Non-Isomorphic Smooth Compactifications of the Moduli Space of Cubic Surfaces

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 MKand 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 MKand 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.