Combinatorial Deformations of Toric Fano Varieties

Definitions and mathematical statement

Let $N$ be a lattice of rank $n$, let $M=\operatorname{Hom}_{{\mathbb{Z}}}(N,{\mathbb{Z}})$, and set $N_{{\mathbb{R}}}=N\otimes_{{\mathbb{Z}}}{\mathbb{R}}$ and $M_{{\mathbb{R}}}=M\otimes_{{\mathbb{Z}}}{\mathbb{R}}$. A full-dimensional lattice polytope $P\subset N_{{\mathbb{R}}}$ is reflexive if $0$ lies in its interior and its polar \[ P^{\vee}=\{u\in M_{{\mathbb{R}}}:\langle u,v\rangle\geq -1\text{ for every }v\in P\} \] is a lattice polytope. Its face fan is \[ \Sigma_P=\{\{0\}\}\cup\{\operatorname{cone}(F):F\text{ is a nonempty proper face of }P\}. \] Write $X_P=X(\Sigma_P)$ for the associated toric Fano variety. The source imposes no dimension restriction and does not specify a ground field or its characteristic; the ground field in this notation remains an unresolved parameter.

In the usual algebraic interpretation, a deformation of $X_P$ is a flat family $\pi:\mathcal X\to B$ over a pointed base $(B,b_0)$, together with an identification $\mathcal X_{b_0}\cong X_P$. A smoothing is such a family with smooth geometric general fibers. Here general fibers refer to fibers over a suitable dense open subset of the relevant irreducible component of the base. These conventions explain the terminology; the source does not specify the permitted bases, properness or projectivity of the family, or whether additional conditions such as ${\mathbb{Q}}$-Gorenstein deformation are required.

Problem. For a reflexive polytope $P$, characterize the combinatorial data on $P$ that yield deformations or smoothings of $X_P$. For families constructed from such data, determine what can be said about their general fibers $X_t=\pi^{-1}(t)$. The source does not specify a particular type of combinatorial datum, an equivalence relation on families or fibers, or whether an exhaustive classification of all deformations is intended. These aspects of the requested characterization remain unresolved.

Short English statement

Describe combinatorial information on a reflexive polytope that constructs deformations or smoothings of its associated toric Fano variety, and describe the general fibers of the resulting families. The field, permitted deformation class, and intended classification scope require clarification.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30004249$, supplied source code catalogue entry, title "Combinatorial Deformations of Toric Fano Varieties." The supplied record attributes the problem to Oberwolfach Reports workshop discussions, records proposed year $2019$, and names no individual proposer.
• Supplied report citation: Toric Geometry (2020), Oberwolfach Reports, pp. 2671--2726, DOI 10.4171/owr/2019/43. https://doi.org/10.4171/owr/2019/43. No problem-specific page or named contributor is supplied.
• Supplied literature reference: Smoothing Gorenstein toric Fano threefolds. https://arxiv.org/abs/2412.06500. No authors or precise result identifiers are supplied in the record.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved" and states that combinatorial criteria produce smoothings for broad classes, including Gorenstein toric Fano threefolds, while a general deformation theory and classification of general fibers for toric varieties remain incomplete. This is supplied dated curation, not an independently verified assessment; the record explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying explanation says that notation, typography, and sentence structure were repaired while mathematical content was preserved. This supplied editorial label does not establish independent mathematical verification of this draft.