Classification of full exceptional collections on smooth toric Fano varieties with Picard rank two

TL;DR

This paper classifies all full exceptional collections on certain smooth toric Fano varieties with Picard rank two, confirming their fullness and providing a partial answer to existing conjectures in the field.

Contribution

It offers a complete classification of full exceptional collections on smooth toric Fano threefolds and fourfolds with Picard rank two, advancing understanding of their derived categories.

Findings

All maximal length exceptional collections are full

Provided explicit classifications for specific toric Fano varieties

Partially confirmed conjectures on exceptional collections

Abstract

In this paper, we give the complete classification of full exceptional collections on smooth toric Fano threefolds and fourfolds with Picard rank two. To be precise, we give a partial answer to the conjecture in \cite{Kuz} and \cite{LYY}: we provide all the exceptional collections of maximal length and prove that they are in fact full exceptional collections.