Maximal lengths of exceptional collections of line bundles

TL;DR

This paper constructs examples of toric Fano varieties with Picard number three that lack full exceptional collections of line bundles, disproving a conjecture and establishing bounds on the maximal length of such collections.

Contribution

It provides the first known examples of toric Fano varieties without full exceptional collections of line bundles and establishes a maximal constant for the length of exceptional collections.

Findings

Constructed infinitely many toric Fano varieties with no full exceptional collections.

Proved the maximal constant for the length of exceptional collections is 3/4.

Established existence of long exceptional collections on certain toric stacks.

Abstract

In this paper we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King's conjecture for toric Fano varieties. More generally, we prove that for any constant $c>\frac34$ there exist infinitely many toric Fano varieties $Y$ with Picard number three, such that the maximal length of exceptional collection of line bundles on $Y$ is strictly less than $c\rk K_0(Y).$ To obtain varieties without exceptional collections of line bundles, it suffices to put $c=1.$ On the other hand, we prove that for any toric nef-Fano DM stack $Y$ with Picard number three, there exists a strong exceptional collection of line bundles on $Y$ of length at least $\frac34 \rk K_0(Y).$ The constant $\frac34$ is thus maximal with this property.