Representation type of surfaces in $\mathbb{P}^3$

TL;DR

This paper classifies the representation types of surfaces in projective 3-space, showing that most such surfaces have very complex, 'wild' categories of arithmetically Cohen-Macaulay sheaves, especially for degrees four and higher.

Contribution

It proves that surfaces of degree at least four in P^3 are of wild representation type under certain conditions, and constructs large families of indecomposable aCM vector bundles.

Findings

Surfaces of degree ≥4 with a regular point are of wild representation type.

Non-integral aCM schemes of dimension ≥2 are also very wild.

Constructs large families of non-isomorphic aCM sheaves.

Abstract

The goal of this article is to prove that every surface with a regular point in the three-dimensional projective space of degree at least four, is of wild representation type under the condition that either $X$ is integral or $\mathrm{Pic}(X) \cong \langle \Oo_X(1) \rangle$; we construct families of arbitrarily large dimension of indecomposable pairwise non-isomorphic aCM vector bundles. On the other hand, we prove that every non-integral aCM scheme of arbitrary dimension at least two, is also very wild in a sense that there exist arbitrarily large dimensional families of pairwise non-isomorphic aCM non-locally free sheaves of rank one.