Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 30 min ago
PreprintCC BY-NC-SA 4.0arXiv

Symbolic powers of the ideal ofn general points in Pn−1

  • Ralf Fröberg
  • Boris Shapiro
Open at sourcePDFDOICite

Abstract

Problem L of Fröberg--Lundqvist--Oneto--Shapiro asks for the difference between the Hilbert series of ordinary and symbolic powers of the ideal of general points in projective space. We solve this completely for n general points of PPn−1. Besides a closed formula for HS(S/Im)−HS(S/I(m)), we determine all minimal monomial generators of I(m), and describe the symbolic Rees algebra. We also show that containment I(m)⊆ Ir is detected solely by initial degrees. This gives the exact containment threshold, the Waldschmidt constant walpha, the resurgence Res, and the asymptotic resurgence aRes: walpha(I)=n/(n−1), Res(I)=aRes(I)=(2(n−1))/n. We also take the first step beyond n points: for n+1 general points of PPn−1 --- again a rigid, non-monomial configuration --- we identify the defining quadrics, resolve the case n=3 completely (a complete intersection, with J(m)=Jm for all m and resurgence 1), and propose an exact Waldschmidt-constant formula walpha=(n+1)/(n−1) for all n, verified computationally in every case we could check.

Related

  1. Generic μ-power ideals, fat points, and symbolic powers
    Boris Shapiro
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. Uniform bounds on symbolic powers in regular rings
    Takumi Murayama
    Journal für die reine und angewandte Mathematik (Crelles Journal)Peer-reviewedLicence not statedFull text16 citedAlgebra & number theoryOpenAlex
  3. Symbolic Powers and Asymptotic Invariants of GL-Invariant Ideals
    Sankhaneel Bisui, Alexandra Seceleanu
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  4. On the Super-Schottky Ideal and Powers of the Classical Schottky Ideal for g ≥ 5
    Yuanyuan Shen
    arXivPreprintarXiv licenceFull textAlgebra & number theoryMathematical physicsarXiv
  5. Regularity of symbolic powers of complementary edge ideals
    Truong Thi Hien, Manohar Kumar
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{froberg2026symbolic,
  title = {{Symbolic powers of the ideal of$n$ general points in $P^n-1$}},
  author = {Ralf Fröberg and Boris Shapiro},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18539},
  eprint = {2609.18539},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18539},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Symbolic powers of the ideal ofn general points in P^n−1
AU  - Ralf Fröberg
AU  - Boris Shapiro
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.18539
UR  - https://arxiv.org/abs/2609.18539
AB  - Problem L of Fröberg--Lundqvist--Oneto--Shapiro asks for the difference between the Hilbert series of ordinary and symbolic powers of the ideal of general points in projective space. We solve this completely for n general points of PP^n−1. Besides a closed formula for HS(S/I^m)−HS(S/I^(m)), we determine all minimal monomial generators of I^(m), and describe the symbolic Rees algebra. We also show that containment I^(m)⊆ I^r is detected solely by initial degrees. This gives the exact containment threshold, the Waldschmidt constant walpha, the resurgence Res, and the asymptotic resurgence aRes: walpha(I)=n/(n−1), Res(I)=aRes(I)=(2(n−1))/n. We also take the first step beyond n points: for n+1 general points of PP^n−1 --- again a rigid, non-monomial configuration --- we identify the defining quadrics, resolve the case n=3 completely (a complete intersection, with J^(m)=J^m for all m and resurgence 1), and propose an exact Waldschmidt-constant formula walpha=(n+1)/(n−1) for all n, verified computationally in every case we could check.
ER  -