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)     

Seshadri constants in two half-periods

We calculate the exact value of the multiple point Seshadri constant in two halfperiods of an abelian surface with Picard number one. Received: 3 December 2003; revised: 9 July 2005     

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].     

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.

     

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 at very general points

We study the local positivity of an ample line bundle at a very general point of a smooth projective variety. We obtain a slight improvement of the result of Ein, Küchle, and Lazarsfeld.

     

An inequality between multipoint Seshadri constants

Let X be a projective variety of dimension n and L be a nef divisor on X. Denote by the d-dimensional Seshadri constant of r very general points in X. We prove that

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 and the generation of jets

In this paper we explore the connection between Seshadri constants and the generation of jets. It is well known that one way to view Seshadri constants is to consider them as measuring the rate of growth of the number of jets that multiples of a line bundle generate. Here we ask, conversely, what we can say about the number of jets once the Seshadri constant is known. As an application of our results, we prove a characterization of projective space among all Fano varieties in terms of Seshadri constants.     

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.     

Fano contact manifolds and nilpotent orbits

A contact structure on a complex manifold M is a corank 1 subbundle F of T_M such that the bilinear form on F with values in the quotient line bundle L = T_M/F deduced from the Lie bracket of vector fields is everywhere non-degenerate. In this paper we consider the case where M is a Fano manifold; this implies that L is ample.?If is a simple Lie algebra, the unique closed orbit in (for the adjoint action) is a Fano contact manifold; it is conjectured that every Fano contact manifold is obtained in this way. A positive answer would imply an analogous result for compact quaternion-Kahler manifolds with positive scalar curvature, a longstanding question in Riemannian geometry.?In this paper we solve the conjecture under the additional assumptions that the group of contact automorphisms of M is reductive, and that the image of the rational map M P(H ₀(M, L)*) sociated to L has maximum dimension. The proof relies on the properties of the nilpotent orbits in a semi-simple Lie algebra, in particular on the work of R. Brylinski and B. Kostant. Received: July 28, 1997     

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     

Remarks on defective Fano manifolds

This note continues our previous work on special secant defective (specifically, conic connected and local quadratic entry locus) and dual defective manifolds. These are now well understood, except for the prime Fano ones. Here we add a few remarks on this case, completing the results in our papers (Russo in Math Ann 344:597–617, 2009; Ionescu and Russo in Compos Math 144:949–962, 2008; Ionescu and Russo in J Reine Angew Math 644:145–157, 2010; Ionescu and Russo in Am J Math 135:349–360, 2013; Ionescu and Russo in Math Res Lett 21:1137–1154, 2014); see also the recent book (Russo, On the Geometry of Some Special Projective Varieties, Lecture Notes of the Unione Matematica Italiana, Springer, 2016).