Mutable Laurent Polynomials and Toric Smoothing Components

Definitions and mathematical statement

A lattice polytope is the convex hull of finitely many points of a lattice. For a lattice $N$ with dual $M=\operatorname{Hom}(N,{\mathbb{Z}})$ and a full-dimensional lattice polytope $F\subset N_{{\mathbb{R}}}$, its cone at height one is \[ \sigma_F=\operatorname{cone}\{(v,1):v\in F\}\subset N_{{\mathbb{R}}}\oplus{\mathbb{R}}. \] Once a ground field $k$ is specified, the standard associated affine toric variety is \[ U_F=\operatorname{Spec}k[\sigma_F^{\vee}\cap(M\oplus{\mathbb{Z}})]. \] Under this construction, $\dim U_F=\dim F+1$. Thus the supplied simultaneous requirements that $F$ be three-dimensional and that $U_F$ be a threefold are incompatible: a three-dimensional $F$ yields a fourfold, whereas a threefold requires a two-dimensional $F$. The intended dimension cannot be determined from the record.

For a nonzero Laurent polynomial $f=\sum_{u\in N}a_u x^u\in k[N]$, with finitely many nonzero coefficients, define \[ \operatorname{Newt}(f)=\operatorname{conv}\{u\in N:a_u\ne0\}. \] In a specified deformation setting, a smoothing component means an irreducible component of the base of a miniversal deformation whose general fiber is smooth. The record does not specify the ground field, the deformation setting, or a definition of $0$-mutability. It also does not specify coefficient normalizations or any equivalence relation on Laurent polynomials.

Supplied conjecture, pending resolution of these qualifications. For every lattice polytope $F$ satisfying the intended dimensional and geometric hypotheses, there is a bijection \[ \left\{\text{smoothing components of the miniversal deformation of }U_F\right\} \;\longleftrightarrow\; \left\{f:\ f\text{ is a }0\text{-mutable Laurent polynomial and }\operatorname{Newt}(f)=F\right\}. \] This displays the proposed correspondence only; its right-hand side remains undefined without the missing definition and conventions, and the dimensional conflict prevents a single unqualified formulation. The supplied background describes the proposal as a generalization of Altmann's deformation construction for relevant toric Gorenstein threefold singularities, but does not specify what restrictions "relevant" imposes.

Short English statement

The proposed conjecture pairs smoothing components of a toric singularity with $0$-mutable Laurent polynomials having a prescribed Newton polytope. The supplied formulation has a dimension conflict and omits the definition of $0$-mutability and essential conventions. It therefore remains a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004428$, exact source problem code catalogue entry, title Mutable Laurent Polynomials and Toric Smoothing Components; collection: Oberwolfach Reports Open Problems. No individual proposer is identified; the supplied proposed year is $2020$.
• Supplied report citation: Mini-Workshop: Superpotentials in Algebra and Geometry (2021), Oberwolfach Reports, pp. 569--600, DOI 10.4171/owr/2020/10. Source URL and supplied literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2020/10. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The report contents were not independently consulted here.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and describes the proposed generalization of Altmann's deformation construction for all relevant toric Gorenstein threefold singularities as a conjectural expectation in the checked literature. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry supplies the statement-status label corrected_verified and says that mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired. This supplied editorial label does not resolve the dimensional inconsistency or establish mathematical correctness.