Let k be a field, let S=k[x_1,…,x_n], and let I⊆ S be the homogeneous ideal of n general points in P^n-1. For every integer m≥ 1, determine explicitly the difference between the Hilbert series of the ordinary and symbolic powers, operatornameHS(S/I^m)-operatornameHS(S/I^(m)).

Problem L of Fröberg–Lundqvist–Oneto–Shapiro

Let $k$ be a field, let $S=k[x_1,\ldots,x_n]$ , and let $I\subseteq S$ be the homogeneous ideal of $n$ general points in $\mathbb{P}^{n-1}$ . For every integer $m\geq 1$ , determine explicitly the difference between the Hilbert series of the ordinary and symbolic powers, $\operatorname{HS}(S/I^m)-\operatorname{HS}(S/I^{(m)})$ .

References

Primary source

arXiv

Additional references

Symbolic powers of the ideal of n general points in P^{n-1} — arXiv — Ralf Fröberg, Boris Shapiro

algebraic-geometry commutative-algebra

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A new paper claims to settle the original problem, but its proposed extension to one extra point remains conjectural.

Problem L concerns symbolic powers of the ideal of $n$ general points and their asymptotic invariants. A paper by Ralf Fröberg and Boris Shapiro claims a complete solution for the $n$ -point case.

September 2026 paper

Fröberg and Shapiro’s paper Symbolic powers of the ideal of $n$ general points in $\mathbb{P}^{n-1}$ claims formulas for minimal symbolic generators, containment thresholds, the Waldschmidt constant, and resurgence in the $n$ -point case. It also proposes formulas for $n+1$ points, but those are conjectural apart from computational checks and the case $n=3$ .

Current status (as of September 2026): The $n$ -point case is claimed solved but remains unverified; the proposed $n+1$ -point formulas remain conjectural except in the stated checked cases.

Sources

No solutions have been posted yet.