Derived Categories of Noncommutative Quadrics and Hilbert Squares

Abstract

A non-commutative deformation of a quadric surface is usually described by a three-dimensional cubic Artin-Schelter regular algebra. In this paper we show that for such an algebra its bounded derived category embeds into the bounded derived category of a commutative deformation of the Hilbert scheme of two points on the quadric. This is the second example in support of a conjecture by Orlov. Based on this example we formulate an infinitesimal version of the conjecture, and provide some evidence in the case of smooth projective surfaces.

Hilbert squares: derived categories and deformations

“…The Hilbert scheme of length-2 subschemes (or Hilbert square) of X is denoted X [2] , and is a smooth projective variety of dimension 2d. We will often make use of the diagram…”

“…where the left square is the blowup square of X × X along its diagonal. The natural involution on X ×X lifts to the blowup and the quotient is canonically isomorphic to the Hilbert square of X , denoted by X [2] . The exceptional divisors in X [2] and Bl ∆ (X × X ) can be compared using the following diagram…”

“…We then proceed to study the deformation theory of X [2] for dim X ≥ 3, where we obtain the following result, showing that the surface case is fundamentally different from the higher-dimensional situation.…”

“…Let X be a smooth projective variety of dimension ≥ 2, such that O X is exceptional. The relative local-to-global Ext spectral sequence (5) E i, j 2 = H i (X [2] , Ext j p X [2] (I, I)) ⇒ Ext i +j X ×X [2] (I, I).…”

Exceptional collections in surface-like categories

“…Note that each of these categories can be realized as a semiorthogonal component of the derived category of a smooth projective variety [Or2], see also [BR2]. More generally, one can ask when a gluing [KL,Or1,Or3] of two triangulated categories is surface-like.…”