Seshadri Constants Research Articles

Seshadri constants on blow‐ups of Hirzebruch surfaces

AbstractLet be integers and let denote the Hirzebruch surface with invariant . We compute the Seshadri constants of an ample line bundle at an arbitrary point of the ‐point blow‐up of when and at a very general point when or . We also discuss several conjectures on linear systems of curves on the blow‐up of at very general points.

Rational curves and Seshadri constants on Enriques surfaces

We prove that classes of rational curves on very general Enriques surfaces are always 2 2 -divisible. As a consequence, we compute the Seshadri constant of any big and nef line bundle on a very general Enriques surface, proving that it coincides with the value of the ϕ \phi -function introduced by Cossec [Math. Ann. 271 (1985), pp. 577–600].

Seshadri constants and AG codes of vector bundles

We give estimates for the minimum distances of the algebraic geometric codes on certain fibered varieties. By means of the Seshadri constants, we also obtain similar results for the codes defined from vector bundles of higher rank.

Seshadri Constants of Curve Configurations on Surfaces

Seshadri Constants of Curve Configurations on Surfaces

Seshadri constants of parabolic vector bundles

Let X be a complex projective variety, and let E_* be a parabolic vector bundle on X with parabolic structure on a divisor. We introduce the notion of parabolic Seshadri constants of E_* . It is shown that these constants are analogous to the classical Seshadri constants of vector bundles, in particular they have parallel definitions and properties. We prove a Seshadri criterion for parabolic ampleness of E_* in terms of the parabolic Seshadri constants of it. We also compute parabolic Seshadri constants for symmetric powers and tensor products of parabolic vector bundles.

Seshadri constants on some blow-ups of projective spaces

Let [Formula: see text] denote the blow-up of [Formula: see text] along [Formula: see text] general lines and [Formula: see text] general points. In this paper, we focus on [Formula: see text]-very ample line bundles on [Formula: see text] and investigate their Seshadri constants with some restrictions on [Formula: see text]. Additionally, we compute the nef cone of [Formula: see text] for [Formula: see text] and study the Seshadri constants of some ample line bundles on it. We also examine the Seshadri constants of some ample line bundles on [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text].

Some results on Seshadri constants of vector bundles

Abstract We study Seshadri constants of certain ample vector bundles on projective varieties. Our main motivation is the following question: Under what conditions are the Seshadri constants of ample vector bundles at least 1 at all points of the variety? We exhibit some conditions under which this question has an affirmative answer. We primarily consider ample bundles on projective spaces and Hirzebruch surfaces. We also show that Seshadri constants of ample vector bundles can be arbitrarily small.

Lower bounds for Seshadri constants via successive minima of line bundles

Given a nef and big line bundle L L on a projective variety X X of dimension d ≥ 2 d \geq 2 , we prove that the Seshadri constant of L L at a very general point is larger than ( d + 1 ) 1 d − 1 (d+1)^{\frac {1}{d}-1} . This slightly improves the lower bound 1 / d 1/d established by Ein, Küchle and Lazarsfeld [J. Differential Geom. 42 (1995), pp. 193–219]. The proof relies on the concept of successive minima for line bundles recently introduced by Ambro and Ito [Adv. Math. 365 (2020), 38 pp.].

Duality for asymptotic invariants of graded families

The starting point of this paper is a duality for sequences of natural numbers which, under mild hypotheses, interchanges subadditive and superadditive sequences and inverts their asymptotic growth constants.We are motivated to explore this sequence duality since it arises naturally in at least two important algebraic-geometric contexts. The first context is Macaulay-Matlis duality, where the sequence of initial degrees of the family of symbolic powers of a radical ideal is dual to the sequence of Castelnuovo-Mumford regularity values of a quotient by ideals generated by powers of linear forms. This philosophy is drawn from an influential paper of Emsalem and Iarrobino. We generalize this duality to differentially closed graded filtrations of ideals.In a different direction, we establish a duality between the sequence of Castelnuovo-Mumford regularity values of the symbolic powers of certain ideals and a geometrically inspired sequence we term the jet separation sequence. We show that this duality underpins the reciprocity between two important geometric invariants: the multipoint Seshadri constant and the asymptotic regularity of a set of points in projective space.

Gaussian Gabor frames, Seshadri constants and generalized Buser–Sarnak invariants

We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from Kähler geometry such as Hörmander’s {overline{partial }}-L^2 estimate with singular weight, Demailly’s Calabi–Yau method for Kähler currents and a Kähler-variant generalization of the symplectic embedding theorem of McDuff–Polterovich for ellipsoids. Our approach is based on the well-known link between sets of interpolation for the Bargmann-Fock space and the frame set of multivariate Gaussian Gabor frames. We state sufficient conditions in terms of a certain extremal type Seshadri constant of the complex torus associated to a lattice to be a set of interpolation for the Bargmann-Fock space, and give also a condition in terms of the generalized Buser-Sarnak invariant of the lattice. In particular, we obtain an effective Gaussian Gabor frame criterion in terms of the covolume for almost all lattices, which is the first general covolume criterion in multivariate Gaussian Gabor frame theory. The recent Berndtsson–Lempert method and the Ohsawa–Takegoshi extension theorem also allow us to give explicit estimates for the frame bounds in terms of certain Robin constant. In the one-dimensional case we obtain a sharp estimate of the Robin constant using Faltings’ theta metric formula for the Arakelov Green functions.

