Twisted genus bounds for subvarieties of generic hypersurfaces

Clemens, C. Herbert; Ran, Ziv

doi:10.1353/ajm.2004.0003

Cited by 22 publications

(38 citation statements)

References 11 publications

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“…Then the latter is proved to be contained in the locus of lines by using the global generation of certain bundles. Finally, let us mention that similar results have also been obtained independently and at the same time in [CR04].…”

Section: A Little History Of the Above Resultssupporting

confidence: 81%

Hyperbolicity of Projective Hypersurfaces

Diverio

Rousseau

2016

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v bundles. This permits to obtain global jet differentials on hypersurfaces of sufficiently high degree [Div09] in every dimension.Then Siu's strategy of exhibiting vector fields is realized on the jet spaces of the universal hypersurfaces [Rou07], [Mer09]. Finally, the full strategy is used to obtain the algebraic degeneracy of entire curves in generic projective hypersurfaces of degree larger than 2 n 5 [DMR10].Last but not least, we would like to warmly thank Alcides Lins Neto, Jorge Vitório Pereira, Paulo Sad and all the people of the IMPA for having organized this course and our stay in Rio. These have been very stimulating, interesting and, why not, funny days.Abstract. In this first chapter we state and describe the basic definitions of complex hyperbolic geometry, basically following [Kob98] and [Dem97]. Then, we state and prove the classical Brody's lemma and Picard's theorem. We conclude by giving a brief account of elementary examples and describing the case of Riemann surfaces.

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Section: A Little History Of the Above Resultssupporting

confidence: 81%

Hyperbolicity of Projective Hypersurfaces

Diverio

Rousseau

2016

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“…Pacienza in ( [6]) obtained this result for n ≥ 6 ; (3) if h is in ( 3n 2 − 1, 2n − 3), the inequality (4.5) implies that X 0 does not contain rational curves other than lines. This is the known result of Clemens and Ran ( [3]), which is implied by their bound of twisted genus; (4) if h = 3n 2 − 1, the inequality (4.5) implies that X 0 contains no rational curves other than lines and quadratic curves. This is our new result.…”

Section: (B 2 ) This Shows That On the Open Setmentioning

confidence: 53%

“…However the detailed discussion of this will be given elsewhere. In this paper we'll only concentrate on the classes (2), (3). Most of the known work and conjectures in this part (Classes (2), (3)) were nicely summarized by Voisin in [9].…”

Section: Applicationsmentioning

confidence: 99%

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Rational curves on hypersurfaces

Wang¹

2017

Michigan Math. J.

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Abstract. In this paper, we prove three related results; (1) Extension of our result in [10] to all generic hypersurfaces. More precisely, the normal sheaf of a generic rational map c 0 to a generic hypersurface X 0 of P n , n ≥ 4 has a vanishing higher cohomology,As applications we give (2) A solution to a Voisin's conjecture [9] on a covering of a generic hypersurface by rational curves (3) A classification of rational curves on hypersurfaces of general type-a solution to another Voisin's conjecture [9].

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“…Notice that, by construction, there exists a positive number c such that for each k ≥ 1, we have (5) f k (0) = c > 0.…”

Section: Let X Be a Projective Manifold And D An Effective Divisor Onmentioning

confidence: 99%

On the logarithmic Kobayashi conjecture

Pacienza

Rousseau

2007

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We study the hyperbolicity of the log variety (P n , X), where X is a very general hypersurface of degree d ≥ 2n + 1 (which is the bound predicted by the Kobayashi conjecture). Using a positivity result for the sheaf of (twisted) logarithmic vector fields, which may be of independent interest, we show that any log-subvariety of (P n , X) is of log-general type, give a new proof of the algebraic hyperbolicity of (P n , X), and exclude the existence of maximal rank families of entire curves in the complement of the universal degree d hypersurface. Moreover, we prove that, as in the compact case, the algebraic hyperbolicity of a log-variety is a necessary condition for the metric one.

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Twisted genus bounds for subvarieties of generic hypersurfaces

Cited by 22 publications

References 11 publications

Hyperbolicity of Projective Hypersurfaces

Hyperbolicity of Projective Hypersurfaces

Rational curves on hypersurfaces

On the logarithmic Kobayashi conjecture

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