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Abstract We determine the cones of effective and nef divisors on the toroidal compactification of the ball quotient model of the moduli space of complex cubic surfaces with a chosen line. From this we also compute the corresponding cones for the moduli space of unmarked cubic surfaces. 

Abstract The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient$${\mathcal {M}}^{\operatorname {GIT}}$$, as a Baily–Borel compactification of a ball quotient$${(\mathcal {B}_4/\Gamma )^*}$$, and as a compactifiedK-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$${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$$, whereas from the ball quotient point of view, it is natural to consider the toroidal compactification$${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$$. The spaces$${\mathcal {M}}^{\operatorname {K}}$$and$${\overline {\mathcal {B}_4/\Gamma }}$$have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in factnotthe case. Indeed, we show the more refined statement that$${\mathcal {M}}^{\operatorname {K}}$$and$${\overline {\mathcal {B}_4/\Gamma }}$$are equivalent in the Grothendieck ring, but notK-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. 

Abstract There are two types of involutions on a cubic threefold: the Eckardt type (which has been studied by the first named and the third named authors) and the non-Eckardt type. Here we study cubic threefolds with a non-Eckardt type involution, whose fixed locus consists of a line and a cubic curve. Specifically, we consider the period map sending a cubic threefold with a non-Eckardt type involution to the invariant part of the intermediate Jacobian. The main result is that the global Torelli Theorem holds for the period map. To prove the theorem, we project the cubic threefold from the pointwise fixed line and exhibit the invariant part of the intermediate Jacobian as a Prym variety of a (pseudo-)double cover of stable curves. The proof relies on a result of Ikeda and Naranjo–Ortega on the injectivity of the related Prym map. We also describe the invariant part of the intermediate Jacobian via the projection from a general invariant line and show that the two descriptions are related by the bigonal construction. 

We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily–Borel and toroidal compactifications of the ball quotient model, due to Allcock–Carlson–Toledo. Our starting point is Kirwan’s method. We then follow by investigating the behavior of the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli space of cubic surfaces is discussed in an appendix.