\rightarrow I_{\Delta }^{n+1}\rightarrow \mathcal{O}_{X\times X}\rightarrow \mathcal{O}_{(n+1)\Delta }\rightarrow 0$| we get an exact sequence

where the fiber over |$x$| of the last arrow is given by the restriction

In particular, the right arrow is surjective if and only if |$L$| separates |$n$|-jets at every point of |$X.$| In this way we also see that |$R_{L}^{(-1)} = H^{0}(L)\otimes \mathcal{O}_{X}$| and |$R_{L}^{(0)}$| is the kernel of the evaluation map (which in [12] is denoted by |$M_{L}$|⁠). On the other hand, applying |$p_{1\ast }(p_{2}^{\ast }L\otimes -)$| to the exact sequence

we get an exact sequence

fitting in the following commutative and exact diagram:

where the surjectivity at the bottom follows from the fact that the sheaf |$I_{\Delta }^{n}/I_{\Delta }^{n+1}$| is supported on the diagonal (e.g., see [7, Proposition 1.3] for details). In particular, from the 5-lemma it follows that |$u$| is surjective as soon as the evaluation map |$\mathrm{ev}_{n}$| is surjective, that is, as soon as |$L$| separates |$n$|-jets at every point of |$X.$|

From the diagram (9) we see that the surjectivity of |$m_{L,M}^{n}$| implies the surjectivity of |$\gamma _{L,M}^{n}.$| Moreover, from (8) and the remark below it, we see that if |$L$| generate |$n$|-jets at every point then the surjectivity of |$m_{L,M}^{n}$| is equivalent to the vanishing of |$H^{1}(R_{L}^{(n)}\otimes M)$| and hence such vanishing implies the surjectivity of |$\gamma _{L,M}^{n}.$| In the case that |$M = L^{k}$| and |$X$| is an abelian variety, this suggests to study the cohomological rank function |$h^{1}(R_{L}^{(n)}\left <-\hspace{0.1cm}\ell \right>),$| where |$\ell $| is the class of |$L$| in |$\mathrm{NS}(X).$|

In the spirit of [12, Section 8] we introduce the following numbers:

Note that, when |$L$| separates |$p$|-jets at every point, the above lemma tells us that |$\mu _{p}(L)$| is the vanishing threshold |$\nu _{\ell }(R_{L}^{(p)})$| (see Definition 3.1) and, in this context, |$\mu _{p}(L)\geq 0.$| Note also that the invariant |$\mu _{0}(L)$| is already considered in [12], where it is denoted by |$s(L).$| In the cited reference it is shown that there is a relation between such number and the base-point freeness threshold, when |$L$| is globally generated. In the following we compute |$\Phi _{\mathcal{P}}(I_{0}^{p+1}\otimes L)$| to show that there is an analogous relation between |$\mu _{p}$| and |$\beta _{p}.$|

Before stating our result, we need to introduce further notation. Let |$L$| be an ample line bundle on an abelian variety |$A$| and |$\alpha \in \hat{A}.$| For |$a,b\in{\mathbb{Z}}_{>0}$| and |$n\in{\mathbb{Z}}_{\geq 0}$| we write

and for |$t = a/b\in{\mathbb{Q}}$| we write

and it is easy to see that this number does not depend on the particular representation |$t=a/b.$| As observed before, when |$L$| separates |$p$|-jets at every point, |$\operatorname{gcorank} m_{L}^{p}(t)$| is no other than |$h^{1}(R_{L}^{(p)}\left <t\ell \right>).$|

The main tool to prove this is the following lemma:

Lemma 5.6.

Let |$A$| be an abelian variety and |$L$| an ample line bundle over |$A.$| We write |$0^{n}$| for the closed subscheme of |$A$| defined by the ideal |$I_{0}^{n}.$| Then we have

Proof.

Let |$m:A\times A\rightarrow A$| be the multiplication map and |$p_{1},p_{2}:A\times A\rightarrow A$| be the projections.

