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Lecture Notes of the Unione Matematica Italiana

On the Geometry of Some Special Projective Varieties

Authors: Russo, Francesco

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  • Winner of the 2015 Book Prize of the Unione Matematica Italiana

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  • ISBN 978-3-319-26765-4
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Softcover $79.99
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About this book

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.

Reviews

“The book under review (awarded the 2015 Book Prize of the Unione Matematica Italiana) gives a survey of some classical and recent results on the geometry of projective varieties and its applications. … The book will be useful to anyone interested in classical algebraic geometry.” (Fyodor L. Zak, Mathematical Reviews, May, 2017)

“The book under review covers fundamental aspects of the theory of secant spaces to varieties, and contains a careful description of many of its recent applications to Algebraic Geometry. Under this respect, it provides a fundamental advanced introduction to recent results and developments of a topic which experienced a rapid evolution in the last years.” (Luca Chiantini, zbMATH 1337.14001, 2016)


Table of contents (7 chapters)

Table of contents (7 chapters)
  • Tangent Cones, Tangent Spaces, Tangent Stars: Secant, Tangent, Tangent Star and Dual Varieties of an Algebraic Variety

    Pages 1-38

    Russo, Francesco

  • The Hilbert Scheme of Lines Contained in a Variety and Passing Through a General Point

    Pages 39-73

    Russo, Francesco

  • The Fulton–Hansen Connectedness Theorem, Scorza’s Lemma and Their Applications to Projective Geometry

    Pages 75-93

    Russo, Francesco

  • Local Quadratic Entry Locus Manifolds and Conic Connected Manifolds

    Pages 95-114

    Russo, Francesco

  • Hartshorne Conjectures and Severi Varieties

    Pages 115-136

    Russo, Francesco

Buy this book

eBook $59.99
price for USA in USD (gross)
  • ISBN 978-3-319-26765-4
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $79.99
price for USA in USD
  • ISBN 978-3-319-26764-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
On the Geometry of Some Special Projective Varieties
Authors
Series Title
Lecture Notes of the Unione Matematica Italiana
Series Volume
18
Copyright
2016
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-26765-4
DOI
10.1007/978-3-319-26765-4
Softcover ISBN
978-3-319-26764-7
Series ISSN
1862-9113
Edition Number
1
Number of Pages
XXVI, 232
Topics