Remarks on two problems by Hassett

TL;DR

This paper investigates log canonical models of moduli spaces of pointed rational curves, focusing on the case n=5 with asymmetric boundary divisors, and relates these models to Deligne-Mostow ball quotients.

Contribution

It generalizes previous results by explicitly describing all weighted moduli spaces as log canonical models of for n=5, and connects these to ball quotient structures.

Findings

All moduli spaces of weighted pointed rational curves are log canonical models of for suitable boundary coefficients.

The study relates these models to Deligne-Mostow ball quotients.

Provides an explicit example for the broader Hassett-Keel program.

Abstract

One of the ultimate goals of the Hassett-Keel program is the determination of the log canonical models of the moduli spaces of pointed rational curves $\overline{M}_{0,n}$. In this paper, we study log canonical models of $\overline{M}_{0,5}$ with \textit{asymmetric} boundary divisors. Our results generalize previous work by Alexeev-Swinarski, Fedorchuk-Smyth, Kiem-Moon and Simpson for the first non-trivial case, namely $n=5$. We prove that all moduli spaces of weighted pointed rational curves $\overline{M}_{0,A}$ arise as log canonical models of $\overline{M}_{0,5}$ for suitable choices of boundary coefficients, thereby also recovering a theorem of Fedorchuk and Moon. In addition, we relate these moduli spaces to Deligne-Mostow ball quotients. We further study log canonical models of the moduli spaces $\overline{M}_{0,n\cdot (1/k)}$ with symmetric weight, which differ from $\overline{M}_{0,n}$. The case $n=5$ can be viewed as an explicit guiding example in a very general program and the paper can thus also serve as an expository introduction.