The tangent space of the punctual Hilbert scheme

TL;DR

This paper investigates the Zariski tangent space of the punctual Hilbert scheme for subschemes supported at a point on a smooth surface, providing bounds and explicit formulas especially for monomial cases.

Contribution

It offers a lower bound on the tangent space dimension and an explicit combinatorial formula for monomial subschemes, advancing understanding of the scheme's local structure.

Findings

Lower bound on tangent space dimension established

Bound is sharp for monomial subschemes in the affine plane

Explicit combinatorial formula derived for monomial subscheme tangent spaces

Abstract

The purpose of this paper is to study the Zariski tangent space of the punctual Hilbert scheme parametrizing subschemes of a smooth surface which are supported at a single point. We give a lower bound on the dimension of the tangent space in general and show the bound is sharp for subschemes of the affine plane cut out by monomials. Furthermore for monomial subschemes we give an explicit combinatorial formula for the dimension of the tangent space.