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**11 - 14**of**14**### The universal rank-(n − 1) bundle on G(1,n) restricted to subvarieties

, 1997

"... En record de Ferran Serrano, extraordinari com a persona i com a matemàtic We classify those smooth (n − 1)-folds in G(1, P n) for which the restriction of the rank-(n − 1) universal bundle has more than n + 1 independent sections. As an aplication, we classify also those (n − 1)-folds for which tha ..."

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En record de Ferran Serrano, extraordinari com a persona i com a matemàtic We classify those smooth (n − 1)-folds in G(1, P n) for which the restriction of the rank-(n − 1) universal bundle has more than n + 1 independent sections. As an aplication, we classify also those (n − 1)-folds for which that bundle splits. It is a classical problem to study which projective varieties of small codimension are not linearly normal (i.e. isomorphically projected from higher projective spaces). The first observation is that any n-dimensional variety can be projected to P 2n+1, but is expected to produce singular points when projected to P 2n. Hence, n-dimensional varieties of codimension at most n are expected to be linearly normal. For n = 2, Severi proved (see his classical paper [6]) that the Veronese surface is the only smooth surface in P 5 that can be isomorphically projected to P 4. In general, projectability (or linear normality) is characterized by the dimension of the secant varieties. A recent thorough study of secant varieties can be found in [7], where a lot of projectability

### Tight immersions and local differential geometry

, 1997

"... Historically, the study of tight immersions of manifolds had its origins in the study of immersions with minimal total absolute curvature. The most significant result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which completely characterizes minima ..."

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Historically, the study of tight immersions of manifolds had its origins in the study of immersions with minimal total absolute curvature. The most significant result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which completely characterizes minimal total absolute curvature immersions (and tight immersions) of spheres into a Euclidean space. An essential ingredient in this characterization was a reformulation of the problem in terms of the Morse theory of linear height functions and the topological characteristics of the manifold being immersed. In this sense, the theorem of Chern and Lashof characterizes tight immersions of the topologically simplest compact manifold. It is very natural to try and characterize tight immersions of other manifolds with topological restrictions. The most natural candidates are highly connected manifolds; i.e., 2k-dimensional manifolds that are (k − 1)-connected, but not k-connected. In the years since the theorem of Chern and Lashof was proved, tight immersions of these manifolds have been studied extensively by Kuiper and others. The purpose of the present paper is to finish a program begun by Kuiper and complete the proof of the following theorem:

### On the dimension of secant varieties

, 2008

"... In this paper we generalize Zak’s theorems on tangencies and on linear normality as well as Zak’s definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety X under suitable regularity assumption on X, ..."

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In this paper we generalize Zak’s theorems on tangencies and on linear normality as well as Zak’s definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety X under suitable regularity assumption on X, and we classify varieties for which the bound is attained.

### 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 algebro-geometric 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 PRONEX-Algebra 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.