## On the moduli space of pairs consisting of a cubic threefold and a hyperplane.(English)Zbl 1411.14042

A Hodge structure is said to be of $$K3$$ type if it is effective, of weight $$2$$, and has $$h^{2,0}=1$$. Such structures arise occasionally as the middle cohomology of a projective variety and are always interesting. It is natural to look at weighted hypersurfaces of this kind. There are famously 95 families of $$K3$$ surfaces that arise in this way [M. Reid, in: Journees de geometrie algebrique, Angers/France 1979, 273–310 (1980; Zbl 0451.14014)]. For fourfolds, one needs to consider degree $$d=\frac{1}{2} (w_1+\cdots+w_5)$$. The first thing that the authors do here (or the last, since it is relegated to an appendix) is to classify the families of such hypersurfaces that contain a Fermat member $$\sum x_i^{n_i}=0$$, which amounts to writing 2 as a sum of integer reciprocals. There are seventeen such families, but for fourteen of them the last two weights are the same, which essentially means that they were on Reid’s list (the other 81 do not contain Fermat hypersurfaces). The first of the three new cases is the family of cubic fourfolds in $$\mathbb{P}^5$$. The others are of degree $$6$$ with weights $$1,2,2,2,2,3$$, which is the case studied here, and degree $$12$$ with weights $$3, 3, 4, 4, 4, 6$$.
Such a degree $$6$$ hypersurface $$Z$$ is a double cover of $$\mathbb{P}^4$$ branched along a smooth cubic threefold $$X$$ and a hyperplane $$H$$, and this is the reason for studying that moduli problem. The first step, though, is to go back to $$Z$$ and find a birational modification $$Y$$ of $$Z$$ that is a smooth cubic fourfold with an Eckardt point $$p$$: by this we mean that $$Y$$ contains a hyperplane section that is a cone on a cubic surface with vertex $$p$$.
The cubic surface is the blow-up of six points in $$\mathbb{P}^2$$ and the exceptional curves, together with the pullback of a general line, give seven classes in $$H^4(Y)\cap H^{2,2}(Y)$$. The lattice they generate in the primitive cohomology is show to be isometric to $$E_6(2)$$. There is a period map associated with its orthogonal complement $$T$$, much as for lattice-polarised $$K3$$ surfaces, and the authors establish all the expected properties, including a Torelli theorem. To understand $$\operatorname{O}(T)$$ they identify $$T$$ (which is isometric with $$2U\oplus 3D_4$$) with the Milnor lattice of the singularity $$\Sigma O_{16}$$, the suspension of the cone on a cubic surface, and use the results of W. Ebeling [Invent. Math. 77, 85–99 (1984; Zbl 0527.14031)].
Inside the period domain (or the moduli space, which is a quotient of it by a suitable orthogonal group) the authors identify two important Heegner divisors, both arising from reflections. The vectors giving rise to them have square $$2$$ and divisor $$1$$, respectively square $$4$$ and divisor $$2$$ (the latter has 36 components). They correspond to nodal cubics and to the case where $$H$$ is tangent to $$X$$. By embedding $$T$$ in $$II_{2,26}$$ and using quasi-pullback of the Borcherds form they get a relation in the Picard group of the moduli space. They extend the period map and show, following a route described by Looijenga, that the Baily-Borel compactification agrees with the GIT moduli space for polarisation of slope $$\frac{1}{3}$$.

### MSC:

 14J15 Moduli, classification: analytic theory; relations with modular forms 14D07 Variation of Hodge structures (algebro-geometric aspects) 14D22 Fine and coarse moduli spaces 14J10 Families, moduli, classification: algebraic theory 14J17 Singularities of surfaces or higher-dimensional varieties 14J35 $$4$$-folds

### Keywords:

cubic fourfolds; Eckardt points; moduli spaces

### Citations:

Zbl 0451.14014; Zbl 0527.14031

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### References:

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