论文标题
在$ \ mathbb {p}^1 $上重新访问Moduli空间8点
Revisiting the moduli space of 8 points on $\mathbb{P}^1$
论文作者
论文摘要
$ \ Mathbb {p}^1 $上的$ 8 $点的模量空间，即所谓的祖先deligne-mostow空间，借助Kondō的作品，也是K3表面的Moduli空间。我们证明，deligne-sostow同构不会提升到吉特（Kirwan）的git商的爆炸与相应球商的独特环形紧凑型之间的形态。此外，我们表明这些空间不是$ k $等效的，即使它们是独特的尖式爆炸，并且具有相同的共同体。这类似于Casalaina-Martin-Grushevsky-Hulek-Laza在立方表面的模量空间上的工作。普通稳定地图的模量空间，即，在$ \ mathbb {p}^1 $上对点的配置空间的富尔顿 - 麦克弗森紧凑型，在证明中起着重要作用。我们将计算进一步与最小模型计划和最近的Odaka工作中的新发展联系起来。我们简要讨论了$ \ mathbb {p}^1 $上的其他点空间的案例，其中可以观察到类似的行为，暗示了一个更一般但尚未完全理解的现象。
The moduli space of $8$ points on $\mathbb{P}^1$, 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 $\mathbb{P}^1$, 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.
