Existence of Rational Curves on Algebraic Varieties, Minimal Rational Tangents, and Applications

Kebekus, Stefan; Conde, Luis Solá

doi:10.1007/3-540-35480-8_10

Cited by 20 publications

(15 citation statements)

References 68 publications

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“…The next example shows how boundary-lc pairs appear as limits of dlt pairs. These limits play an important role in Keel-McKernan's proof of the Miyanishi conjecture for surfaces, [KM99, § 6], and in the last two authors' recent attempts to generalize Shafarevich hyperbolicity to families over higher-dimensional base manifolds, [KK07,KK08b], see also [KS06].…”

Section: Finitely Dominated and Boundary-lc Pairsmentioning

confidence: 99%

Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties

2009

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Given a normal variety Z, a p-form σ defined on the smooth locus of Z and a resolution of singularities π : Z → Z, we study the problem of extending the pull-back π * (σ) over the π-exceptional set E ⊂ Z. For log canonical pairs and for certain values of p, we show that an extension always exists, possibly with logarithmic poles along E. As a corollary, it is shown that sheaves of reflexive differentials enjoy good pull-back properties. A natural generalization of the well-known Bogomolov-Sommese vanishing theorem to log canonical threefold pairs follows.

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Section: Finitely Dominated and Boundary-lc Pairsmentioning

confidence: 99%

Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties

2009

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“…Theorem 4.6 ( [17,29]). Let X be a Fano n-fold of ρ = 1 with n ≥ 3 and K a minimal rational component of X.…”

Section: Spanning Dimensions Of Loci Of Chainsmentioning

confidence: 99%

Lengths of chains of minimal rational curves on Fano manifolds

Watanabe

2011

Journal of Algebra

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Communicated by Steven Dale Cutkosky MSC: 14E30 14J45 14N99 Keywords: Fano manifold Chain of rational curves Minimal rational component Variety of minimal rational tangentsIn this paper, we consider a natural question how many minimal rational curves are needed to join two general points on a Fano manifold X of Picard number 1. In particular, we study the minimal length of such chains in the cases where the dimension of X is at most 5, the coindex of X is at most 3 and X equips with a structure of a double cover. As an application, we give a better bound on the degree of Fano 5-folds of Picard number 1.

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“…[7,Cor. 8.6], [5] or [4,Chapt. 2.1], applies to show that Y is uniruled, contradicting the assumption that κ(Y ) = 0.…”

Section: Introductionmentioning

confidence: 99%

Families of varieties of general type over compact bases

Kebekus

Kovács

2008

Advances in Mathematics

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Let f : X → Y be a smooth family of canonically polarized complex varieties over a smooth base. Generalizing the classical Shafarevich hyperbolicity conjecture, Viehweg conjectured that Y is necessarily of log general type if the family has maximal variation. A somewhat stronger and more precise version of Viehweg's conjecture was shown by the authors in [S. Kebekus, S.J. Kovács, Families of canonically polarized varieties over surfaces, preprint math.AG/0511378; Invent. Math. (2008), doi: 10.1007/s00222-008-0128-8; S. Kebekus, S.J. Kovács, The structure of surfaces mapping to the moduli stack of canonically polarized varieties, arXiv: 0707.2054v1 [math.AG], 2007] in the case where Y is a quasi-projective surface.Assuming that the minimal model program holds, this very short paper proves the same result for projective base manifolds Y of arbitrary dimension.

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Existence of Rational Curves on Algebraic Varieties, Minimal Rational Tangents, and Applications

Abstract: Both authors were supported in full or in part by the priority program "Globale Methoden in der komplexen Geometrie" of the Deutsche Forschungsgemeinschaft, DFG.

Cited by 20 publications

References 68 publications

Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties

Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties

Lengths of chains of minimal rational curves on Fano manifolds

Families of varieties of general type over compact bases

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