Seshadri Constants for Curve Classes

We develop a local positivity theory for movable curves on projective varieties similar to the classical Seshadri constants of nef divisors. We give analogues of the Seshadri ampleness criterion, of a characterization of the augmented base locus of a big and nef divisor, and of the interpretation of Seshadri constants as an asymptotic measure of jet separation. As application, we show in any characteristic that if $C$ is a smooth curve with ample normal bundle in a smooth projective variety then the class of $C$ is in the strict interior of the Mori cone. This was conjectured by Peternell and proved by Ottem and Lau in Characteristic 0.

On Non-Archimedean Curves Omitting Few Components and their Arithmetic Analogues

Let k be an algebraically closed field completewith respect to a non-Archimedean absolute value of arbitrary characteristic. Let D1 , … , Dn be effective nef divisors intersecting transversally in an n-dimensional nonsingular projective variety X. We study the degeneracy of non-Archimedean analytic maps from k into under various geometric conditions. When X is a rational ruled surface and D1 and D2 are ample, we obtain a necessary and sufficient condition such that there is no non-Archimedean analytic map from k into . Using the dictionary between non-Archimedean Nevanlinna theory and Diophantine approximation that originated in earlier work with T. T. H. An, we also study arithmetic analogues of these problems, establishing results on integral points on these varieties over ℤ or the ring of integers of an imaginary quadratic field.

Teissier's problem on inequalities of nef divisors

Teissier has proven remarkable inequalities [Formula: see text] for intersection numbers si = (ℒi ⋅ ℳd-i) of a pair of nef line bundles ℒ, ℳ on a d-dimensional complete algebraic variety over a field. He asks if two nef and big line bundles are numerically proportional if the inequalities are all equalities. In this paper, we show that this is true in the most general possible situation, for nef and big line bundles on a proper irreducible scheme over an arbitrary field k. Boucksom, Favre and Jonsson have recently established this result on a complete variety X over an algebraically closed field of characteristic zero. Their proof involves an ingenious extension of the intersection theory on a variety to its Zariski Riemann Manifold. This extension requires the existence of a direct system of nonsingular varieties dominating X. We make use of a simpler intersection theory which does not require resolution of singularities, and extend volume to an arbitrary field and prove its continuous differentiability, extending results of Boucksom, Favre and Jonsson, and of Lazarsfeld and Mustaţă. A goal in this paper is to provide a manuscript which will be accessible to many readers. As such, subtle topological arguments which are required to give a complete proof in [S. Bouksom et al., J. Algebraic Geometry18 (2009) 279–308] have been written out in this manuscript, in the context of our intersection theory, and over arbitrary varieties.