Demailly's Inequality for Finite Sets of Points in Positive Characteristic
Abstract
Let k be an algebraically closed field of characteristic p>0, n≥1, and let X⊂ ℙnk 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))≥ α(J(t−1))+1 (t≥1). 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 pth root and an induction on the symbolic exponent. The remainder of the proof uses the q=pe Frobenius decomposition and a maximal Hasse derivative.
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BibTeX
@misc{pokora2026demaillys,
title = {{Demailly's Inequality for Finite Sets of Points in Positive Characteristic}},
author = {Piotr Pokora and Tomasz Szemberg},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16337},
eprint = {2609.16337},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16337},
}
RIS
TY - UNPB
TI - Demailly's Inequality for Finite Sets of Points in Positive Characteristic
AU - Piotr Pokora
AU - Tomasz Szemberg
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16337
UR - https://arxiv.org/abs/2609.16337
AB - Let k be an algebraically closed field of characteristic p>0, n≥1, and let X⊂ 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))≥ α(J^(t−1))+1 (t≥1). 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 pth 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.
ER -