Looking out for Frobenius summands on a blown-up surface of $\mathbb{P}^{2}$

Abstract

For an algebraic variety $X$ in characteristic $p>0$, the push-forward $F^{e}_{*}\mathcal{O}_{X}$ of the structure sheaf by an iterated Frobenius endomorphism $F^{e}$ is closely related to the geometry of $X$. We study the decomposition of $F^{e}_{*}\mathcal{O}_{X}$ into direct summands when $X$ is obtained by blowing up the projective plane $\mathbb{P}^{2}$ at four points in general position. We explicitly describe the decomposition of $F^{e}_{*}\mathcal{O}_{X}$ and show that there appear only finitely many direct summands up to isomorphism, when $e$ runs over all positive integers. We also prove that these summands generate the derived category $D^{b}(X)$. On the other hand, we show that there appear infinitely many distinct indecomposable summands of iterated Frobenius push-forwards on a ten-point blowup of $\mathbb{P}^{2}$.