The representation type of rational normal scrolls

Abstract

The goal of this paper is to demonstrate that all non-singular rational normal scrolls \(S(a_0,\ldots ,a_k)\subseteq \mathbb P ^N\), \(N =\sum _{i=0}^k(a_i)+k\), (unless \(\mathbb P ^{k+1}=S(0,\ldots ,0,1)\), the rational normal curve \(S(a)\) in \(\mathbb P ^a\), the quadric surface \(S(1,1)\) in \(\mathbb P ^3\) and the cubic scroll \(S(1,2)\) in \(\mathbb P ^4\)) support families of arbitrarily large rank and dimension of simple Ulrich (and hence indecomposable ACM) vector bundles. Therefore, they are all of wild representation type unless \(\mathbb P ^{k+1}\), \(S(a)\), \(S(1,1)\) and \(S(1,2)\) which are of finite representation type.