The representation type of determinantal varieties

TL;DR

This paper constructs large families of indecomposable sheaves on determinantal schemes, demonstrating their wild representation type and analyzing conditions for Ulrich sheaves to exhibit wildness, advancing understanding of their algebraic complexity.

Contribution

It introduces methods to construct indecomposable sheaves of high rank on determinantal schemes and characterizes when these schemes are of wild representation type, especially concerning Ulrich sheaves.

Findings

Standard determinantal schemes are of wild representation type under certain degree conditions.

Explicit construction of indecomposable sheaves as iterated extensions.

Identification of conditions for Ulrich sheaves to exhibit wild representation type.

Abstract

This work is entirely devoted to construct huge families of indecomposable arithmetically Cohen-Macaulay (resp. Ulrich) sheaves E of arbitrary high rank on a general standard (resp. linear) determinantal scheme X\subset \PP^n of codimension c \ge 1, n-c \ge 1 and defined by the maximal minors of a t \times (t+c-1) homogeneous matrix A. The sheaves E are constructed as iterated extensions of sheaves of lower rank. As applications: (1) we prove that any general standard determinantal scheme X\subset \PP^n is of wild representation type provided the degrees of the entries of the matrix 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 \PP^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.