Fano varieties with large Seshadri constants in positive characteristic

TL;DR

This paper characterizes Fano varieties in positive characteristic with large Seshadri constants, showing they are isomorphic to projective space or classifying those with specific constants, and establishes boundedness results in dimension three.

Contribution

It proves that Fano varieties with Seshadri constants exceeding the dimension are isomorphic to projective space and classifies those with constants equal to the dimension, also providing boundedness in dimension three.

Findings

Fano varieties with Seshadri constant > n are isomorphic to .

Classification of Fano varieties with Seshadri constant = n.

Boundedness of anti-canonical degrees in dimension 3 for certain Seshadri constants.

Abstract

We prove that a Fano variety (with arbitrary singularities) of dimension $n$ in positive characteristic is isomorphic to $\mathbb{P}^n$ if the Seshadri constant of the anti-canonical divisor at some smooth point is greater than $n$ and classify Fano varieties whose anti-canonical divisors have Seshadri constants $n$. In characteristic $p>5$ and dimension $3$, we also show that Fano varieties $X$ with Seshadri constants $\epsilon(-K_X,x)>2+\epsilon$ at some smooth point $x\in X$ (for some fixed $\epsilon>0$) have bounded anti-canonical degrees.