Recursive combinatorial aspects of compactified moduli spaces

TL;DR

This paper explores the combinatorial structure of compactified moduli spaces, highlighting the relationship between algebraic and polyhedral moduli spaces, especially for curves, covers, Jacobians, and Neron models.

Contribution

It provides an overview of the connection between algebraic and polyhedral moduli spaces, emphasizing the combinatorial properties of their boundaries.

Findings

Identifies the combinatorial boundary structures of moduli spaces

Connects skeleta of algebraic objects to their moduli spaces

Highlights applications to curves, covers, Jacobians, and Neron models

Abstract

A connection between moduli spaces of algebro-geometric objects and moduli spaces of polyhedral objects has been under investigation in recent years. Loosely speaking, the skeleton of an algebro-geometric moduli space is expressed as the modul space of the skeleta of the objects it parametrizes. The connection is based on the combinatorial properties of the boundary of the algebro-geometric moduli spaces. This paper overviews this topic with focus on moduli of curves, admissible covers, Jacobians and their Neron models.