正特征有限点集的Demailly不等式
Demailly's Inequality for Finite Sets of Points in Positive Characteristic
AI总结：
本文在正特征代数闭域上证明了有限点集的Demailly不等式，利用Frobenius–Hasse导数方法和符号幂初始度的严格增长引理，通过微分或p次根归纳完成论证。
AI中文摘要：
设 $k$ 为特征 $p>0$ 的代数闭域，$n\ge1$，并设 $X\subset \mathbb P^n_k$ 为有限非空的不同点集，其定义理想为 $I=I(X)$。我们给出了正特征下Demailly不等式的一个证明。该论证基于Hà和Sivakumar在此背景下使用的Frobenius–Hasse导数方法。关键的额外观察是仿射点集符号幂初始度的一个严格增长引理：$$ \alpha(J^{(t)})\ge \alpha(J^{(t-1)})+1\qquad(t\ge1). $$ 在特征 $p$ 中，若一个最小次数多项式具有非零的一阶普通导数，则通过微分即可得到该结论；若所有一阶普通导数均为零，则完美性给出一个 $p$ 次根，并对符号指数进行归纳。证明的其余部分使用 $q=p^e$ Frobenius 分解和极大Hasse导数。
英文摘要：
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.
