Quasiphantom categories on a family of surfaces isogenous to a higher product

TL;DR

This paper constructs exceptional collections of line bundles on certain surfaces isogenous to a higher product, leading to new examples of quasiphantom categories with specific geometric properties.

Contribution

It introduces the first known exceptional collections of line bundles of maximal length on these surfaces, producing new quasiphantom categories.

Findings

Constructed exceptional collections of line bundles of length 4

Derived new quasiphantom categories from these collections

Analyzed surfaces with specific group actions and geometric invariants

Abstract

We construct exceptional collections of line bundles of maximal length 4 on $S=(C \times D)/G$ which is a surface isogenous to a higher product with $p_g=q=0$ where $G=G(32,27)$ is a finite group of order 32 having number 27 in the list of Magma library. From these exceptional collections, we obtain new examples of quasiphantom categories as their orthogonal complements.