Corrigendum: On the construction of Weakly Ulrich bundles

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.