A Pardon Algebra for Zero-cycles

TL;DR

This paper introduces a new algebraic framework called the Pardon algebra for zero-cycles in higher-dimensional varieties, extending previous work on 1-cycles and offering fresh insights into enumerative geometry problems.

Contribution

It generalizes the Pardon homology algebra to zero-cycles in d-folds, providing a novel perspective on point-counting and related enumerative conjectures.

Findings

Provides a new algebraic structure for zero-cycles in d-folds.

Offers a fresh approach to the degree zero MNOP conjecture.

Extends Pardon’s work from 1-cycles to 0-cycles.

Abstract

Recently, John Pardon proved the MNOP conjecture (on the GW-DT correspondence for CY3s) by introducing a new mathematical gadget, which we call the Pardon homology algebra of 1-cycles in 3-folds. We work out an analogous construction for 0-cycles in d-folds. This gives a new point of view on enumerative problems involving point-counting, such as, for example, the degree zero MNOP conjecture on the Hilbert scheme of points in projective 3-folds.