Weak Approximation for Fano Complete Intersections in Positive Characteristic

[L’approximation faible pour les intersections complètes de type Fano en caractéristique positive]

Starr, Jason M.¹ ; Tian, Zhiyu² ; Zong, Runhong³

¹ Department of Mathematics Stony Brook University Stony Brook, NY 11794 (USA)
² Beijing International Center for Mathematical Research Peking University No.5 Yiheyuan Road Beijing, 100871 (China)
³ Department of Mathematics Nanjing University 204 Meng Minwei Building No. 22 Hankou Road Nanjing, Jiangsu, 210093 (China)

Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1503-1534.

Abstract (VO)

For a smooth curve $B$ over a field $k=\overline{k}$ with $\mathrm{char}\left(k\right)=p$, for every complete intersection $X_B$ in $B{\times }_{\mathrm{Spec}k}{\mathbb{P}}_k^n$ of type $(d_1,\cdots ,d_c)$, we prove weak approximation of adelic points of $X_B$ by $k\left(B\right)$-points at all places of (strong) potentially good reduction, if the Fano index is $\ge 2$ and if $p>max(d_1,\cdots ,d_c)$. This also applies to specializations of complex Fano manifolds with Picard rank $1$ and Fano index $1$ away from “bad primes”.

Résumé

Pour une courbe lisse $B$ sur un corps $k=\overline{k}$ de caractéristique positive $p$, pour chaque intersection complète $X_B$ dans $B{\times }_{\mathrm{Spec}k}{\mathbb{P}}_k^n$ de type $(d_1,\cdots ,d_c)$, nour prouvons l’approximation faible des points adeliques de $X_B$ par des $k\left(B\right)$-points sur toutes les places de forte réduction potentiellement bonne, si l’indice de Fano est au moins deux et si $p>max(d_1,\cdots ,d_c)$. Cela s’applique également aux spécialisations des variétés de Fano complexes de nombre de Picard de rang $1$ et d’indice de Fano $1$ en dehors de l’ensemble des mauvais nombres premiers.

Détail

Reçu le : 2019-01-04
Révisé le : 2020-08-04
Accepté le : 2020-11-19
Publié le : 2022-09-12

DOI : 10.5802/aif.3495

Classification : 14M22, 14D10, 14G12
Keywords: weak approximation, separably rationally connected, stability
Mots-clés : approximation faible, séparablement rationnellement connexe, stabilité

Licence : CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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