Lazy tournaments and multidegrees of a projective embedding of $\overline{M}_{0,n}$

TL;DR

This paper introduces a new geometric interpretation of multidegrees for a specific embedding of the moduli space of stable genus 0 curves, using a combinatorial algorithm called lazy tournaments, and connects it to parking functions.

Contribution

It provides a novel geometric perspective and a combinatorial algorithm for enumerating multidegrees, linking boundary points of the moduli space to lazy tournaments and parking functions.

Findings

Enumerates multidegrees via boundary points using lazy tournaments.

Proves total degree equals the odd double factorial.

Establishes a bijection with parking functions.

Abstract

We provide a new geometric interpretation of the multidegrees of the (iterated) Kapranov embedding $\Phi_n:\overline{M}_{0,n+3}\hookrightarrow \mathbb{P}^1\times \mathbb{P}^2\times \cdots \times \mathbb{P}^n$, where $\overline{M}_{0,n+3}$ is the moduli space of stable genus $0$ curves with $n+3$ marked points. We enumerate the multidegrees by disjoint sets of boundary points of $\overline{M}_{0,n+3}$ via a combinatorial algorithm on trivalent trees that we call a lazy tournament. These sets are compatible with the forgetting maps used to derive the recursion for the multidegrees proven in 2020 by Gillespie, Cavalieri, and Monin. The lazy tournament points are easily seen to total $(2n-1)!!=(2n-1)\cdot (2n-3) \cdots 5 \cdot 3 \cdot 1$, giving a natural proof of the fact that the total degree of $\Phi_n$ is the odd double factorial. This fact was first proven using an insertion algorithm on certain parking functions, and we additionally give a bijection to those parking functions.