Derived categories of noncommutative quadrics and Hilbert squares

TL;DR

This paper demonstrates that the derived category of a noncommutative deformation of a quadric surface embeds into that of a commutative deformation of its Hilbert scheme, supporting Orlov's conjecture.

Contribution

It provides the first example of such an embedding for noncommutative quadrics and formulates an infinitesimal version of Orlov's conjecture.

Findings

Derived category embedding supports Orlov's conjecture

Provides evidence for the conjecture in the case of smooth projective surfaces

Formulates an infinitesimal version of the conjecture

Abstract

A noncommutative 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.