Non-isomorphic smooth compactifications of the moduli space of cubic surfaces

verfasst von

Sebastian Casalaina-Martin, Samuel Grushevsky, Klaus Hulek, Radu Laza

Abstract

The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient M^GIT, as a Baily–Borel compactification of a ball quotient (B₄/Γ)^∗, and as a compactified K -moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup M^K → M^GIT, whereas from the ball quotient point of view, it is natural to consider the toroidal compactification B₄/Γ → (B₄/Γ)^∗. The spaces M^K and B₄/Γ have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in fact not the case. Indeed, we show the more refined statement that M^K and B₄/Γ are equivalent in the Grothendieck ring, but not K -equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.

Organisationseinheit(en)

Institut für Algebraische Geometrie

Externe Organisation(en)

University of Colorado Boulder
Stony Brook University (SBU)

Typ

Artikel

Journal

Nagoya Mathematical Journal

Band

254

Seiten

315-365

Anzahl der Seiten

51

ISSN

0027-7630

Publikationsdatum

06.2024

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Allgemeine Mathematik

Elektronische Version(en)

https://doi.org/10.48550/arXiv.2207.03533 (Zugang: Offen)
https://doi.org/10.1017/nmj.2023.27 (Zugang: Offen)