On the classification of polarized surfaces with weakly Ulrich or initialized aCM tangent bundles

Abstract

Counterexamples to the Eisenbud-Schreyer conjecture on the existence of Ulrich bundles have appeared recently (see Anghel [A1] and the third author [Lo]). As weakly Ulrich bundles are a useful generalization of Ulrich ones, in this paper, we classify smooth surfaces $S\subseteq \mathbb P^N$ with weakly Ulrich tangent bundle. They are Del-Pezzo surfaces of degree at least 5 and we classify the possible embeddings. An interesting feature is the appearance of two infinite series of polarizations, which is quite different from the Ulrich case [BMPT, LR]. We also achieve the classification of surfaces with initialized aCM tangent bundles and, as a consequence, of surfaces with almost Ulrich tangent bundles.