Corrigendum: On the construction of Weakly Ulrich bundles

TL;DR

This paper corrects a previous theorem on weakly Ulrich bundles, generalizes key results, and proves new properties of algebraic surfaces and their embeddings, ensuring the validity of earlier findings and extending their scope.

Contribution

It provides a corrected and more general theorem on weakly Ulrich bundles, including new results on embeddings of algebraic surfaces and their associated bundles.

Findings

Corrects a previous theorem on weakly Ulrich bundles.

Proves that certain algebraic surfaces are embedded via pluricanonical systems.

Establishes that specific pushforward bundles are almost Ulrich for these embeddings.

Abstract

This note corrects a mistake in Theorem~4.1 of https://doi.org/10.1016/j.aim.2021.107598 (arXiv:1901.03395). The error noted here does not affect any other results of loc. cit. To correct the error, here I prove a more general result (Theorem 3.1) and deduce Theorem~3.3 and also the correct version of the previously announced theorem in Theorem~3.4. Theorem~3.5 supplements Theorem~3.3. In it, I prove that if $k$ is an algebraically closed field of characteristic $p\geq 3$ and $X/k$ is any smooth, projective, minimal surface of general type and with $Pic(X)=\mathbb{Z}$, then for all integers $r\geq 5$, $X$ is embedded as a smooth surface by its pluricanonical linear system $X\hookrightarrow |\omega_X^r|$, and $E=F_*(\omega_X^{r+1})(1)$ is an almost Ulrich bundle for the pluricanonical embedding $X\hookrightarrow|\omega_X^r|$ of $X$ and for the ample line bundle provided by this embedding. Corollary~3.6 generalizes Theorem 3.1 of https://doi.org/10.1016/j.aim.2021.107598 (arXiv:1901.03395). In particular this note corrects and extends the results of loc. cit.