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.