On the Seshadri constants of adjoint line bundles

In the present paper we study the possible values of Seshadri constants. While in general every positive rational number appears as the local Seshadri constant of some ample line bundle, we point out that for adjoint line bundles there are explicit lower bounds depending only on the dimension of the underlying variety. In the surface case, where the optimal lower bound is 1/2, we characterize all possible values in the range between 1/2 and 1??there are surprisingly few. As expected, one obtains even more restrictive results for the Seshadri constants of adjoints of very ample line bundles. Finally, we study Seshadri constants of adjoint line bundles in the multi-point setting.     

Seshadri constants and degrees of defining polynomials

In this paper, we study a relation between Seshadri constants and degrees of defining polynomials. In particular, we compute the Seshadri constants on Fano varieties obtained as complete intersections in rational homogeneous spaces of Picard number one.     

Seshadri constants on ruled surfaces: the rational and the elliptic cases

We study the Seshadri constants on geometrically ruled surfaces. The unstable case is completely solved. Moreover, we give some bounds for the stable case. We apply these results to compute the Seshadri constant of the rational and elliptic ruled surfaces. Both cases are completely determined. The elliptic case provides an interesting picture of how particular is the behavior of the Seshadri constants.     

Seshadri constants on Jacobian of curves

We compute the Seshadri constants on the Jacobian of hyperelliptic curves, as well as of curves with genus three and four. For higher genus curves we conclude that if the Seshadri constants of their Jacobian are less than 2, then the curves must be hyperelliptic.

     

Seshadri constants on rational surfaces with anticanonical pencils

We study a Seshadri constant at a general point on a rational surface whose anticanonical linear system contains a pencil. First, we describe a Seshadri constant of an ample line bundle on such a rational surface explicitly by the numerical data of the ample line bundle. Second, we classify log del Pezzo surfaces which are special in terms of the Seshadri constants of the anticanonical divisors when the anticanonical degree is between 4 and 9.     

Okounkov bodies and Seshadri constants

Okounkov bodies, which are closed convex sets defined for big line bundles, have rich information on the line bundles. On the other hand, Seshadri constants are invariants which measure the positivity of line bundles. In this paper, we prove that Okounkov bodies give lower bounds of Seshadri constants.     

Lower bounds for Seshadri constants

One of Demailly's characterization of Seshadri constants on ample line bundles works with Lelong numbers of certain positive singular hermitian metrics. In this note this is translated into algebraic terms by using sections of multiples of the line bundle. The resulting formula for Seshadri constants is applied to compute Seshadri constants on blown up products of curves, to disprove a conjectured characterization of maximal rationally connected quotients and to introduce a new approach to Nagata's conjecture. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Frobenius variant of Seshadri constants

We define and study a version of Seshadri constant for ample line bundles in positive characteristic. We prove that lower bounds for this constant imply the global generation or very ampleness of the corresponding adjoint line bundle. As a consequence, we deduce that the criterion for global generation and very ampleness of adjoint line bundles in terms of usual Seshadri constants holds also in positive characteristic.     

Seshadri constants and Fano manifolds

In memory of Meeyoung Kim In this paper, we give a lower bound of Seshadri constants on smooth Fano varieties. More precisely, we show that on a smooth Fano manifold of dimension n whose anticanonical system is base point free, Seshadri constants of ample divisors are bounded from below by one over n–2. As a corollary we recover the earlier result on Fano threefolds. Mathematics Subject Classification (2000):14J45, 14N30.The author was supported in part by KOSEF Grant R14-2002-007-01001-0(2002).     

A lower bound on Seshadri constants of hyperplane bundles on threefolds

We give the lower bound on Seshadri constants for the case of very ample line bundles on threefolds. We consider the situation when the Seshadri constant is strictly less than 2 and give a version of Bauer’s theorem (Math Ann 313(3):547–583, 1999, Theorem 2.1) for singular surfaces so we can prove the same result for smooth threefolds.     

Algebro-geometric characterization of Cayley polytopes

In this paper, we give an algebro-geometric characterization of Cayley polytopes. As a special case, we also characterize lattice polytopes with lattice width one by using Seshadri constants.     

Seshadri fibrations of algebraic surfaces

We refine results of [6] and [10] which relate local invariants – Seshadri constants – of ample line bundles on surfaces to the global geometry – fibration structure. We show that the same picture emerges when looking at Seshadri constants measured at any finite subset of the given surface (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Seshadri constants and Okounkov bodies revisited

In recent years, the interaction between the local positivity of divisors and Okounkov bodies has attracted considerable attention, and there have been attempts to find a satisfactory theory of positivity of divisors in terms of convex geometry of Okounkov bodies. Many interesting results in this direction have been established by Choi–Hyun–Park–Won [4] and Küronya–Lozovanu [17], [18], [19] separately. The first aim of this paper is to give uniform proofs of these results. Our approach provides not only a simple new outlook on the theory but also proofs for positive characteristic in the most important cases. Furthermore, we extend the theorems on Seshadri constants to graded linear series setting. Finally, we introduce the integrated volume function to investigate the relation between Seshadri constants and filtered Okounkov bodies introduced by Boucksom–Chen [3].     

Base loci of linear series are numerically determined

We introduce a numerical invariant, called a moving Seshadri constant, which measures the local positivity of a big line bundle at a point. We then show how moving Seshadri constants determine the stable base locus of a big line bundle.

     

Seshadri constants on surfaces of general type

We study Seshadri constants of the canonical bundle on minimal surfaces of general type. First, we prove that if the Seshadri constant ε(K _X , x) is between 0 and 1, then it is of the form (m − 1)/m for some integer m ≥ 2. Secondly, we study values of ε(K _X , x) for a very general point x and show that small values of the Seshadri constant are accounted for by the geometry of X.     

Seshadi constants via functions on Newton–Okounkov bodies

The aim of this note is to establish a somewhat surprising connection between functions on Newton–Okounkov bodies and Seshadri constants of line bundles on algebraic surfaces.     

General blow-ups of the projective plane

We study linear series on a projective plane blown up in a bunch of general points. Such series arise from plane curves of fixed degree with assigned fat base points. We give conditions (expressed as inequalities involving the number of points and the degree of the plane curves) on these series to be base point free, i.e. to define a morphism to a projective space. We also provide conditions for the morphism to be a higher order embedding. In the discussion of the optimality of obtained results we relate them to the Nagata Conjecture expressed in the language of Seshadri constants and we give a lower bound on these invariants.     

Seshadri constants in a family of surfaces

The aim of this note is to study local and global Seshadri constants for a family of smooth surfaces with prescribed polarization. Received: 16 April 2001 / Published online: 26 April 2002     

Seshadri constants and the spaces of curves with prescribed singularities and positions

We will give optimal bounds for Seshadri constants of an ample line bundle at multiple points on a complex projective surface X.We also present a solution to the long-studied classical problem on the existence of curves on X with given topological singularities at r arbitrary points p1,...,pr.Namely,we obtain a universal lower bound on the degree of curves for the existence.It is independent of the position of the singular points.     

A Second Main Theorem of Nevanlinna Theory for Closed Subschemes in Subgeneral Position

In this paper, by using Seshadri constants for subschemes, the author establishes a second main theorem of Nevanlinna theory for holomorphic curves intersecting closed subschemes in (weak) subgeneral position. As an application of his second main theorem,he obtain a Brody hyperbolicity result for the complement of nef effective divisors. He also give the corresponding Schmidt’s subspace theorem and arithmetic hyperbolicity result in Diophantine approximation.