Seshadri Constants and Fujita's Conjecture via Positive Characteristic Methods

TL;DR

This paper introduces a novel approach to Fujita's conjecture using Seshadri constants and positive characteristic methods, successfully deriving new results for complex varieties with singularities without relying on vanishing theorems.

Contribution

It presents a new technique employing positive characteristic methods and Frobenius-Seshadri constants to study Fujita's conjecture, extending results to singular complex varieties.

Findings

Recovered some known results over complex numbers without vanishing theorems

Proved new results for complex varieties with singularities

Characterized projective space using Seshadri constants in positive characteristic

Abstract

In 1988, Fujita conjectured that there is an effective and uniform way to turn an ample line bundle on a smooth projective variety into a globally generated or very ample line bundle. We study Fujita's conjecture using Seshadri constants, which were first introduced by Demailly in 1992 with the hope that they could be used to prove cases of Fujita's conjecture. While examples of Miranda seemed to indicate that Seshadri constants could not be used to prove Fujita's conjecture, we present a new approach to Fujita's conjecture using Seshadri constants and positive characteristic methods. Our technique recovers some known results toward Fujita's conjecture over the complex numbers, without the use of vanishing theorems, and proves new results for complex varieties with singularities. Instead of vanishing theorems, we use positive characteristic techniques related to the Frobenius-Seshadri constants introduced by Musta\c{t}\u{a}-Schwede and the author. As an application of our results, we give a characterization of projective space using Seshadri constants in positive characteristic, which was proved in characteristic zero by Bauer and Szemberg.