Details
| Original language | English |
|---|---|
| Pages (from-to) | 315-365 |
| Number of pages | 51 |
| Journal | Nagoya Mathematical Journal |
| Volume | 254 |
| Early online date | 3 Oct 2023 |
| Publication status | Published - Jun 2024 |
Abstract
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
In: Nagoya Mathematical Journal, Vol. 254, 06.2024, p. 315-365.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Non-isomorphic smooth compactifications of the moduli space of cubic surfaces
AU - Casalaina-Martin, Sebastian
AU - Grushevsky, Samuel
AU - Hulek, Klaus
AU - Laza, Radu
N1 - Publisher Copyright: © The Author(s), 2023. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal.
PY - 2024/6
Y1 - 2024/6
N2 - 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.
AB - 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.
KW - 14L24 14F25 14J26
UR - http://www.scopus.com/inward/record.url?scp=85173708583&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2207.03533
DO - 10.48550/arXiv.2207.03533
M3 - Article
VL - 254
SP - 315
EP - 365
JO - Nagoya Mathematical Journal
JF - Nagoya Mathematical Journal
SN - 0027-7630
ER -