Generic μ-power ideals, fat points, and symbolic powers
Abstract
Problem F of \cite{FLOS18} asks whether an ideal generated by r generic μ-power forms, where μ⊢ d is a non-pure partition, has the same Hilbert series Gn,d,r=[(1−td)r(1−t)−n]+ 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 PPn−1, the homogeneous system LL(q;mr) 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≤ n+1. For ternary forms, the obstruction produces counterexamples precisely for 5≤ r≤ 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≤ 4 and r=9, and, assuming Nagata's conjecture, for every non-pure μ when r≥ 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.
Related
- Symbolic Powers and Asymptotic Invariants of GL-Invariant Ideals
- Regularity of symbolic powers of complementary edge ideals
- Comparisons between ordinary and symbolic powers of edge ideals with respect to regularity and depth
- Structural properties of admissible subgraphs and the depth of symbolic powers of cover ideals
- Regularity of Symbolic Powers of Polymatroidal Ideals and Supermodular Prime Intersections
BibTeX
@misc{shapiro2026generic,
title = {{Generic $μ$-power ideals, fat points, and symbolic powers}},
author = {Boris Shapiro},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27965},
eprint = {2609.27965},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27965},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generic μ-power ideals, fat points, and symbolic powers
AU - Boris Shapiro
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.27965
UR - https://arxiv.org/abs/2609.27965
AB - Problem F of \cite{FLOS18} asks whether an ideal generated by r generic μ-power forms, where μ⊢ d is a non-pure partition, has the same Hilbert series G_n,d,r=[(1−t^d)^r(1−t)^−n]_+ 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≤ n+1. For ternary forms, the obstruction produces counterexamples precisely for 5≤ r≤ 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≤ 4 and r=9, and, assuming Nagata's conjecture, for every non-pure μ when r≥ 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.
ER -