Fano varieties with large Seshadri constants in positive characteristic

Abstract

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∈X (for some fixed ϵ>0) have bounded anti-canonical degrees.