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

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 (B 4 /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 , while from the ball quotient point of view it is natural to consider the toroidal compactification B 4 /Γ → (B 4 /Γ) * . The spaces M K and B 4 /Γ 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 M K and B 4 /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.

Revisiting the moduli space of 8 points on $\mathbb{P}^1$

“…Secondly, we show that the corresponding divisors intersect generically transversally in the toroidal compactification of the 5-dimensional ball quotient. Here is a major difference to [CMGHL22]. This is because we cannot use Naruki's compactification.…”

“…It was shown by Casalaina-Martin-Grushevsky-Hulek-Laza [CMGHL22] that the Kirwan blow-up and the toroidal compactification of the moduli space of (non-marked) smooth cubic surfaces are not isomorphic. In this paper, we prove analogous results for the moduli space of unordered 8 points on P 1 , denoted by M GIT .…”

“…In this paper, we prove analogous results for the moduli space of unordered 8 points on P 1 , denoted by M GIT . The proof we give here is inspired by that of [CMGHL22], but requires further ideas. As we shall discuss in Section 6, the behaviour observed here is shared by other ball quotients as well, thus pointing towards a much more general, and yet not fully understood, phenomenon.…”

“…One obstruction to this could be that the varieties are topologically different. Indeed, the topology of these varieties is of independent interest (and indeed this was the starting point of [CMGHL19] and [CMGHL22] in the case of cubic threefolds and cubic surfaces). We compute the cohomology of these varieties, according to the Kirwan method [Ki84, Ki85,Ki89] and Casalaina-Martin-Grushevsky-Hulek-Laza [CMGHL19].…”

“…The strategy of the proof of Theorem 1.1 is as follows. As in [CMGHL22] the argument is divided into two steps. We first prove that the discriminant divisor and the boundary divisor intersect non-transversally in the Kirwan blow-up.…”