Details
| Originalsprache | Englisch |
|---|---|
| Aufsatznummer | 110126 |
| Seitenumfang | 41 |
| Fachzeitschrift | Advances in Mathematics |
| Jahrgang | 463 |
| Frühes Online-Datum | 28 Jan. 2025 |
| Publikationsstatus | Veröffentlicht - März 2025 |
Abstract
The moduli space of 8 points on P1, a so-called ancestral Deligne-Mostow space, is, by work of Kondō, 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 P1, play an important role in the proof. We further relate our computations to new developments in the minimal model program and the recent work of Odaka. We briefly discuss other cases of moduli space of points on P1 where a similar behaviour can be observed, hinting at a more general, but not yet fully understood phenomenon.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Allgemeine Mathematik
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in: Advances in Mathematics, Jahrgang 463, 110126, 03.2025.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Revisiting the moduli space of 8 points on P1
AU - Hulek, Klaus
AU - Maeda, Yota
N1 - Publisher Copyright: © 2025 The Author(s)
PY - 2025/3
Y1 - 2025/3
N2 - The moduli space of 8 points on P1, a so-called ancestral Deligne-Mostow space, is, by work of Kondō, 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 P1, play an important role in the proof. We further relate our computations to new developments in the minimal model program and the recent work of Odaka. We briefly discuss other cases of moduli space of points on P1 where a similar behaviour can be observed, hinting at a more general, but not yet fully understood phenomenon.
AB - The moduli space of 8 points on P1, a so-called ancestral Deligne-Mostow space, is, by work of Kondō, 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 P1, play an important role in the proof. We further relate our computations to new developments in the minimal model program and the recent work of Odaka. We briefly discuss other cases of moduli space of points on P1 where a similar behaviour can be observed, hinting at a more general, but not yet fully understood phenomenon.
KW - Ball quotients
KW - Deligne-Mostow varieties
KW - Moduli spaces
UR - http://www.scopus.com/inward/record.url?scp=85216109643&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2211.00052
DO - 10.48550/arXiv.2211.00052
M3 - Article
VL - 463
JO - Advances in Mathematics
JF - Advances in Mathematics
SN - 0001-8708
M1 - 110126
ER -