Maximum Simple Singularities of Plane Curves

Definitions and mathematical statement

An $A_k$ singularity, for an integer $k\geq 1$, is a complex plane curve germ analytically equivalent to \[ \{(u,v)\in{\mathbb{C}}^2:u^2+v^{k+1}=0\} \] at the origin. Equivalently, after an invertible local analytic change of coordinates, a local defining equation is a nowhere-vanishing analytic factor times $u^2+v^{k+1}$. This specifies the entire local curve germ, rather than merely one branch of a larger germ.

The supplied problem asks for the largest possible $k$ for a plane curve of degree $d$, and, more generally, for a curve in ${\mathbb{A}}^2({\mathbb{C}})$ of bidegree $(a,b)$, where $(a,b)\in{\mathbb{N}}^2$.

A precise conditional affine formulation can be given as follows. For a nonzero polynomial $f\in{\mathbb{C}}[x,y]$, let $V(f)$ denote its zero curve with the local defining equations furnished by $f$. Define \[ \begin{aligned} \mathcal K_d^{\mathrm{aff}} ={}&\bigl\{k\in{\mathbb{Z}}_{\geq1}:\exists f\in{\mathbb{C}}[x,y]\setminus\{0\},\ \exists p\in{\mathbb{C}}^2,\\ &\hspace{35mm}\deg f=d,\ f(p)=0,\ (V(f),p)\text{ has type }A_k\bigr\}. \end{aligned} \] Under the convention that bidegree means the exact pair of separate polynomial degrees $(\deg_x f,\deg_y f)$, define \[ \begin{aligned} \mathcal K_{a,b}^{\mathrm{bi}} ={}&\bigl\{k\in{\mathbb{Z}}_{\geq1}:\exists f\in{\mathbb{C}}[x,y]\setminus\{0\},\ \exists p\in{\mathbb{C}}^2,\\ &\hspace{12mm}\deg_x f=a,\ \deg_y f=b,\ f(p)=0,\ (V(f),p)\text{ has type }A_k\bigr\}. \end{aligned} \] The requested extremal quantities, under these conventions, are \[ \max\mathcal K_d^{\mathrm{aff}} \qquad\text{and}\qquad \max\mathcal K_{a,b}^{\mathrm{bi}}, \] whenever the corresponding sets have largest elements. If a set is empty, there is no such singularity and no maximum is assigned here.

These formulas are conditional interpretations, not a resolution of the source's missing conventions. The first question does not explicitly specify the ground field or whether its plane is affine or projective; the complex affine formulation above is suggested by the second question. A complex projective formulation would instead quantify over degree-$d$ homogeneous equations and points of ${\mathbb{P}}^2({\mathbb{C}})$. The record also does not specify whether curves must be globally reduced or irreducible, whether bidegree is exact or bounded, or whether $0\in{\mathbb{N}}$. The displayed polynomial formulation imposes no global reducedness or irreducibility condition, although the prescribed local germ is reduced. These choices require confirmation before a canonical formulation is accepted.

Short English statement

Determine how large an $A_k$ singularity can occur on a plane curve of degree $d$, and more generally on a complex affine plane curve of prescribed bidegree $(a,b)$. The precise curve class and degree conventions remain unresolved. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003866$, title "Maximum Simple Singularities of Plane Curves," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified in the supplied record; its proposed year is 2018.
• Supplied report citation: Subgroups of Cremona Groups (2019), Oberwolfach Reports, pp. 1685--1743, DOI 10.4171/owr/2018/28. Supplied link: https://doi.org/10.4171/owr/2018/28. The background attributes the problem to workshop open-problem discussions documented in this report.
• Supplied literature reference: Schneider, Plane curves and $A_k$-singularities, https://arxiv.org/abs/1808.05133.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the cited work provides formulas and tables in small bidegrees, while the requested general extremal function for arbitrary $a$ and $b$ is not determined. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.