By [16, (3.10)] we have that

Now, let |$\mu :A\times A\rightarrow A\times A$| be the map given by |$\mu (x,y) = (m(x,y),y),$| whose inverse is given by |$\nu (x,y) = (\delta (x,y),y),$| where |$\delta :A\times A\rightarrow A$| is the difference map. Then we have:

where in the last line we used the fact that |$\delta ^{\ast }\mathcal{O}_{0^{n}} = \mathcal{O}_{n\Delta }$| and the definition of the natural map in (7).

Proof of the Theorem.

 

We have that |$I_{0}^{p+1}\otimes L$| is isomorphic in |$D^{b}(A)$| to the complex |$L\rightarrow L\otimes \mathcal{O}_{0^{p+1}}$| (concentrated in degrees 0 and 1). By Lemma 5.6 above, this implies that in |$D^{b}(A)$| we have the following isomorphism:

$$ \begin{align*} &\varphi_{L}^{\ast}\Phi_{\mathcal{P}}(I_{0}^{p+1}\otimes L) \simeq \mathrm{EV}_{p}^{\bullet}\otimes L^{\vee},\end{align*} $$

where |$\mathrm{EV}_{p}^{\bullet }$| is the complex (concentrated in degrees 0 and 1) given by

By the transformation formula (Proposition 2.11) it follows that for |$s\in{\mathbb{Q}}^{-}$| we have

$$ \begin{align*} &h^{1}\left((I_{0}^{p+1}\otimes L)\left<s\ell\right>\right) = \frac{(-s)^{g}}{h^{0}(L)}\cdot h^{1}\left((\mathrm{EV}_{p}^{\bullet}\otimes L^{\vee})\left<-\frac{1}{s}\ell\right>\right).\end{align*} $$

Now, by Lemma 5.3 above, we have that both |$(P^{p}(L)\otimes L^{\vee })\left <-\frac{1}{s}\ell \right>$| and |$L^{\vee }\left <-\frac{1}{s}\ell \right>$| are IT(0) for |$s\in (-1,0).$| In this context, writing |$s=-a/b$| with |$b>a>0,$| the spectral sequence [9, Remark 2.67]

$$ \begin{align*} &E_{1}^{r,q} = H^{q}(\lambda_{a}^{\ast}(\mathrm{EV}_{p}^{r}\otimes L^{\vee})\otimes L^{ab}\otimes P_{\alpha})\implies H^{r+q}(\lambda_{a}^{\ast}(\mathrm{EV}_{p}^{\bullet}\otimes L^{\vee})\otimes L^{ab}\otimes P_{\alpha})\end{align*} $$

says that

$$ \begin{align*} &H^{1}(\lambda_{a}^{\ast}(\mathrm{EV}_{p}^{\bullet}\otimes L^{\vee})\otimes L^{ab}\otimes P_{\alpha})\simeq\operatorname{Coker} m_{L,\alpha}^{p}(b-a,a)\end{align*} $$

and thus

$$ \begin{align*} &h^{1}\left((\mathrm{EV}_{p}^{\bullet}\otimes L^{\vee})\left<-\frac{1}{s}\ell\right>\right) = \operatorname{gcorank} m_{L}^{p}\left(-\left(1+\frac{1}{s}\right)\right).\end{align*} $$

Finally, as |$h^{1}((I_{0}^{p+1}\otimes L)\left <s\ell \right>) = h^{1}(I_{0}^{p+1}\left <(1+s)\ell \right>),$| setting |$y = 1+s,$| the result follows.

An important consequence of Theorem 5.5 is that we can ensure the surjectivity of certain Gauss–Wahl maps when we know that |$\beta _{p}$| is small. We discuss this application in detail in the next section. For the moment, we limit ourselves to discuss formal consequences of the above theorem and its relation with the Seshadri constant.

First, we note that, unlike the jets separation thresholds, a priori it is not clear how |$\mu _{p}(L)$| changes when we replace |$L$| by a multiple of it. However, using the previous theorem, we are able to give such a formula:

Combining with the results from previous section we get an expression of the Seshadri constant in terms of the thresholds |$\mu _{p}.$|

Using the results of this section, we get the following:

6 Effective Surjectivity of Gauss–Wahl Maps

In this section we establish Theorem 1.4 from the introduction.

