Strong approximation over function fields

Chen, Qile; Zhu, Yi

doi:10.1090/jag/706

Cited by 5 publications

(7 citation statements)

References 27 publications

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“…The result for conics strengthens some key results in [dJS06] which is used to prove the (strongly) rational simple connectedness of low degree complete intersections, see [dJS06, Theorem 1.2] for details. The result for conics also leads a proof of strong approximation 5 for low degree affine complete intersections over function fields by Q. Chen and Y. Zhu, see [CZ15]. Therefore, we expect that there will be more applications of our main theorem in arithmetic (eg.…”

Section: Introductionmentioning

confidence: 76%

Spaces of Low-Degree Rational Curves on Fano Complete Intersections

Pan

2019

International Mathematics Research Notices

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For a smooth complete intersection X, we consider a general fiber F of the following evaluation map ev of the Kontsevich moduli space ev : M 0,m (X, m) → X m and the forgetful map F : F → M 0,m . We prove that a general fiber of the map F is a smooth complete intersection if X is of low degree. The result sheds some light on the arithmetic and geometry of Fano complete intersections.Date: November 12, 2018.1 This condition is to guarantee that the locus of Ft parametrizing the stable maps of maximal degeneration type is of dimension at least 2, cf. Corollary 5.7 2 The map Φ is introduced in the proof of [dJS06, Lemma 6.4]. 3 We denote by P(V )/P(W ) the projective space P(V /W ) for a flag (W ⊆ V ) of a vector space V and denote by Span(D) the smallest projective subspaces containing the algebraic set D(⊆ P n ). 4 The fiber Ft is possibly empty. However, in Section 5, we will see the conditions (1.2) andLemma 2.8 guarantee that Ft is a non-empty, cf. Proposition 5.6. 5 Strong approximation is considered to be more difficult to show than weak approximation.

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Section: Introductionmentioning

confidence: 76%

Spaces of Low-Degree Rational Curves on Fano Complete Intersections

Pan

2019

International Mathematics Research Notices

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“…Remark 6.6. Proposition 6.5 could be proved alternatively by a test curve computation (see [CZ15,Lemma 4.12]).…”

Section: An Integral Cycle Relationmentioning

confidence: 99%

Spaces of conics on low degree complete intersections

Pan

2017

Trans. Amer. Math. Soc.

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Abstract. Let X be a smooth complete intersection contained in P n C and of low degree. We consider conics contained in X and passing through two general points of X. We show that the moduli space of these conics is a smooth complete intersection in a projective space. The main ingredients of the proof are a criterion for characterizing when a smooth projective variety is a complete intersection in a projective space, the Grothendieck-Riemann-Roch theorem, and the geometry of spaces of conics.

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“…. , x n ) = c, avec c = 0 et f homogène de degré d non singulière, (approximation forte établie dans [1] pour d 2 ≤ n + 1) à l'approximation forte pour les ouverts de P n K d'équation f (x 0 , . .…”

Section: L'approximation Forte Pour Les Groupes Semi-simples Et Leurs...unclassified

“…Depuis une dizaine d'années, les techniques de déformation de courbes en géométrie algébrique complexe sont appliquées à l'étude de ces questions pour les variétés rationnellement connexes définies sur un corps K = C(Γ) de fonctions d'une variable sur les complexes, situation a priori plus simple que celle des variétés sur les corps de nombres. Dans l'article récent [1], Chen et Zhu établissent l'approximation forte sur K pour le complémentaire d'hypersurfaces lisses dans les intersections complètes lisses dans un espace projectif, sous certaines conditions sur les multidegrés.…”

Section: Introductionunclassified

Approximation forte pour les espaces homogènes de groupes semisimples sur le corps des fonctions d'une courbe algébrique complexe

Colliot-Thélène¹

2016

Preprint

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Sur un corps de fonctions d'une variable sur le corps des complexes, l'approximation forte hors d'un ensemble fini non vide de places vaut pour pour tout espace homogène d'un groupe semi-simple simplement connexe. En particulier l'approximation forte hors d'un ensemble fini non vide de places vaut pour les quadriques affines lisses de dimension au moins 2 sur un tel corps. L'approximation forte hors d'un ensemble fini de places ne vaut pas pour le groupe multiplicatif.

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Strong approximation over function fields

Abstract: By studying A 1 -curves on varieties, we propose a geometric approach to strong approximation problem over function fields of complex curves. We prove that strong approximation holds for smooth, low degree affine complete intersections with the boundary smooth at infinity.

Cited by 5 publications

References 27 publications

Spaces of Low-Degree Rational Curves on Fano Complete Intersections

Spaces of Low-Degree Rational Curves on Fano Complete Intersections

Spaces of conics on low degree complete intersections

Approximation forte pour les espaces homogènes de groupes semisimples sur le corps des fonctions d'une courbe algébrique complexe

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