{
  "abstract": "Problem F of \\cite{FLOS18} asks whether an ideal generated by $r$ generic $μ$-power forms, where $μ\\vdash d$ is a non-pure partition, has the same Hilbert series $G_{n,d,r}=\\big[(1-t^d)^r(1-t)^{-n}\\big]_+$ as an ideal generated by $r$ generic forms of degree $d$. We show that the answer is negative, and we determine exactly when the natural obstruction is available. Our main tool is an apolarity inclusion that converts a distinguished large part of $μ$ into a fat-point linear system in the dual projective space. This yields a single \\emph{master obstruction} (Theorem \\ref{thm:master}): a counterexample exists whenever, for $r$ general points of $\\PP^{n-1}$, the homogeneous system $\\LL(q;m^r)$ is non-empty while $\\LL(q;(m-a)^r)$ has non-positive virtual dimension. We complement this with a positive result (Theorem \\ref{thm:small-r}): by Stanley's theorem, Problem F has a positive answer for \\emph{every} $μ$ as soon as $r\\le n+1$. For ternary forms, the obstruction produces counterexamples precisely for $5\\le r\\le 8$, with extremal curves given by the conic through five points and products of $(-1)$-curves on del Pezzo surfaces. It is vacuous unconditionally for $r\\le 4$ and $r=9$, and, assuming Nagata's conjecture, for every non-pure $μ$ when $r\\ge 10$. We also obtain counterexamples in four variables for $r=9$. An exact finite-field certificate shows that $d=14$, $μ=(13,1)$ is the \\emph{first} failure for five ternary generators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27965",
  "author_total": 1,
  "authors": [
    {
      "name": "Boris Shapiro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/12257v7pxzmbj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27965",
  "fulltext_url": null,
  "id": "12257v7pxzmbj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27965",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7214164594",
  "pdf_url": "https://arxiv.org/pdf/2609.27965",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/12257v7pxzmbj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.27965",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27965"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Generic $μ$-power ideals, fat points, and symbolic powers",
  "title_text": "Generic μ-power ideals, fat points, and symbolic powers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/12257v7pxzmbj",
  "venue": "arXiv",
  "versions": []
}