Towards a Combinatorial Description of the Intersection Product on $\mathbb{P}^{2[N]}$

TL;DR

This paper presents an algorithm for computing intersection classes in the Chow ring of the Hilbert scheme of points in the projective plane, advancing towards a combinatorial understanding of intersection products.

Contribution

It introduces an algorithm to compute intersection classes with a fixed divisor, aiding the development of a combinatorial framework for the intersection product.

Findings

Algorithm successfully computes intersection classes involving a fixed divisor.

Progress made towards a combinatorial description of the intersection product.

Enhances computational tools for the Chow ring of the Hilbert scheme.

Abstract

We prove that there is an algorithm to compute the class of the intersection of the divisor of schemes incident to a fixed line with any other class of a basis of the Chow ring $A^*(\mathbb{P}^{2[N]})$ due to Mallavibarrena and Sols. This is progress towards a combinatorial description of the intersection product on the Hilbert scheme of points in the projective plane.