Categorical cones and quadratic homological projective duality

TL;DR

This paper introduces categorical cones and explores their role in homological projective duality, providing resolutions for singular quadrics and proving duality conjectures for Gushel-Mukai varieties.

Contribution

It defines categorical cones, links them to homological duality, and applies these concepts to resolve singular quadrics and prove duality conjectures.

Findings

Categorical cones provide resolutions for singular quadrics.

The quadratic homological projective duality theorem is established.

Duality conjecture for Gushel-Mukai varieties is proved.

Abstract

We introduce the notion of a categorical cone, which provides a categorification of the classical cone over a projective variety, and use our work on categorical joins to describe its behavior under homological projective duality. In particular, our construction provides well-behaved categorical resolutions of singular quadrics, which we use to obtain an explicit quadratic version of the main theorem of homological projective duality. As applications, we prove the duality conjecture for Gushel-Mukai varieties, and produce interesting examples of conifold transitions between noncommutative and honest Calabi-Yau threefolds.