Prime Fano threefolds of genus 8 in positive characteristic

TL;DR

This paper proves that prime Fano threefolds of genus 8 in positive characteristic are linear sections of Gr(2,6), establishing their irrationality and global F-regularity under certain conditions.

Contribution

It demonstrates the isomorphism of prime Fano threefolds of genus 8 with linear sections of Gr(2,6) in positive characteristic, extending known results to this setting.

Findings

Prime Fano threefolds of genus 8 are isomorphic to linear sections of Gr(2,6).

Such threefolds are irrational.

They are globally F-regular if the characteristic exceeds two.

Abstract

We prove that a prime Fano threefold of genus 8 over an algebraically closed field of positive characteristic is isomorphic to a linear section of the Grassmannian variety Gr(2, 6). As applications, it is shown that a prime Fano threefold of genus 8 is irrational and, if the characteristic is larger than two, then it is globally F-regular.