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

TL;DR

This paper compares different compactifications of the moduli space of cubic surfaces, showing they are not isomorphic or K-equivalent, but share the same cohomology, and develops techniques for analyzing their singularities and canonical classes.

Contribution

It demonstrates that the Kirwan blowup and toroidal compactification are not isomorphic or K-equivalent, despite having the same cohomology, and introduces new methods for studying their singularities.

Findings

Kirwan blowup and toroidal compactification are not isomorphic.

Both spaces are equivalent in the Grothendieck ring.

The paper develops techniques for singularities and canonical classes.

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 $K$-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, 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 $K$-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.