Weak boundedness of Fano threefolds with large Seshadri constants in characteristic $p>5$

TL;DR

This paper proves that in characteristic p>5, Fano threefolds with large Seshadri constants at some smooth point have bounded anticanonical volume, extending understanding of their geometric properties.

Contribution

It establishes an upper bound on the anticanonical volume of Fano threefolds with large Seshadri constants in characteristic p>5, regardless of singularities.

Findings

Anticanonical volume is bounded above under given conditions.

Large Seshadri constant implies geometric boundedness.

Results hold for Fano threefolds with arbitrary singularities.

Abstract

Given $\epsilon>0$, we show that over an algebraically closed field of characteristic $p>5$, the anticanonical volume of a Fano threefold $X$ (with arbitrary singularities) whose anticanonical divisor has Seshadri constant $\epsilon(-K_X,x)>2+\epsilon$ at some smooth point $x\in X$ is bounded from above.