Parametrizations of Ideals in K[x,y] and K[x,y,z]

TL;DR

This paper develops a parametrization of Artinian ideals in polynomial rings with fixed initial ideals, extending to zero-dimensional subschemes in the projective plane, and determines Betti strata and their codimensions.

Contribution

It introduces a new parametrization method for ideals with fixed initial ideals, extending from affine to projective cases, and computes Betti strata codimensions.

Findings

Parametrization of Artinian ideals with fixed initial ideals in K[x,y].

Extension of parametrization to zero-dimensional subschemes in the projective plane.

Verification of Iarrobino's formula for Betti strata codimension in specific cases.

Abstract

We parametrize the affine space of Artinian affine ideals of K[x,y] which have a given initial ideal with respect to the degree reverse lexicographic term order. The fact that the term order is degree compatible allows us to extend the parametrization to the projective case, namely zero-dimensional subschemes of the projective plane, with some extra assumption and to determine the Betti strata of these Groebner cells. This allows us to prove a formula due to A. Iarrobino for the codimension of the Betti strata of codimension two punctual schemes in the projective plane.