Finally, we use multiplier ideals in order to bound the jets separation thresholds |$\beta _{p}(\ell )$| in terms of the Seshadri constant of |$\ell .$| To do this, we will need the following:

Now, if we fix |$c\ll 1$| and |$m\gg 0$| as in the previous lemma then considering |$\Delta = \frac{g+p}{mu_{0}}D_{0}$| and |$r\in{\mathbb{Q}}$| with |$r\geq \frac{g+p}{\varepsilon (L)}\left (1-\frac{c}{2}\right ),$| we obtain:

Now, as |$\mu $| is a log-resolution of |$D_{0}$| we have that |$\mu _{\ast }^{-1}D_{0}+E$| is snc and hence, as |$m\gg 0,$| it follows that

In this way we can compute the multiplier ideal

On the other hand,

which is ample because of the way in we chose |$r.$| Therefore, the following inequality directly follows from Nadel’s vanishing:

Now, from Proposition 6.3 and Theorem 6.1 we are able to prove the following:

To illustrate the usage of this result, let |$L,M$| be algebraically equivalent ample line bundles as in the theorem. Consider |$f,h:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}_{\geq 0}$| two functions and suppose that we want to ensure the surjectivity of |$\gamma _{cL,dM}^{p}$| for all |$c\geq f(p)$| and |$d\geq h(p).$| In order to apply the corollary we would just need

or, equivalently,

As an example, we contrast with [22, Theorem 2.2]. In such reference it is proved that |$\gamma _{cL,dM}^{p}$| is surjective for all |$c,d\geq 2(p+1)$| with |$c+d\geq 4p+5.$| Now, if we consider |$f(p) = h(p) = 2p+1$| we get that |$\gamma _{cL,dM}^{p}$| is surjective for all |$c,d\geq 2(p+1)$| (no matter |$c+d$|⁠) whenever

As |$(g+p)/(p+1/2)\to 1$| as |$p\to \infty ,$| from Nakamaye’s theorem we get that if |$(A,\ell )$| is not as in (3) then (15) holds for |$p\gg 0.$| In this sense, Theorem 6.1 above asymptotically improves [22, Theorem 2.2].

7 Further Questions

1. In Corollary 5.9 we saw that a special kind of polarized abelian varieties, namely the products

are characterized by the equality |$\beta _{p}(\ell ) = p+1$| for all  |$p.$| Now, by [28, Proposition 2] we know that if |$(A,\theta )$| is a principally polarized abelian surface (p.p.a.s) then |$\varepsilon (\theta ) = 4/3$| whenever |$(A,\theta )$| is not a product. From Proposition 6.3 this means that for indecomposable p.p.a.s and |$p\geq 2$| we have

and thus, a posteriori, the polarized surfaces of the form (16) are characterized by the equality |$\beta _{2}(\theta ) = 3.$|

More generally, we may ask:

2. Note that, except for Proposition 6.3, whose proof uses Nadel’s vanishing theorem, the results presented in this article hold over an algebraically closed field of any characteristic. We may then ask the following:

On the other hand, in [19] and in [18], there are introduced the Frobenius–Seshadri constants  |$\varepsilon _{F}^{k}(L,x)$| of a line bundle |$L$| at a point |$x$| of a smooth variety |$X$| over an algebraically closed field of positive characteristic and it is shown that the following inequalities hold:

where |$\varepsilon (L,x)$| is the usual Seshadri constant at |$x$|⁠. This, together with our Theorem 1.1 3), suggests to study the following:

For instance, if we are able to prove that

then Question 7.3 would have a positive answer.

Funding

This work was supported by the MIUR Excellence Department Project MatMod@TOV awarded to the Department of Mathematics of the University of Rome Tor Vergata.

Acknowledgments

This work is part of the author’s Phd thesis. The author would like to thank his advisor, Giuseppe Pareschi, for his invaluable guidance and help to improve both the exposition and the mathematical content of this article. The author also thanks to the anonymous referees for their comments, which were very useful to improve the presentation of this article and to clarify some proofs.

Communicated by Enrico Arbarello

References