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Inductive characterizations of hyperquadrics
 Math. Ann
, 2008
"... Abstract. We give two characterizations of hyperquadrics: one as nondegenerate smooth projective varieties swept out by large dimensional quadric subvarieties passing through a point; the other as LQELmanifolds with large secant defects. 1. ..."
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Abstract. We give two characterizations of hyperquadrics: one as nondegenerate smooth projective varieties swept out by large dimensional quadric subvarieties passing through a point; the other as LQELmanifolds with large secant defects. 1.
On the classification of OADP varieties
 SCIENCE CHINA Mathematics, Special
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Universidade Federal de Pernambuco
"... The aim of these notes is to provide an introduction to some classical and recent results and techniques in projective algebraic geometry. We treat the geometrical properties of varieties embedded in projective space, their secant and tangent lines, the behaviour of tangent linear spaces, the algebr ..."
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The aim of these notes is to provide an introduction to some classical and recent results and techniques in projective algebraic geometry. We treat the geometrical properties of varieties embedded in projective space, their secant and tangent lines, the behaviour of tangent linear spaces, the algebrogeometric and topological obstructions to their embedding into smaller projective spaces, the classification in the extremal cases. These are classical themes in algebraic geometry and the renewed interest at the beginning of the ’80 of the last century came from some conjectures posed by Hartshorne, [H2], from an important connectedness theorem of Fulton and Hansen, [FH], and from its new and deep applications to the geometry of algebraic varieties, as shown by Fulton, Hansen, Deligne, Lazarsfeld and Zak, [FH], [FL], [D2], [Z2]. We shall try to illustrate these themes and results during the course and with more details through these notes, also pointing out simple proofs of some important theorems and some new results via the theory of deformations of rational curves on algebraic varieties (Mori’s Theory) and via the theory of degenerations, see [CMR], [CR], [Ru2], [IR1], [IR2]. A standard reference on some topics treated here is [Z2], which influenced the presentation of some parts of the book, altough the proofs and the general philosophy of important classification results differ substantially from Zak’s original ones. Ringraziamenti Innanzitutto sono molto riconoscente agli organizzatori della Scuola/Workshop, Fernando Cukierman e Ciro Ciliberto per avermi invitato ad offrire questo corso. Il CNPq (Conselho Nacional de Desenvolmimento Cientifico e Tecnologico do Brasil) e il PRONEXAlgebra Comutativa e Geometria Algebrica hanno finanziato fin troppo generosamente negli ultimi otto anni vari miei progetti di ricerca su questi argomenti, sia come borsista, sia con fondi diretti e grant di vario tipo. Esprimo qui la mia gratitudine per la fiducia concessa, spero almeno parzialmente ricambiata.