The representation type of determinantal varieties

Abstract

This work is entirely devoted to construct huge families of indecomposable arithmetically Cohen-Macaulay (resp. Ulrich) sheaves $\mathcal{E}$ of arbitrary high rank on a general standard (resp. linear) determinantal scheme $X \subset \mathbb{P}^n$ of codimension $c \geq 1, n-c \geq 1$ and defined by the maximal minors of a $t \times(t+c-1)$ homogeneous matrix $\mathcal{A}$. The sheaves $\mathcal{E}$ are constructed as iterated extensions of sheaves of lower rank. As applications: (1) we prove that any general standard determinantal scheme $X \subset \mathbb{P}^n$ is of wild representation type provided the degrees of the entries of the matrix $\mathcal{A}$ satisfy some weak numerical assumptions; and (2) we determine values of $t, n$ and $n-c$ for which a linear standard determinantal scheme $X \subset \mathbb{P}^n$ is of wild representation type with respect to the much more restrictive category of its indecomposable Ulrich sheaves, i.e. $X$ is of Ulrich wild representation type.