Fano foliations with small algebraic ranks

In this paper we study the algebraic ranks of foliations on Q-factorial normal projective varieties. We start by establishing a Kobayashi-Ochiai's theorem for Fano foliations in terms of algebraic rank. We then investigate the local positivity of the anti-canonical divisors of foliations, obtaining a lower bound for the algebraic rank of a foliation in terms of Seshadri constant. We describe those foliations whose algebraic rank slightly exceeds this bound and classify Fano foliations on smooth projective varieties attaining this bound. Finally we construct several examples to illustrate the general situation, which in particular allow us to answer a question asked by Araujo and Druel on the generalised indices of foliations.

Seshadri constants and K-stability of Fano manifolds

We give a lower bound of the δ-invariants of ample line bundles in terms of Seshadri constants. As applications, we prove the uniform K-stability of infinitely many families of Fano hypersurfaces of arbitrarily large index, as well as the uniform K-stability of most families of smooth Fano threefolds of Picard number one.

Seshadri functions on abelian surfaces

So far, Seshadri constants on abelian surfaces are completely understood only in the cases of Picard number one and on principally polarized abelian surfaces with real multiplication. Beyond that, there are partial results for products of elliptic curves. In this paper, we show how to compute the Seshadri constant of any nef line bundle on any abelian surface over the complex numbers. We develop an effective algorithm depending only on the basis of the Néron-Severi group to compute not only the Seshadri constants but also the numerical data of their Seshadri curves. Access to the Seshadri curves allows us to plot Seshadri functions and better understand their structure. We show that already in the case of Picard number two the complexity of Seshadri functions can vary to a great degree. Our results indicate that aside from finitely many cases the complexity of the Seshadri function is at least as high as in the Cantor function.

Seshadri constants over fields of characteristic zero

Let X be a smooth projective variety defined over a field k of characteristic 0 and let L be a nef line bundle defined over k. We prove that if x∈X is a k-rational point then the Seshadri constant ε(X,L,x) over k‾ is the same as that over k. We show, by constructing families of examples, that there are varieties whose global Seshadri constant ε(X) is zero. We also prove a result on the existence of a Seshadri curve with a natural (and necessary) hypothesis.

The Gromov Width of Bott-Samelson Varieties

We prove that the Gromov width of any Bott-Samelson variety associated to a reduced expression and equipped with a rational Kähler form equals the symplectic area of a minimal curve. From this, we derive an estimate for the Seshadri constants of ample line bundles on Bott-Samelson varieties.

Seshadri constants on abelian and bielliptic surfaces—potential values and lower bounds

In this note we contribute to the study of Seshadri constants on abelian and bielliptic surfaces. We specifically focus on bounds that hold on all such surfaces, depending only on the self-intersection of the ample line bundle under consideration. Our result improves previous bounds and it provides rational numbers as bounds, which are potential Seshadri constants.

Fano varieties with large Seshadri constants in positive characteristic

We prove that a Fano variety (with arbitrary singularities) of dimension n in positive characteristic is isomorphic to ℙ n if the Seshadri constant of the anti-canonical divisor at some smooth point is greater than n and classify Fano varieties whose anti-canonical divisors have Seshadri constants n. In characteristic p>5 and dimension 3, we also show that Fano varieties X with Seshadri constants ϵ(-K X ,x)>2+ϵ at some smooth point x∈X (for some fixed ϵ>0) have bounded anti-canonical degrees.

Generation of jets and Fujita's jet ampleness conjecture on toric varieties

Jet ampleness of line bundles generalizes very ampleness by requiring the existence of enough global sections to separate not just points and tangent vectors , but also their higher order analogues called jets. We give sharp bounds guaranteeing that a line bundle on a projective toric variety is k -jet ample in terms of its intersection numbers with the invariant curves, in terms of the lattice lengths of the edges of its polytope , in terms of the higher concavity of its piecewise linear function and in terms of its Seshadri constant. For example, the tensor power k + n − 2 of an ample line bundle on a projective toric variety of dimension n ≥ 2 always generates all k -jets, but might not generate all ( k + 1 ) -jets. As an application, we prove the k -jet generalizations of Fujita's conjectures on toric varieties with arbitrary singularities.

Seshadri constants on Bott towers

For a positive integer n, let Xn→Xn−1→…→X2→X1→X0 be a Bott tower of height n, and let L be a nef line bundle on Xn. We compute Seshadri constants ε(Xn,L,x) of L at any point x∈Xn under some conditions.

On Seshadri constants of non-simple abelian varieties

Let (A, L) be a polarized abelian variety of dimension n. We prove two results for the Seshadri constants of abelian varieties. The first one is that if the Seshadri constant \(\epsilon (A,L)\) is relatively small with respect to \(L^n\), there exists a proper abelian subvariety B such that \(\epsilon (A,L)\) is equal to \(\epsilon (B,L|_B)\). Second, we prove that if there exists a codimension one abelian subvariety D satisfying certain numerical conditions, then \(\epsilon (D,L|_D)\) is equal to \(\epsilon (A,L)\). As an application of these results, we investigate the structure of low dimensional polarized abelian varieties whose Seshadri constants is sufficiently small with respect to \(L^n\).