Fano varieties with large Seshadri constants in positive characteristic

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 $\epsilon(-K_X,x)>2+\epsilon$ at some smooth point $x\in X$ (for some fixed $\epsilon>0$) have bounded anti-canonical degrees.