Rational curves of minimal degree and characterizations of projective spaces

Araujo, Carolina

doi:10.1007/s00208-006-0775-2

Cited by 39 publications

(56 citation statements)

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“…We refer the reader to the article [ADK08] which reviews these matters. Notice that our results generalize Mori's (see [Mor79]), Wahl's (see [Wah83] and [Dru04]), AndreattaWiśniewski's (see [AW01] and [Ara06]) and Araujo-Druel-Kovács's (see [ADK08]) characterizations of projective spaces and hyperquadrics. K. Ross recently posted a somewhat related result (see [ROS10]).…”

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confidence: 81%

Characterizations of projective spaces and hyperquadrics

Druel

Paris

2013

Asian Journal of Mathematics

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Abstract. In this paper we prove that if the r-th tensor power of the tangent bundle of a smooth projective variety X contains the determinant of an ample vector bundle of rank at least r, then X is isomorphic either to a projective space or to a smooth quadric hypersurface. Our result generalizes Mori's, Wahl's, Andreatta-Wiśniewski's and Araujo-Druel-Kovács's characterizations of projective spaces and hyperquadrics.Key words. Algebraic geometry, rational varieties, projective spaces, quadric hypersurfaces.AMS subject classifications. 14M20.1. Introduction. Starting with Mori's seminal paper [Mor79] where the author characterized projective spaces as the only smooth projective varieties with ample tangent bundle, the study of the relation of the positivity of the tangent bundle with the geometry of the variety has become a very active subject in the classification theory of smooth projective variety.In [CS95], the authors prove that if X is a smooth complex projective variety of dimension 3 with ∧ 2 T X ample, then X is isomorphic to a projective space or an hyperquadric.The aim of this paper is to provide a new characterization of projective spaces and hyperquadrics in terms of positivity properties of the tangent bundle. We refer the reader to the article [ADK08] which reviews these matters. Notice that our results generalize Mori's (see

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confidence: 81%

Characterizations of projective spaces and hyperquadrics

Druel

Paris

2013

Asian Journal of Mathematics

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“…In higher dimensions one can define positivity of a complex manifold in different ways: for instance one can simply assume that a locally free subsheaf E ⊂ TX is ample. Mori's proof extends also in this apparently more general set up and it was proved that this is the case if and only if X = P n ( [AW01]; see also [Ara06]). …”

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confidence: 76%

A Conjecture of Mukai Relating Numerical Invariants of Fano Manifolds

Andreatta¹

2009

Milan J. Math.

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A complex manifold X of dimension n such that the anticanonical bundle −K X := det TX is ample is called a Fano manifold. Besides the dimension, other two integers play an essential role in the classification of these manifolds, namely the pseudoindex of X, i X , which is the minimal anticanonical degree of rational curves on X, and the Picard number ρ X , the dimension of N 1 (X), the vector space generated by irreducible complex curves modulo numerical equivalence . A (generalization of a) conjecture of Mukai says that ρ X (i X − 1) ≤ n. In this paper we present some partial steps towards the conjecture, we show how one can interpretate and possibly solve it with the use of families of rational curves on a uniruled variety, and more generally with the instruments of Mori theory. We consider also other related problems: the description of some Fano manifolds which are at the border of the Mukai relations and how the pseudoindex changes via (some) birational transformation.

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“…Remark The assumptions of the above theorem imply that every curve of the family

scriptM

is free (see, for instance [, Remark 2.3]), so that the proof of [, Proposition 2.7] tells us that every curve is standard , i.e. the pullback of the tangent bundle of X via the normalization of the curve is isomorphic to