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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 thisembedded 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.
List of contents
Preface.-Introduction.- 1.Tangent cones, tangent spaces, tangent stars; secant, tangent and tangent star varieties to an algebraic variety.- 2.Basics of Deformation Theory of Rational Curves on Projective Varieties.- 3.Fulton-Hansen Connectedness Theorem, Scorza Lemma and their applications to projective geometry.- 4.Local quadratic entry locus manifolds and conic connected manifolds.- 5.Hartshorne Conjectures and Severi varieties.- 6.Varieties n-covered by curves of a fixed degree and the XJC.- 7. Hypersurfaces with vanishing hessian.-Bibliography
Summary
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 thisembedded 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.
Additional text
“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)
Report
"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)
