Abelian surfaces with two plane cubic curve fibrations and Calabi-Yau threefolds

(1,d)-polarized abelian surfaces in P^(d-1) with two plane cubic curve fibrations lie in two elliptic P^2-scrolls. The union of these scrolls form a reducible Calabi-Yau 3-fold. In this paper we show that this occurs when d<10 and analyse the family of such surfaces and 3-folds in detail when d=6. In particular, the reducible Calabi-Yau 3-folds deform in that case to irreducible ones with non-normal singularities.Comment: 41 page

Variety of power sums and divisors in the moduli space of cubic fourfolds

We show that a cubic fourfold F that is apolar to a Veronese surface has the property that its variety of power sums VSP(F,10) is singular along a K3 surface of genus 20. We prove that these cubics form a divisor in the moduli space of cubic fourfolds and that this divisor is not a Noether-Lefschetz divisor. We use this result to prove that there is no nontrivial Hodge correspondence between a very general cubic and its VSP.Comment: 42 pages, expanded and revised version to appear in Documenta Mathematic

Conic bundles in projective fourspace

P. Ellia and G.Sacchiero have shown that if $S$ is a smooth surface in \Pn 4 which is ruled in conics, then $S$ has degree 4 or 5. In this paper we give a proof of this result combining the ideas of Ellia and Sacchiero as they are used in the paper of the second author on plane curve fibrations and the recent work of G. Fl\o ystad and the first author bounding the degree of smooth surfaces in \Pn 4 not of general type.Comment: 7 pages, Plain-Te

On the convex hull of a space curve

The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We express the degree of this surface in terms of the degree, genus and singularities of the curve. We present algorithms for computing their defining polynomials, and we exhibit a wide range of examples.Comment: 19 pages, 4 figures, minor change

The Convex Hull of a Variety

We present a characterization, in terms of projective biduality, for the hypersurfaces appearing in the boundary of the convex hull of a compact real algebraic variety.Comment: 12 pages, 2 figure