On compactifications of $\mathcal{M}_{g,n}$ with colliding markings

TL;DR

This paper classifies all modular compactifications of the moduli space of n-pointed smooth algebraic curves by allowing markings to collide, introducing the concept of a collision complex to control these collisions.

Contribution

It introduces the collision complex as a combinatorial tool to classify and construct all modular compactifications of rme9 space with colliding markings, generalizing previous models.

Findings

Collision complexes control marking collisions in compactifications.

Any collision complex can be realized by a modular compactification.

The framework includes known compactifications and new constructions.

Abstract

In this paper, we study all ways of constructing modular compactifications of the moduli space $\mathcal{M}_{g,n}$ of $n$-pointed smooth algebraic curves of genus $g$ by allowing markings to collide. We find that for any such compactification, collisions of markings are controlled by a simplicial complex which we call the collision complex. Conversely, we identify modular compactifications of $\mathcal{M}_{g,n}$ with essentially arbitrary collision complexes, including complexes not associated to any space of weighted pointed stable curves. These moduli spaces classify the modular compactifications of $\mathcal{M}_{g,n}$ by nodal curves with smooth markings as well as the modular compactifications of $\mathcal{M}_{1,n}$ with Gorenstein curves and smooth markings. These compactifications generalize previous constructions given by Hassett, Smyth, and Bozlee--Kuo--Neff.