Demailly's Inequality for Finite Sets of Points in Positive Characteristic

Let $k$ be an algebraically closed field of characteristic $p>0$, $n\ge1$, and let $X\subset \mathbb P^n_k$ be a finite nonempty set of distinct points with the defining ideal $I=I(X)$. We give a proof of Demailly's inequality in positive characteristic. The argument is based on the Frobenius--Hasse derivative method used in this context by Hà and Sivakumar. The key additional observation is a strict-growth lemma for the initial degrees of symbolic powers of a finite set of affine points: $$ α(J^{(t)})\ge α(J^{(t-1)})+1\qquad(t\ge1). $$ In characteristic $p$, if a minimum-degree polynomial has a nonzero first ordinary derivative, this follows by differentiation; if all first ordinary derivatives vanish, perfectness gives a $p$th root and an induction on the symbolic exponent. The remainder of the proof uses the $q=p^e$ Frobenius decomposition and a maximal Hasse derivative.