A support theorem for Hilbert schemes of planar curves, II

TL;DR

This paper explores the cohomological properties of Hilbert schemes of points on singular, reducible planar curves, revealing how their cohomologies relate to those of compactified Jacobians through a perverse filtration.

Contribution

It establishes a support theorem linking the cohomologies of Hilbert schemes to compactified Jacobians for singular planar curves, extending previous results to more general cases.

Findings

Cohomologies of Hilbert schemes are encoded in Jacobians via perverse Leray filtration.

The study applies to singular, reducible, and locally planar curves.

Provides a framework for understanding the topology of Hilbert schemes in complex algebraic geometry.

Abstract

We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration.