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Quartic double solids with ordinary singularities. (English) Zbl 1209.14012
A double solid is a double cover of $$\mathbb{P}^3$$ branched along a surface of even degree. This paper deals with ordinary double solids, which are those where the ramification surface has at worst ordinary singularities. In local holomorphic coordinates $$u,v,w,t$$, the singularities of such a double solid are thus of three types: $$t^2 = uv$$ (type $$A$$), $$t^2 = uvw$$ (type $$T$$), and $$t^2 = u^2 -vw^2$$ (type $$D$$). The authors describe the mixed Hodge structures on the homology groups of the more general class of ADT threefolds $$X$$: they are pure with the exception of $$H^3(X)$$, which is an extension of a Hodge structure of weight $$3$$ by a Hodge structure of type $$(1,1)$$, with extension data determined by the Abel-Jacobi mapping to the intermediate Jacobian of the natural resolution of singularities of $$X$$. They then study in detail the cyclide double solid (ramified along an irreducible quartic surface whose singular locus is a smooth plane conic), and show that the Torelli mapping, sending $$X$$ to the polarized mixed Hodge structure on $$H_3(X)$$, is six-to-one.

##### MSC:
 14E20 Coverings in algebraic geometry 14J30 $$3$$-folds 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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##### References:
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