{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16337",
  "author_total": 2,
  "authors": [
    {
      "name": "Piotr Pokora"
    },
    {
      "name": "Tomasz Szemberg"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1adrbcghg04y7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16337",
  "fulltext_url": null,
  "id": "1adrbcghg04y7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16337",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7213390808",
  "pdf_url": "https://arxiv.org/pdf/2609.16337",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1adrbcghg04y7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16337",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16337"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Demailly's Inequality for Finite Sets of Points in Positive Characteristic",
  "title_text": "Demailly's Inequality for Finite Sets of Points in Positive Characteristic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/1adrbcghg04y7",
  "venue": "arXiv",
  "versions": []
}