# Homological projective duality for quadrics

## TL;DR

This paper explores homological projective duality for smooth quadrics and their double covers, revealing it as a combination of classical duality and branched cover interchange operations.

## Contribution

It demonstrates that homological projective duality for quadrics involves two fundamental operations: classical duality and branched cover interchange, clarifying the duality structure.

## Key findings

- Homological projective duality for quadrics is characterized by two operations.
- Classical projective duality is one of the key operations involved.
- Double covers branched over quadrics are dual to the quadrics themselves.

## Abstract

We show that over an algebraically closed field of characteristic not equal to 2, homological projective duality for smooth quadric hypersurfaces and for double covers of projective spaces branched over smooth quadric hypersurfaces is a combination of two operations: one interchanges a quadric hypersurface with its classical projective dual and the other interchanges a quadric hypersurface with the double cover branched along it.