Semicontinuity of the $M$-Number in Plane-Curve Deformations

Definitions and mathematical statement

The supplied record does not define the $M$-number or specify the category of deformations. Consequently, the following is a conditional formulation of the question; it does not supply the missing definition of $M$.

Let $(T,t_0)$ be a pointed parameter space for an admissible deformation $(C_t)_{t\in T}$ of plane-curve singularities, with special member $C_{t_0}$. Here the topology on $T$, admissibility of the deformation, and the meaning of its singularity members must be specified from the source. Write $M(C_t)$ for the intended numerical invariant. If this invariant is real-valued, upper semicontinuity at $t_0$ means \[ \forall\varepsilon>0\quad\exists U\text{ an open neighborhood of }t_0\quad\forall t\in U,\qquad M(C_t)<M(C_{t_0})+\varepsilon. \] The general question asks whether this condition holds at every parameter value in every admissible deformation. The particular question asks whether it holds whenever the deformation is $\delta$-constant, meaning that, on a neighborhood $V$ of $t_0$, \[ \delta(C_t)=\delta(C_{t_0})\qquad\text{for every }t\in V. \]

Under the conventional interpretation in terms of reduced complex analytic plane-curve germs $(C_t,p_t)$, the local $\delta$-invariant is \[ \delta(C_t,p_t)=\dim_{{\mathbb{C}}}\!\left(\widetilde{\mathcal O}_{C_t,p_t}/\mathcal O_{C_t,p_t}\right), \] where $\widetilde{\mathcal O}_{C_t,p_t}$ is the integral closure of the local ring in its total ring of fractions. This interpretation is conditional: the supplied record does not state reducedness, the ground field, or an analytic category. It also does not say whether invariants are taken at a selected singular point or summed over singular points appearing in a nearby fiber. Thus the displayed formulation requires these choices, together with the definition and normalization of $M$, before it becomes a fully specified mathematical problem.

The supplied dated literature assessment says that upper semicontinuity fails without the $\delta$-constant hypothesis, while only partial bounds are known for the $\delta$-constant question and that question remains unresolved. These are supplied curation claims, not independently verified conclusions of this draft.

Short English statement

The question asks whether the $M$-number can increase near a special plane-curve singularity under deformation, especially when the $\delta$-invariant stays constant. The supplied assessment reports failure in general and an unresolved $\delta$-constant case with partial bounds. A precise unconditional statement still requires a definition of $M$, deformation hypotheses, and a convention for fibers with several singular points.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001345$, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. No individual proposer is identified; the supplied proposed year is 2009.
• Supplied source citation: Singularities (2010), Oberwolfach Reports, pp. 2405--2470, DOI 10.4171/owr/2009/43. Original report: https://doi.org/10.4171/owr/2009/43. The record describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference, labeled "deformations of plane curve singularities": https://arxiv.org/abs/0907.4129.
• Supplied literature reference, labeled "official publication record": https://doi.org/10.18910/50809.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved, reports failure of upper semicontinuity without the $\delta$-constant hypothesis, and describes the $\delta$-constant case as unresolved with partial bounds. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; the cited claims and time-sensitive status require independent verification.
• The supplied statement label is corrected_verified; its accompanying verification describes meaning-preserving semantic, typographic, and LaTeX cleanup without added source-dependent content. This editorial label does not establish mathematical correctness or resolve the missing definitions.