Exceptional sequences of 8 line bundles on (P^1)^3

TL;DR

This paper studies maximal exceptional sequences of line bundles on the threefold product of projective lines, showing that for the case r=3, these sequences are always full and generate the entire derived category.

Contribution

It proves that all maximal exceptional sequences of line bundles on (P^1)^3 are full, providing a complete understanding for this specific case in the discrete setting.

Findings

Maximal exceptional sequences of line bundles on (P^1)^3 are always full.

The study is conducted within a discrete framework based on the Picard group.

The results confirm the generation of the derived category by these sequences.

Abstract

We investigate maximal exceptional sequences of line bundles on (P^1)^3, i.e. those consisting of 2^r elements. For r=3 we show that they are always full, meaning that they generate the derived category. Everything is done in the discrete setup: Exceptional sequences of line bundles appear as special finite subsets s of the Picard group Z^r of (P^1)^r, and the question of generation is understood like a process of contamination of the whole Z^r out of an infectious seed s.