Revisiting the moduli space of 8 points on $\mathbb{P}^1$
Abstract
The moduli space of $8$ points on $\mathbb{P}^1$, a socalled ancestral DeligneMostow space, is, by work of Kondō, also a moduli space of K3 surfaces. We prove that the DeligneMostow isomorphism does not lift to a morphism between the Kirwan blowup 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 blowups at the unique cusps and have the same cohomology. This is analogous to the work of CasalainaMartinGrushevskyHulekLaza on the moduli space of cubic surfaces. The moduli spaces of ordinary stable maps, that is the FultonMacPherson 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2022
 DOI:
 10.48550/arXiv.2211.00052
 arXiv:
 arXiv:2211.00052
 Bibcode:
 2022arXiv221100052H
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory;
 14G35;
 11F03;
 14E05;
 14F25
 EPrint:
 v2: We describe the relationship with the minimal model program and prove that the Kirwan blowup in the unordered case is not a semitoric compactification in Subsection 4.3. We also refer to the relationship with the work of KudlaRapoport in the introduction