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

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Authors

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

Research Organisations

External Research Organisations

  • University of Colorado Boulder
  • Stony Brook University (SBU)
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Details

Original languageEnglish
Pages (from-to)315-365
Number of pages51
JournalNagoya Mathematical Journal
Volume254
Early online date3 Oct 2023
Publication statusPublished - Jun 2024

Abstract

The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient MGIT, 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 MK → MGIT, whereas from the ball quotient point of view, it is natural to consider the toroidal compactification B4/Γ → (B4/Γ). The spaces MK 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 MK 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.

Keywords

    14L24 14F25 14J26

ASJC Scopus subject areas

Cite this

Non-isomorphic smooth compactifications of the moduli space of cubic surfaces. / Casalaina-Martin, Sebastian; Grushevsky, Samuel; Hulek, Klaus et al.
In: Nagoya Mathematical Journal, Vol. 254, 06.2024, p. 315-365.

Research output: Contribution to journalArticleResearchpeer review

Casalaina-Martin S, Grushevsky S, Hulek K, Laza R. Non-isomorphic smooth compactifications of the moduli space of cubic surfaces. Nagoya Mathematical Journal. 2024 Jun;254:315-365. Epub 2023 Oct 3. doi: 10.48550/arXiv.2207.03533, 10.1017/nmj.2023.27
Casalaina-Martin, Sebastian ; Grushevsky, Samuel ; Hulek, Klaus et al. / Non-isomorphic smooth compactifications of the moduli space of cubic surfaces. In: Nagoya Mathematical Journal. 2024 ; Vol. 254. pp. 315-365.
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N1 - Publisher Copyright: © The Author(s), 2023. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal.

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