Hyperplane sections and derived categories

Abstract

Abstract. We give a generalization of the theorem of Bondal and Orlov about the derived categories of coherent sheaves on intersections of quadrics revealing its relation to projective duality. As an application we describe the derived categories of coherent sheaves on Fano 3-folds of index 1 and degrees 12, 16 and 18.

Lefschetz decompositions and categorical resolutions of singularities

“…Let f : X → S be a morphism of algebraic varieties. A triangulated subcategory A ⊂ D b (X) is called S-linear [13] if it is stable with respect to tensoring by pullbacks of perfect complexes on S, i.e. A ⊗ f * (D perf (S)) ⊂ A.…”

“…A vector bundle E on X is said to be exceptional over S if it has finite Tor-dimension over S (see [13]) and f * (E * ⊗ E) ∼ = O S . Note that if on X there exists a bundle of finite Tor-dimension over S then X itself has finite Tor-dimension over S. on P N .…”

Geometry of the genus 9 Fano 4-folds

“…They described the link between some moduli spaces of vector bundles in terms of linear sections of the dual variety of G w , with main application to the genus 9 Fano threefold case. Note also that many of the results of this section are obtained in a universal way in derived categories by A. Kuznetsov in [11], but we detail this short description to use it in the next sections.…”

Formal deformations and their categorical general fibre

“…More precisely, one could apply Kuznetsov's discussion to the cartesian triangle given by X n / / X over Spec.R n / / / Spf.R/. Corollary 2.23 in [12] shows that from the flatness of W X / / Spf.R/ one cannot only deduce the standard flat base change, but also the above assertion (see also [7,Chapter 3,Remark 3.33]). For flat base change in our more general context see [17].…”