Rationally connected varieties

Araujo, Carolina

doi:10.1090/conm/388/07254

Cited by 5 publications

(4 citation statements)

References 18 publications

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“…Before sketching a proof of Theorem 1.10, we recall a few facts and notations associated with the rationally connected quotient of a normal variety, introduced by Campana and Kollár-Miyaoka-Mori, [Cam92,KMM92b]. See [Deb01], [Kol96] or [Ara05] for an introduction. THEOREM 1.14 (Campana, and Kollár-Miyaoka-Mori).…”

Section: Rationally Connected Foliationsmentioning

confidence: 99%

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

Kebekus

Conde

Global Aspects of Complex Geometry

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Section: Rationally Connected Foliationsmentioning

confidence: 99%

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

Kebekus

Conde

Global Aspects of Complex Geometry

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“…• For any finite set of points F ⇢ X there exists a rational curve C F. We refer to [1] for these facts. Now let us turn to the proof of Theorem 1.9 from the Introduction.…”

Section: Rational Connectivity Of the Exceptional Components Of The Lmentioning

confidence: 99%

“…Both these chains contain the point f | 1 0 0 (a). Therefore C := C 1 [ C 2 is connected. At the same time by construction C i 3 a i .…”

Section: Rational Connectivity Of the Exceptional Components Of The Lmentioning

confidence: 99%

Weak normality of families of meromorphic mappings and bubbling in higher dimensions

Ivashkovich

Neji²

2015

Annali Scuola Normale Superiore - Classe Di Scienze

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Our primary goal in this paper is to understand whether the sets of normality of families of meromorphic mappings between general complex manifolds are pseudoconvex or not. It turns out that the answer crucially depends on the type of convergence one is interested in. We examine three natural types of convergence introduced by one of us earlier and prove pseudoconvexity of sets of normality for a large class of target manifolds for the so called weak and gamma convergencies. Furthermore we determine the structure of the exceptional components of the limit of a weakly/gamma but not strongly converging sequence, they turn to be rationally connected. This observation allows to determine effectively when a weakly/gamma converging sequence fails to converge strongly. An application to the Fatou sets of meromorphic self-maps of compact complex surfaces is given.

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“…For a nice survey on rationality and unirationality problems with a focus on their relation with the notion of rational connection, we refer to A. Verra’s paper [Ver08]. We recall that a projective variety is rationally connected if any two of its points can be joined by a rational curve and refer to C. Araujo’s paper [Ara05] for a survey on the subject.…”

Section: Introductionmentioning

confidence: 99%

Quartic and Quintic Hypersurfaces with Dense Rational Points

Massarenti

2023

Forum of Mathematics, Sigma

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Let $X_4\subset \mathbb {P}^{n+1}$ be a quartic hypersurface of dimension $n\geq 4$ over an infinite field k. We show that if either $X_4$ contains a linear subspace $\Lambda $ of dimension $h\geq \max \{2,\dim (\Lambda \cap \operatorname {\mathrm {Sing}}(X_4))+2\}$ or has double points along a linear subspace of dimension $h\geq 3$ , a smooth k-rational point and is otherwise general, then $X_4$ is unirational over k. This improves previous results by A. Predonzan and J. Harris, B. Mazur and R. Pandharipande for quartics. We also provide a density result for the k-rational points of quartic $3$ -folds with a double plane over a number field, and several unirationality results for quintic hypersurfaces over a $C_r$ field.

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Rationally connected varieties

Abstract: The aim of these notes is to provide an introduction to the theory of rationally connected varieties, as well as to discuss a recent result by T.

Cited by 5 publications

References 18 publications

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

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

Weak normality of families of meromorphic mappings and bubbling in higher dimensions

Quartic and Quintic Hypersurfaces with Dense Rational Points

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