Overview
This study investigates the Hilbert Scheme of points on $\mathbb{CP}^{2}$ in the context of holomorphic foliations. Specifically, it focuses on the relationship between singular subschemes of these foliations and elements within the Hilbert Scheme. The research examines affine varieties that parameterize a partition of $N=7$ and cover the Hilbert Scheme of seven points. These varieties possess a Hilbert function corresponding to that of degree $d=2$ holomorphic foliations. Additionally, the paper presents some results relevant to degree $3$ foliations.
Research Context
Holomorphic foliations of degree $d$ on $\mathbb{CP}^{2}$ with isolated singularities are characterized by their singular subschemes. These subschemes correspond to elements within the Hilbert Scheme of $N=d^{2}+d+1$ points. The research aims to analyze specific components of this Hilbert Scheme.
Approach
The approach involves analyzing affine varieties that parametrize a partition of $N=7$. These affine varieties are designed to cover the Hilbert Scheme of seven points. A key property of these affine varieties is that the Hilbert function of their elements aligns with that of a holomorphic foliation of degree $d=2$. The study also extends to present certain results pertaining to degree $3$ foliations.
Findings
- The research analyzes affine varieties that parameterize a partition of $N=7$.
- These affine varieties cover the Hilbert Scheme of seven points.
- The Hilbert function of elements within these analyzed affine varieties corresponds to that of holomorphic foliations of degree $d=2$.
- Results for degree $3$ foliations are also presented.