On the geometry of some special projective varieties / / by Francesco Russo

• Record Nr.: UNINA9910254085803321
• Autore: Russo Francesco
• Pubbl/distr/stampa: Cham : , : Springer International Publishing : , : Imprint : Springer, , 2016
• ISBN: 3-319-26765-5
• Edizione: [1st ed. 2016.]
• Descrizione fisica: 1 online resource (257 p.)
• Collana: Lecture Notes of the Unione Matematica Italiana, , 1862-9113 ; ; 18
• Disciplina: 510
• Soggetti: Geometry, Algebraic; Commutative algebra; Commutative rings; Geometry; Algebraic Geometry; Commutative Rings and Algebras
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: 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.
• Sommario/riassunto: 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.