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

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 , its bounded derived category of coherent sheaves is generated by the Thomsen collection of line bundles obtained as direct summands of the pushforward of 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 with a -divisor as an auxiliary input, which recovers Thomsen's oringinal collection by setting . We introduce the notion of a generating system and prove a theorem on the generation of 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic Geometry and Number Theory · Algebraic structures and combinatorial models · Polynomial and algebraic computation
