A graphical algorithm for the integration of monomials in the Chow ring of the moduli space of stable marked curves of genus zero

TL;DR

This paper introduces a graphical, linear-time algorithm for integrating monomials in the Chow ring of the moduli space of genus zero stable curves, based on tree representations and quadratic relations.

Contribution

It presents a novel graphical algorithm that efficiently computes integrals of monomials in the Chow ring using tree structures and quadratic relation criteria.

Findings

Algorithm is linear in the size of the tree

Provides a tree-based representation for monomials

Simplifies computation of Chow ring integrals

Abstract

The Chow group of zero cycles in the moduli space of stable pointed curves of genus zero is isomorphic to the integer additive group. Let $M$ be monomial in this Chow group. If no two factors of $M$ fulfill a particular quadratic relation, then the monomial can be represented equivalently by a specific tree; otherwise, $M$ is mapped to zero under the stated isomorphism. Starting from this tree representation, we introduce a graphical algorithm for computing the corresponding integer for $M$ under the aforementioned isomorphism. The algorithm is linear with respect to the size of the tree.