The tilting property for $F_*^e\mathcal O_X$ on Fano surfaces and threefolds

TL;DR

This paper investigates whether Frobenius pushforwards of the structure sheaf are tilting bundles on certain Fano varieties, showing they are not tilting for specific del Pezzo surfaces and Fano threefolds in characteristic p.

Contribution

It provides new results demonstrating that $F_*^e\mathcal{O}_X$ is not tilting on certain Fano surfaces and threefolds, answering a natural question in positive characteristic geometry.

Findings

$F_*^e\mathcal{O}_X$ is not tilting for del Pezzo surfaces of degree ≤ 3.

$F_*^e\mathcal{O}_X$ is not tilting for Fano threefolds with volume of $K_X$ less than 24.

Non-vanishing of certain Ext groups proves the non-tilting property.

Abstract

Let $X$ be a smooth variety over a field of characteristic $p$. It is a natural question whether the Frobenius pushforwards $F_*^e\mathcal O_X$ of the structure sheaf are tilting bundles. We show if $X$ is a smooth del Pezzo surface of degree $\leq 3$ or a Fano threefold with $\mathrm{vol}(K_X)<24$ over a field of characteristic $p$, then $\mathrm{Ext}^i(F_*^e\mathcal O_X,F^e_*\mathcal O_X)\neq 0$ and thus $F_*^e\mathcal O_X$ is not tilting.