On the first steps of the minimal model program for the moduli space of stable pointed curves

TL;DR

This paper investigates the initial steps of the minimal model program for the moduli space of stable pointed curves, providing modular interpretations and geometric insights, and recovering known models like Hassett-Keel.

Contribution

It offers a modular interpretation of the first steps in the minimal model program for these moduli spaces and constructs new birational morphisms to alternative compactifications.

Findings

First steps admit modular interpretation

Recovered Hassett-Keel log canonical models

Produced new birational morphisms

Abstract

The aim of this paper is to study all the natural first steps of the minimal model program for the moduli space of stable pointed curves. We prove that they admit a modular interpretation and we study their geometric properties. As a particular case, we recover the first few Hassett-Keel log canonical models. As a by-product, we produce many birational morphisms from the moduli space of stable pointed curves to alternative modular projective compactifications of the moduli space of pointed curves.