Fano threefolds in positive characteristic II

TL;DR

This paper investigates the properties of Fano threefolds over algebraically closed fields of positive characteristic, establishing bounds on their genus and the non-existence of certain Fano blowups.

Contribution

It proves genus bounds for smooth Fano threefolds with very ample anticanonical bundle in positive characteristic and shows the non-existence of specific Fano blowups along smooth curves.

Findings

Genus g satisfies 3 ≤ g ≤ 12, g ≠ 11.

No smooth curve exists on X along which the blowup is Fano.

Results extend classification to positive characteristic settings.

Abstract

Let $X$ be a smooth Fano threefold over an algebraically closed field of positive characteristic. Assume that $|-K_X|$ is very ample and each of the index and the Picard number is equal to one. We prove that $3 \leq g \leq 12$ and $g \neq 11$ for the genus $g$ of $X$. Moreover, we show that there exists no smooth curve on $X$ along which the blowup is Fano.