Generating systems, generalized Thomsen collections and derived categories of toric varieties

TL;DR

This paper extends Bondal's claim that the derived category of smooth toric varieties is generated by certain line bundles, introducing a generalized collection with a divisor and proving generation results.

Contribution

It introduces a generalized Thomsen collection with a divisor, establishes a new generation theorem, and verifies Bondal's claim for specific toric varieties using novel methods.

Findings

Generalized Thomsen collection recovers original collection when divisor is zero

Proved generation of the derived category using the new collection

Verified Bondal's claim for some toric varieties with different arguments

Abstract

Bondal claims that for a smooth toric variety $X$, its bounded derived category of coherent sheaves $D_{c}^{b}(X)$ is generated by the Thomsen collection $T(X)$ of line bundles obtained as direct summands of the pushforward of $\mathcal{O}_{X}$ along a Frobenius map with sufficiently divisible degree. The claim is confirmed recently. In this article, we consider a generalized Thomsen collection of line bundles $T(X,D)$ with a $\mathbb{Q}$-divisor $D$ as an auxiliary input, which recovers Thomsen's oringinal collection by setting $D=0$. We introduce the notion of a generating system and prove a theorem on the generation of $\mathcal{O}_{X}$ using many line bundles arising from the generating system. As an application, we verify Bondal's claim for some toric varieties, using a different argument from existing works.