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On derived categories of arithmetic toric varieties. (English) Zbl 1458.14022

Summary: 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 V. E. Voskresenskij and A. A. Klyachko [Math. USSR, Izv. 24, 221–244 (1985; Zbl 0572.14029); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 2, 237–263 (1984)], 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.

MSC:

14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14G27 Other nonalgebraically closed ground fields in algebraic geometry
19E08 \(K\)-theory of schemes

Citations:

Zbl 0572.14029
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