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

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.