Resurgence Bounds for Linear Arrangements in Multiprojective Spaces

Definitions and mathematical statement

The supplied record poses three research questions rather than a quantified conjecture with specified arrangements or numerical bounds. The following definitions give a conditional framework for interpreting them; the source does not fix all of its hypotheses.

Let $k$ be a field, let $s\geq 1$, and let \[ P=\prod_{i=1}^{s}{\mathbb{P}}_k^{n_i},\qquad R=k[x_{i,j}:1\leq i\leq s,\ 0\leq j\leq n_i], \] where $n_i\geq 1$ and $R$ has its usual ${\mathbb{Z}}^s$-grading, with $\deg(x_{i,j})$ the $i$-th standard basis vector. No characteristic or algebraic-closure assumption on $k$ is supplied; these must be specified for any proposed result.

For a reduced closed subscheme $X\subseteq P$ with nonzero proper radical multihomogeneous defining ideal $I=I(X)\subset R$, use the symbolic-power convention \[ I^{(m)}=\bigcap_{\mathfrak p\in\operatorname{Min}(R/I)} \left(I^mR_{\mathfrak p}\cap R\right),\qquad m\in{\mathbb{Z}}_{\geq 1}, \] where $\operatorname{Min}(R/I)$ denotes the minimal prime ideals over $I$, and $I^r$ is the ordinary ideal power. Its resurgence is \[ \rho(I)=\sup\left\{\frac{m}{r}:m,r\in{\mathbb{Z}}_{\geq 1},\ I^{(m)}\not\subseteq I^r\right\}. \] Thus a finite upper bound $\rho(I)\leq B$ amounts to proving \[ \forall m,r\in{\mathbb{Z}}_{\geq 1},\qquad \frac{m}{r}>B\ \Longrightarrow\ I^{(m)}\subseteq I^r. \] This definition covers reduced arrangements once their components and ambient space are specified. It does not settle the symbolic-power convention for any nonreduced subschemes intended by the source.

1. For which further arrangements of lines can one obtain useful upper bounds on the resurgence of their defining ideals?
2. Which analogous symbolic-to-ordinary power containment results and resurgence bounds extend to arrangements involving higher-dimensional linear spaces in products of projective spaces?
3. For specified concrete subschemes of multiprojective spaces, what upper bounds can be obtained for the resurgence of their defining ideals?

The record does not identify the previously treated line arrangements, the results to be extended, the intended meaning of linear spaces in a product of projective spaces, or the concrete subschemes in the third question. It also supplies neither a numerical bound nor a criterion for a bound to be useful. Consequently, these questions cannot be sharpened into a single precise conjecture without additional source information. The reduced setting above is an explicitly conditional interpretation, not an assertion that the source restricts all three questions to reduced subschemes.

Short English statement

The problem asks for resurgence bounds for further line arrangements, extensions of containment and resurgence results to higher-dimensional arrangements in products of projective spaces, and upper bounds for particular multiprojective subschemes. Resurgence measures failures of containment of symbolic powers in ordinary powers. The intended examples, hypotheses, comparison results, and target bounds are unspecified, so this remains a research programme requiring source clarification and human mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30002789$, source problem code catalogue entry, title Resurgence Bounds for Linear Arrangements in Multiprojective Spaces; collection: Oberwolfach Reports Open Problems. The record gives proposed year 2015 and does not name an individual proposer.
• Supplied report citation: Mini-Workshop: Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems (2015), Oberwolfach Reports, pp. 489--532, DOI 10.4171/owr/2015/9. Supplied source URL and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2015/9. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; it supplies no problem-specific page or named contribution.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that no general classification or bound resolving these three broad questions was verified. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified; its accompanying explanation says that mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. This is source metadata, not independent verification of this draft or of mathematical correctness.