Non-isomorphic smooth compactifications of the moduli space of cubic surfaces
Abstract
The moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient, as a Baily--Borel compactification of a ball quotient, and as a compactified -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, while from the ball quotient point of view it is natural to consider the toroidal compactification. Both these spaces have the same cohomology and 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 both spaces are equivalent in the Grothendieck ring, but not -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
Cite
@article{arxiv.2207.03533,
title = {Non-isomorphic smooth compactifications of the moduli space of cubic surfaces},
author = {Sebastian Casalaina-Martin and Samuel Grushevsky and Klaus Hulek and Radu Laza},
journal= {arXiv preprint arXiv:2207.03533},
year = {2024}
}
Comments
v2: Relation to work of Kudla-Rapoport and Zheng provided in the introduction; new Remark 4.7 gives a short independent proof of the result of Gallardo, Kerr, Scheffler on the isomorphism between Naruki's cross ratio variety and the toroidal compactification of the ball quotient. v3: final version. Nagoya Math J, to appear