Seshadri Constants for Curve Classes

Abstract

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.

Polar transform and local positivity for curves

“…Independently, M. Fulger [15] has also studied Seshadri constants for movable curve classes, and his work has substantial overlap with the present paper. The starting point of his work is the equivalent definition for S x in Remark 1.3.…”

“…Remark 4.8. -In [15], Fulger noticed the following estimate: let L ∈ Nef 1 (X) and let α = L n−1 . Then S x (α) s x (L) n−1 .…”

“…-In[15], Fulger shows that if α is a movable curve class, then x → S x (α) is lower semicontinuous in the countable Zariski topology. Using this and the definition of S x as the polar transform of n x , one easily sees that if L is a divisor class, then x → n x (L) is actually upper semicontinuous in this same topology, an improvement over Cor.…”