Varieties with ample Frobenius-trace kernel

TL;DR

This paper explores the positivity properties of the Frobenius trace kernel on smooth projective varieties, aiming to find analogs of classical theorems in the context of positive characteristic algebraic geometry.

Contribution

It establishes conditions under which the Frobenius trace kernel is ample, particularly characterizing Fano varieties of Picard rank one as the cases where ampleness occurs.

Findings

The Frobenius trace kernel is ample for projective spaces.

Ampleness of the kernel occurs only for Fano varieties of Picard rank 1 in certain dimensions.

Abstract

In the search of a projective analog of Kunz's theorem and a Frobenius-theoretic analog of Mori--Hartshorne's theorem, we investigate the positivity of the kernel of the Frobenius trace (equivalently, the negativity of the cokernel of the Frobenius endomorphism) on a smooth projective variety over an algebraically closed field of positive characteristic. For instance, such kernel is ample for projective spaces. Conversely, we show that for curves, surfaces, and threefolds the Frobenius trace kernel is ample only for Fano varieties of Picard rank $1$.