Waldschmidt Constants of Plane Curve Singular Sets

Definitions and mathematical statement

The supplied record does not specify a ground field or its characteristic. The following notation gives a conditional formulation over a fixed algebraically closed field $k$; it does not assert that every characteristic is intended.

Let $C\subset{\mathbb{P}}^2_k$ be a reduced plane curve, and let $Z=\operatorname{Sing}(C)_{\mathrm{red}}$ be its finite set of singular points, endowed with the reduced structure. For $Z\ne\varnothing$, put $S=k[x_0,x_1,x_2]$ and let \[ I_Z=\bigcap_{p\in Z}\mathfrak p_p\subset S, \qquad I_Z^{(m)}=\bigcap_{p\in Z}\mathfrak p_p^{\,m} \quad(m\in{\mathbb{Z}}_{\ge1}), \] where $\mathfrak p_p$ is the homogeneous ideal of the point $p$. Thus $I_Z^{(m)}$ consists of homogeneous forms vanishing to order at least $m$ at every point of $Z$. For a nonzero homogeneous ideal $J\subset S$, define its initial degree by \[ \alpha(J)=\min\{d\in{\mathbb{Z}}_{\ge0}:J_d\ne0\}. \] The Waldschmidt constant is \[ \widehat\alpha(I_Z) =\lim_{m\to\infty}\frac{\alpha(I_Z^{(m)})}{m} =\inf_{m\ge1}\frac{\alpha(I_Z^{(m)})}{m}. \]

The source asks whether, over its intended ground field, every reduced plane curve with nonempty singular set satisfies \[ \widehat\alpha\left(I_{\operatorname{Sing}(C)_{\mathrm{red}}}\right)\in{\mathbb{Q}}, \] and whether this constant can be computed. It specifically asks for its computation when $C$ is the union of the $21$ lines in Klein's arrangement, with $Z$ the full singular set of that union.

The ground field, the meaning of computability, and a defining realization of the named arrangement are absent from the supplied record. Consequently, the computational request is retained as a request to determine the constant, without imposing an unspecified algorithmic model. The source also does not address smooth curves: if one adopts $I_{\varnothing}=S$, the same degree convention gives $\widehat\alpha(I_{\varnothing})=0$. This is an explicit convention for that boundary case, not an additional source hypothesis.

Short English statement

For the reduced set of singular points of a reduced plane curve, ask whether the asymptotic minimum degree per unit of imposed vanishing is always rational and whether it can be computed, particularly for Klein's arrangement of $21$ lines. The intended ground field and the precise computational requirement remain unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004330$, title Waldschmidt Constants of Plane Curve Singular Sets, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as $2019$.
• Supplied report citation: Mini-Workshop: Seshadri Constants (2020), Oberwolfach Reports, pp. 3271--3313, DOI 10.4171/owr/2019/53. https://doi.org/10.4171/owr/2019/53. The record describes these reports as documenting mathematical workshops and their open-problem discussions. The relationship between the citation year $2020$ and the recorded proposed year $2019$ is not explained.
• Supplied literature source, labeled "OWR report on Seshadri constants": https://ems.press/content/serial-article-files/46833.
• Supplied literature source, labeled "Bauer et al., symmetric blowups and Waldschmidt constants": https://arxiv.org/abs/1609.08648. No fuller bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that rationality and computability in the Klein-arrangement test case remain unresolved in the primary literature, with later work providing estimates rather than a general solution. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.