Revisiting the moduli space of 8 points on $\mathbb{P}^1$

TL;DR

This paper investigates the complex relationships between different compactifications of the moduli space of 8 points on the projective line, revealing non-equivalence and non-liftability phenomena with implications for the minimal model program.

Contribution

It demonstrates that the Deligne-Mostow isomorphism does not lift to a morphism between specific compactifications and shows these spaces are not K-equivalent, advancing understanding of moduli space structures.

Findings

Deligne-Mostow isomorphism does not lift to a morphism.

The studied spaces are not K-equivalent despite similar cohomology.

Connections to minimal model program and recent developments.

Abstract

The moduli space of $8$ points on $\mathbb{P}^1$, a so-called ancestral Deligne-Mostow space, is, by work of Kond\={o}, also a moduli space of K3 surfaces. We prove that the Deligne-Mostow isomorphism does not lift to a morphism between the Kirwan blow-up of the GIT quotient and the unique toroidal compactification of the corresponding ball quotient. Moreover, we show that these spaces are not $K$-equivalent, even though they are natural blow-ups at the unique cusps and have the same cohomology. This is analogous to the work of Casalaina-Martin-Grushevsky-Hulek-Laza on the moduli space of cubic surfaces. The moduli spaces of ordinary stable maps, that is, the Fulton-MacPherson compactification of the configuration space of points on $\mathbb{P}^1$, play an important role in the proof. We further relate our computations to new developments in the minimal model program and recent work of Odaka. We briefly discuss other cases of moduli space of points on $\mathbb{P}^1$ where a similar behaviour can be observed, hinting at a more general, but not yet fully understood phenomenon.