Fano varieties with large Seshadri constants in positive characteristic

Zhuang, Ziquan

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

Nous prouvons qu’une variété de Fano (avec des singularités arbitraires) de dimension $n$ en caractéristique positive est isomorphe à ${\mathbb{P}}^n$, 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 ${\mathbb{P}}^n$ 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.

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

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