Deformations of polarizations of powers of the maximal ideal

TL;DR

This paper investigates two types of polarizations of powers of the maximal ideal in polynomial rings, demonstrating their geometric properties and component dimensions within the Hilbert scheme.

Contribution

It characterizes the smoothness and component dimensions of standard and box polarizations of maximal ideal powers, revealing their distinct positions in the Hilbert scheme.

Findings

Polarizations correspond to smooth points in the Hilbert scheme.

Different polarizations lie on different components.

All maximal polarizations are smooth when d=2.

Abstract

In this paper, we study the two natural polarizations, namely the standard polarization and the box polarization, of the d-th power of the maximal ideal in a polynomial ring. We show that these polarizations correspond to smooth points in the Hilbert scheme, and we calculate the dimension of their component which shows that they lie on different components. When d=2, we show that all maximal polarizations are smooth points, and we give a simple method for calculating the dimension of their component.