Vol. 4, No. 2, 2019

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On derived categories of arithmetic toric varieties

Matthew Ballard, Alexander Duncan and Patrick McFaddin

Vol. 4 (2019), No. 2, 211–242
Abstract

We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections, making it possible to give concrete descriptions of their derived categories. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskiĭ and Klyachko, and toric varieties associated to Weyl fans of type A. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly nontoric) varieties over nonclosed fields.

Keywords
derived categories, exceptional collections, Galois descent, toric varieties
Mathematical Subject Classification 2010
Primary: 14F05, 14M25
Secondary: 14G27, 19E08
Milestones
Received: 13 April 2018
Revised: 3 January 2019
Accepted: 18 January 2019
Published: 16 June 2019
Authors
Matthew Ballard
Department of Mathematics
University of South Carolina
Columbia, SC
United States
Alexander Duncan
Department of Mathematics
University of South Carolina
Columbia, SC
United States
Patrick McFaddin
Department of Mathematics
University of South Carolina
Columbia, SC
United States