The structure of exceptional sequences on toric varieties of Picard rank two

TL;DR

This paper classifies all maximal-length exceptional sequences of invertible sheaves on smooth projective toric varieties with Picard rank two, revealing their structural properties and confirming their fullness in the derived category.

Contribution

It provides a complete classification of maximal exceptional sequences on Picard rank two toric varieties, highlighting their stability under reordering and their lattice constraints.

Findings

Maximal exceptional sequences are stable under lexicographical reordering.

They satisfy strong height constraints in the Picard lattice.

These sequences generate the entire derived category.

Abstract

For a smooth projective toric variety of Picard rank two we classify all exceptional sequences of invertible sheaves which have maximal length. In particular, we prove that unlike non-maximal sequences, they (a) remain exceptional under lexicographical reordering (b) satisfy strong height constraints in the Picard lattice (c) are full, that is, they generate the derived category of the variety.