84 research outputs found

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

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    (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

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    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

    On the convex hull of a space curve

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    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

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    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

    Surfaces of Degree 10 in the Projective Fourspace via Linear Systems and Linkage

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    The paper discusses the classification of surfaces of degree 10 and sectional genus 9 and 10. The surfaces of degree at most 9 are described through classical work dating from the last century up to recent years, while surfaces of degree 10 and other sectional genera are studied elsewhere. We use relations between multisecants, linear systems, syzygies and linkage to describe the geometry of each surface. We want in fact to stress the importance of multisecants and syzygies for the study of these surfaces. Adjunction, which provided efficient arguments for the classification of surfaces of smaller degrees, here appears to be less effective and will play almost no role in the proofs. We show that there are 8 different families of smooth surfaces of degree 10 and sectional genus 9 and 10. The families are determined by numerical data such as the sectional genus, the Euler characteristic, the number of 6-secants to the surface and the number of 5-secants to the surface which meet a general plane. For each type we describe the linear system giving the embedding in P^4, the resolution of the ideal, the geometry of the surface in terms of curves on the surface and hypersurfaces containing the surface, and the liaison class; in particular we find minimal elements in the even liaison class. Each type corresponds to an irreducible, unirational component of the Hilbert scheme, and the dimension is computed.Comment: 52 pages, plain Te