Initial Ideals, Low-degree Foliations and the Hilbert Scheme of Points on ${\mathbb{CP}}^2$: A First Approach

Abstract

The holomorphic foliations of degree $d$ on ${\mathbb{CP}}^2$ with isolated singularities are determined by their singular subschemes, which correspond to elements in the Hilbert Scheme of $N=d^2+d+1$ points. In this paper, we analyze the affine varieties parametrized by a partition of $N=7$ that cover the Hilbert Scheme of seven points, with the property that the Hilbert function of the elements in these affine varieties corresponds to that of holomorphic foliation of degree $d=2$. Moreover, we present some results for degree $3$.