Infinitesimal deformations of double covers of smooth algebraic varieties

Abstract

The goal of this paper is to give a method to compute the space of infinitesimal deformations of a double cover of a smooth algebraic variety. The space of all infinitesimal deformations has a representation as a direct sum of two subspaces. One is isomorphic to the space of simultaneous deformations of the branch locus and the base of the double covering. The second summand is the subspace of deformations of the double covering which induce trivial deformations of the branch divisor. The main result of the paper is a description of the effect of imposing singularities in the branch locus.As a special case we study deformations of Calabi–Yau threefolds which are non‐singular models of double cover of the projective 3‐space branched along an octic surface. We show that in that case the number of deformations can be computed explicitly using computer algebra systems. This gives a method to compute the Hodge numbers of these Calabi–Yau manifolds. In this case the transverse deformations are resolutions of deformations of double covers of projective space but not double covers of a blow‐up of projective space. In the paper we gave many explicit examples. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The monodromy groups of Dolgachev's CY moduli spaces are Zariski dense

“…For the n = 3 case, see Lemma 2.1 in [17], whose proof is based on [6]. This proof goes through in general case verbatim.…”

Fiber Products of Elliptic Surfaces With Section and Associated Kummer Fibrations

“…The results of [7] give another method to compute the dimension h 1,2 (X) = h 1,2 (Ŷ ) of the space of deformations ofX. Let us use the method in our context.…”

“…By the methods of [7] it remains to compute the equisingular deformation of the surface D. As we have assumed, D is the sum of two quartic cones Q 1 and Q 2 . Proof.…”

“…The above reasoning as it stands works only for Kummer fibrations satisfying the case (a) from Corollary 2.12. However the method of [7] should also work in the remaining cases provided we prove that the surrounding weighted projective spaces admit rigid resolutions. This is not hard as we have a description of such spaces as a hypersurface and a complete intersection respectively.…”

“…We need only to allow the cones to pass through each other vertices. To this more general picture we can also use results of [7], the only difference is that we have fivefold points, which are resolved in a more complicated but also allowed way. Remark 2.23.…”