Tangent bundle of punctual Hilbert scheme and distinguishing products of varieties

TL;DR

This paper analyzes the tangent bundle structure of punctual Hilbert schemes on surfaces and applies this to classify product varieties and determine isomorphisms between symmetric powers.

Contribution

It characterizes indecomposable components of tangent bundles and proves a conjecture on classifying products of punctual Hilbert schemes.

Findings

Identified indecomposable components of tangent bundles of punctual Hilbert schemes.

Proved a conjecture on classification of products of punctual Hilbert schemes.

Determined conditions for isomorphisms between products of symmetric powers.

Abstract

We describe the indecomposable components of the tangent bundle of the punctual Hilbert scheme of a smooth projective surface. As an application, we prove a recent conjecture about classification of products of punctual Hilbert schemes of smooth projective surfaces. We also determine when two products of symmetric powers of a smooth variety can be isomorphic.