Seshadri Constants Research Articles

On the integrality of Seshadri constants of abelian surfaces

We consider the question of when Seshadri constants on abelian surfaces are integers. Our first result concerns self-products $$E\times E$$ of elliptic curves: if E has complex multiplication in $${\mathbb {Z}}[i]$$ or in $${\mathbb {Z}}[(1\,{+}\,i\sqrt{3})/2]$$ or if E has no complex multiplication at all, then it is known that for every ample line bundle L on , the Seshadri constant $$\varepsilon (L)$$ is an integer. We show that, contrary to what one might expect, these are in fact the only elliptic curves for which this integrality statement holds. Our second result answers the question how—on any abelian surface—integrality of Seshadri constants is related to elliptic curves.

Seshadri constants and special configurations of points in the projective plane

In the present note, we focus on certain properties of special curves that might be used in the theory of multi-point Seshadri constants for ample line bundles on the complex projective plane. In particular, we provide three Ein-Lazarsfeld-Xu-type lemmas for plane curves and a lower bound on the multi-point Seshadri constant of $\mathcal {O}_{\mathbb {P}^{2}}(1)$ under the assumption that the chosen points are not very general. In the second part, we focus on certain arrangements of points in the plane which are given by line arrangements. We show that, in some cases, the multi-point Seshadri constants of $\mathcal {O}_{\mathbb {P}^{2}}(1)$ centered at singular loci of line arrangements are computed by lines from the arrangement having some extremal properties.

On Seshadri constants of varieties with large fundamental group

Let $X$ be a smooth variety and let $L$ be an ample line bundle on $X$. If $\pi^{alg}_{1}(X)$ is large, we show that the Seshadri constant $\epsilon(p^{*}L)$ can be made arbitrarily large by passing to a finite etale cover $p:X'\rightarrow X$. This result answers affirmatively a conjecture of J.-M. Hwang. Moreover, we prove an analogous result when $\pi_{1}(X)$ is large and residually finite. Finally, under the same topological assumptions, we appropriately generalize these results to the case of big and nef line bundles. More precisely, given a big and nef line bundle $L$ on $X$ and a positive number $N>0$, we show that there exists a finite etale cover $p: X'\rightarrow X$ such that the Seshadri constant $\epsilon(p^{*}L; x)\geq N$ for any $x\notin p^{*}\textbf{B}_{+}(L)=\textbf{B}_{+}(p^{*}L)$, where $\textbf{B}_{+}(L)$ is the augmented base locus of $L$.

Fano varieties with large Seshadri constants

We show that the set of Fano varieties (with arbitrary singularities) whose anticanonical divisors have large Seshadri constants satisfies certain weak and birational boundedness. We also classify singular Fano varieties of dimension n whose anticanonical divisors have Seshadri constants at least n, generalizing an earlier result of Liu and the author.

Minimal plane valuations

We consider the value μ ^ ( ν ) = lim m → ∞ m − 1 a ( m L ) \hat {\mu } (\nu ) = \lim _{m \rightarrow \infty } m^{-1} a(mL) , where a ( m L ) a(mL) is the last value of the vanishing sequence of H 0 ( m L ) H^0(mL) along a divisorial or irrational valuation ν \nu centered at O P 2 , p \mathcal {O}_{\mathbb {P}^2,p} , L L (respectively, p p ) being a line (respectively, a point) of the projective plane P 2 \mathbb {P}^2 over an algebraically closed field. This value contains, for valuations, similar information as that given by Seshadri constants for points. It is always true that μ ^ ( ν ) ≥ 1 / v o l ( ν ) \hat {\mu } (\nu ) \geq \sqrt {1 / \mathrm {vol}(\nu )} and minimal valuations are those satisfying the equality. In this paper, we prove that the Greuel–Lossen-Shustin Conjecture implies a variation of the Nagata Conjecture involving minimal valuations (that extends the one stated in [Comm. Anal. Geom. 25 (2017), pp. 125–161] to the whole set of divisorial and irrational valuations of the projective plane) which also implies the original Nagata Conjecture. We also provide infinitely many families of minimal very general valuations with an arbitrary number of Puiseux exponents and an asymptotic result that can be considered as evidence in the direction of the above-mentioned conjecture.

Seshadri constants on hyperelliptic surfaces

We prove new results on single point Seshadri constants for ample line bundles on hyperelliptic surfaces, motivated by the results of Farnik [Arch. Math. 107 (2016), pp. 227–237]. Given a hyperelliptic surface X X and an ample line bundle L L on X X , we show that the least Seshadri constant ε ( L ) \varepsilon (L) of L L is a rational number when X X is not of type 6. We also prove new lower bounds for the Seshadri constant ε ( L , 1 ) \varepsilon (L,1) of L L at a very general point.

Diophantine Approximation Constants for Varieties over Function Fields

By analogy with the program of McKinnon and Roth [10], we define and study approximation constants for points of a projective variety X defined over K, the function field of an irreducible and nonsingular in codimension 1 projective variety defined over an algebraically closed field of characteristic zero. In this setting, we use Wang’s theorem, which is an effective version of Schmidt’s subspace theorem, to give a sufficient condition for such approximation constants to be computed on a proper K-subvariety of X. We also indicate how our approximation constants are related to volume functions and Seshadri constants.

Local positivity of linear series on surfaces

We study asymptotic invariants of linear series on surfaces with the help of Newton-Okounkov polygons. Our primary aim is to understand local positivity of line bundles in terms of convex geometry. We work out characterizations of ample and nef line bundles in terms of their Newton-Okounkov bodies, treating the infinitesimal case as well. One of the main results is a description of moving Seshadri constants via infinitesimal Newton-Okounkov polygons. As an illustration of our ideas we reprove results of Ein-Lazarsfeld on Seshadri constants on surfaces.

Canonical growth conditions associated to ample line bundles

We propose a new construction which associates to any ample (or big) line bundle $L$ on a projective manifold $X$ a canonical growth condition (i.e. a choice of a psh function well-defined up to a bounded term) on the tangent space $T_p X$ of any given point $p$. We prove that it encodes such classical invariants as the volume and the Seshadri constant. Even stronger, it allows you to recover all the infinitesimal Okounkov bodies of $L$ at $p$. The construction is inspired by toric geometry and the theory of Okounkov bodies; in the toric case the growth condition is "equivalent" to the moment polytope. As in the toric case the growth condition says a lot about the K\"ahler geometry of the manifold. We prove a theorem about K\"ahler embeddings of large balls, which generalizes the well-known connection between Seshadri constants and Gromov width established by McDuff and Polterovich.

Frobenius–Seshadri constants and characterizations of projective space

We introduce higher-order variants of the Frobenius-Seshadri constant due to Musta\c{t}\u{a} and Schwede, which are defined for ample line bundles in positive characteristic. These constants are used to show that Demailly's criterion for separation of higher-order jets by adjoint bundles also holds in positive characteristic. As an application, we give a characterization of projective space using Seshadri constants in positive characteristic, which was proved in characteristic zero by Bauer and Szemberg. We also discuss connections with other characterizations of projective space.

Single point Seshadri constants on rational surfaces

Motivated by a similar result of Dumnicki, Küronya, Maclean and Szemberg under a slightly stronger hypothesis, we exhibit irrational single-point Seshadri constants on a rational surface X obtained by blowing up very general points of PC2, assuming only that all prime divisors on X of negative self-intersection are smooth rational curves C with C2=−1. (This assumption is a consequence of the SHGH Conjecture, but it is weaker than assuming the full conjecture.)

Asymptotic base loci via Okounkov bodies

An Okounkov body is a convex subset of Euclidean space associated to a divisor on a smooth projective variety with respect to an admissible flag. In this paper, we recover the asymptotic base loci from the Okounkov bodies by studying various asymptotic invariants such as the asymptotic valuations and the moving Seshadri constants. Consequently, we obtain the nefness and ampleness criteria of divisors in terms of the Okounkov bodies. Furthermore, we compute the divisorial Zariski decomposition by the Okounkov bodies, and find upper and lower bounds for moving Seshadri constants given by the size of simplexes contained in the Okounkov bodies.

Characterization of projective spaces by Seshadri constants

We prove that an n-dimensional complex projective variety is isomorphic to {mathbb {P}}^n if the Seshadri constant of the anti-canonical divisor at some smooth point is greater than n. We also classify complex projective varieties with Seshadri constants equal to n.

The Toledo invariant, and Seshadri constants of fake projective planes

The purpose of this paper is to explicitly compute the Seshadri constants of all ample line bundles on fake projective planes. The proof relies on the theory of the Toledo invariant, and more precisely on its characterization of $\mathbb{C}$-Fuchsian curves in complex hyperbolic spaces.

Multi-point Seshadri constants on ruled surfaces

Let $X$ be a surface and let $L$ be an ample line bundle on $X$. We first obtain a lower bound for the Seshadri constant $\varepsilon(X,L,r)$, when $r \ge 3$. We then assume that $X$ is a ruled surface and study Seshadri constants on $X$ in greater detail. We also make precise computations of Seshadri constants on $X$ in some cases.

Infinitesimal Newton–Okounkov bodies and jet separation

In this article we explore the connection between asymptotic base loci and Newton–Okounkov bodies associated to infinitesimal flags. Analogously to the surface case, we obtain complete characterizations of augmented and restricted base loci. Interestingly enough, an integral part of the argument is a study of the relationship between certain simplices contained in Newton–Okoukov bodies and jet separation; our results also lead to a convex geometric description of moving Seshadri constants.

Kähler packings and Seshadri constants on projective complex surfaces

In analogy to the relation between symplectic packings and symplectic blow ups we show that multiple point Seshadri constants on projective complex surfaces can be calculated as the supremum of radii of multiple Kähler ball embeddings. We exemplify this connection on toric surfaces, also discussing how toric moment maps reflect the packing.

Restrictions on Seshadri Constants on Surfaces

Starting with the pioneering work of Ein and Lazarsfeld restrictions on values of Seshadri constants on algebraic surfaces have been studied by many authors. In the present note we show how approximation involving continued fractions combined with recent results of Kuronya and Lozovanu on Okounkov bodies of line bundles on surfaces lead to effective statements considerably restricting possible values of Seshadri constants. These results in turn provide strong aditional evidence to a conjecture governing the Seshadri constants on algebraic surfaces with Picard number 1.

Very general monomial valuations of $\mathbb{P}^2$ and a Nagata-type conjecture

It is well known that multi-point Seshadri constants for a small number t of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for t≥9 points. Tackling the problem in the language of valuations one can make sense of t points for any real t≥1. We show somewhat surprisingly that a Nagata-type conjecture should be valid for t≥8+1/36 points and we compute explicitly all Seshadri constants (expressed here as the asymptotic maximal vanishing element) for t≤7+1/9. In the range 7+1/9≤t≤8+1/36 we are able to compute some sporadic values.

Seshadri constants on surfaces with Picard number 1

Let X be a smooth projective surface with Picard number 1. Let L be the ample generator of the Neron–Severi group of X. Given an integer \(r\ge 2\), we prove lower bounds for the Seshadri constant of L at r very general points in X.