Surfaces of minimal degree of tame representation type and mutations of Cohen-Macaulay modules

TL;DR

This paper classifies certain algebraic surfaces and bundles, showing examples of tame Cohen-Macaulay types and describing rigid Ulrich bundles via Fibonacci extensions and braid group actions.

Contribution

It provides explicit examples of tame CM type surfaces and classifies rigid ACM bundles on specific rational normal scrolls using braid group symmetries.

Findings

Parameter spaces of indecomposable ACM bundles are either points or lines.

Rigid Ulrich bundles are characterized as Fibonacci extensions.

Complete classification of rigid ACM bundles on S(2,3) and S(3,3).

Abstract

We provide two examples of smooth projective surfaces of tame CM type, by showing that any parameter space of isomorphism classes of indecomposable ACM bundles with fixed rank and determinant on a rational quartic scroll in projective 5-space is either a single point or a projective line. For surfaces of minimal degree and wild CM type, we classify rigid Ulrich bundles as Fibonacci extensions. For the rational normal scrolls S(2,3) and S(3,3), a complete classification of rigid ACM bundles is given in terms of the action of the braid group in three strands.