aCM vector bundles on projective surfaces of nonnegative Kodaira dimension

TL;DR

This paper constructs large families of indecomposable arithmetically Cohen-Macaulay vector bundles on various polarized surfaces, demonstrating their complexity and wild representation type, and introduces a framework for studying their classification.

Contribution

It provides new methods for constructing indecomposable aCM vector bundles of arbitrary rank on polarized surfaces and explores their complexity and classification.

Findings

Existence of families of indecomposable aCM vector bundles of any rank

Many such bundles are of wild representation type

Construction of bundles that are aCM for all ample line bundles

Abstract

In this paper we contribute to the construction of families of arithmetically Cohen-Macaulay (aCM) indecomposable vector bundles on a wide range of polarized surfaces $(X,\Oo_X(1))$ for $\Oo_X(1)$ an ample line bundle. In many cases, we show that for every positive integer $r$ there exists a family of indecomposable aCM vector bundles of rank $r$, depending roughly on $r$ parameters, and in particular they are of \emph{wild representation type}. We also introduce a general setting to study the complexity of a polarized variety $(X,\Oo_X(1))$ with respect to its category of aCM vector bundles. In many cases we construct indecomposable vector bundles on $X$ which are aCM for all ample line bundles on $X$.