Homological projective duality for linear systems with base locus

TL;DR

This paper develops a method to generate new homological projective duals by blowing up varieties at base loci of linear systems, extending the theory to noncommutative and singular cases, with practical examples.

Contribution

It introduces a procedure to produce new HP duals via blowups at base loci, including noncommutative and singular scenarios, broadening the scope of homological projective duality.

Findings

Blowup of $X$ in $X_L$ is HP dual to $Y_L$

Extension to cases with multiple base loci and rational singularities

Examples with noncommutative $Y$ leading to geometric HP duals

Abstract

We show how blowing up varieties in base loci of linear systems gives a procedure for creating new homological projective duals from old. Starting with a HP dual pair $X,Y$ and smooth orthogonal linear sections $X_L,Y_L$, we prove that the blowup of $X$ in $X_L$ is naturally HP dual to $Y_L$. The result does not need $Y$ to exist as a variety, i.e. it may be "noncommutative". We extend the result to the case where the base locus $X_L$ is a multiple of a smooth variety and the universal hyperplane has rational singularities; here the HP dual is a categorical resolution of singularities of $Y_L$. Finally we give examples where, starting with a noncommutative $Y$, the above process nevertheless gives geometric HP duals.