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On the Geometry of Some Special Projective Varieties

  • Francesco Russo

Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 18)

About this book

Introduction

Providing an introduction to both classical and modern techniques in projective algebraic geometry, this monograph treats the geometrical properties of varieties embedded in projective spaces, their secant and tangent lines, the behavior of tangent linear spaces, the algebro-geometric and topological obstructions to their embedding into smaller projective spaces, and the classification of extremal cases. It also provides a solution of Hartshorne’s Conjecture on Complete Intersections for the class of quadratic manifolds and new short proofs of previously known results, using the modern tools of Mori Theory and of rationally connected manifolds.

The new approach to some of the problems considered can be resumed in the principle that, instead of studying a special embedded manifold uniruled by lines, one passes to analyze the original geometrical property on the manifold of lines passing through a general point and contained in the manifold. Once this embedded manifold, usually of lower codimension, is classified, one tries to reconstruct the original manifold, following a principle appearing also in other areas of geometry such as projective differential geometry or complex geometry.

Keywords

14N05,14M07,14M10,14M22,14E30,14J70,14E05 Secant varieties Hartshorne Conjecture on Complete intersection varieties with (local) quadratic entry locus [ (L)QEL-manifold] special uniruled and/or rationally connected manifolds hypersurfaces with vanishing hessian

Authors and affiliations

  • Francesco Russo
    • 1
  1. 1.Mathematics and InforrmaticUniversità degli Studi di CataniaSant' Agata Li Battiati (CT)Italy

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-26765-4
  • Copyright Information Springer International Publishing Switzerland 2016
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-26764-7
  • Online ISBN 978-3-319-26765-4
  • Series Print ISSN 1862-9113
  • Series Online ISSN 1862-9121
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