Representation type of surfaces in ℙ³

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 Pic(𝑋) ≅ ⟨𝒪_𝑋(1)⟩; 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.