Fano varieties with large Seshadri constants in positive characteristic

Zhuang, Ziquan

doi:10.5802/aif.3477

Fano varieties with large Seshadri constants in positive characteristic
[Variétés de Fano avec grandes constantes de Seshadri en caractéristique positive]

Zhuang, Ziquan ¹

¹ Department of Mathematics, MIT Cambridge, MA, 02139, USA

Annales de l'Institut Fourier, Tome 72 (2022) no. 2, pp. 685-725.

Résumé
Abstract

Nous prouvons qu’une variété de Fano (avec des singularités arbitraires) de dimension $n$ en caractéristique positive est isomorphe à $ℙ^{n}$ , si la constante de Seshadri du diviseur anti-canonique en un point lisse est supérieure à $n$ . Nous classons les variétés de Fano dont les constantes de Seshadri des diviseurs anti-canoniques sont égales à $n$ . En caractéristique $p > 5$ et en dimension $3$ , nous montrons également que les degrés anti-canoniques d’une variété de Fano $X$ avec des constantes de Seshadri $ϵ (- K_{X}, x) > 2 + ϵ$ en un certain point lisse $x$ (pour un certain $ϵ > 0$ fixé) sont bornés.

We prove that a Fano variety (with arbitrary singularities) of dimension $n$ in positive characteristic is isomorphic to $ℙ^{n}$ if the Seshadri constant of the anti-canonical divisor at some smooth point is greater than $n$ and classify Fano varieties whose anti-canonical divisors have Seshadri constants $n$ . In characteristic $p > 5$ and dimension $3$ , we also show that Fano varieties $X$ with Seshadri constants $ϵ (- K_{X}, x) > 2 + ϵ$ at some smooth point $x \in X$ (for some fixed $ϵ > 0$ ) have bounded anti-canonical degrees.

Reçu le : 2019-05-19
Révisé le : 2020-05-15
Accepté le : 2020-08-03
Publié le : 2022-07-07

Zbl

DOI : 10.5802/aif.3477

Classification : 14J45, 14E99, 14C20, 14G17
Keywords: Fano variety, Seshadri constant, positive characteristic, boundedness, classification
Mot clés : Variétés de Fano, constante de Seshadri, caractéristique positive, bornitude, classification

Affiliations des auteurs :

Zhuang, Ziquan ¹

¹ Department of Mathematics, MIT Cambridge, MA, 02139, USA

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Fano varieties with large {Seshadri} constants in positive characteristic},
     journal = {Annales de l'Institut Fourier},
     pages = {685--725},
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Zhuang, Ziquan. Fano varieties with large Seshadri constants in positive characteristic. Annales de l'Institut Fourier, Tome 72 (2022) no. 2, pp. 685-725. doi : 10.5802/aif.3477. https://aif.centre-mersenne.org/articles/10.5802/aif.3477/

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