Laurent polynomials and deformations of non-isolated Gorenstein toric singularities

Filip, Matej (author)

We establish a correspondence between one-parameter deformations of an affine Gorenstein toric pair $(X_P, \partial X_P)$, defined by a polytope $P$, and mutations of a Laurent polynomial $f$ with Newton polytope $\Delta(f) = P$. For a Laurent polynomial $f$ in two variables, we construct a formal deformation of the three-dimensional Gorenstein toric pair $(X_{\Delta(f)}, \partial X_{\Delta(f)})$ over ${\mathbb C}[[{\mathbf T}_f]]$, where ${\mathbf T}_f$ is the set of deformation parameters arising from mutations. The general fibre of this deformation is smooth if and only if $f$ is $0$-mutable. The Kodaira-Spencer map of the constructed deformation is injective, and if $f$ is maximally mutable, then the deformation cannot be nontrivially extended to a larger smooth base space.

• Language: English
• Keywords: deformation theory, toric singularities, Laurent polynomials, mirror symmetry, Fano manifolds
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FE - Faculty of Electrical Engineering
• Publication status: Published
• Publication version: Version of Record
• Year: 2026
• Number of pages: 56 str.
• Numbering: Vol. 487, art. 110775
• PID: 20.500.12556/RUL-182861
• UDC: 514.7
• ISSN on article: 0001-8708
• DOI: 10.1016/j.aim.2025.110775
• COBISS.SI-ID: 267113219
• Publication date in RUL: 26.05.2026

Record is a part of a journal

• Title: Advances in mathematics
• Shortened title: Adv. math.
• Publisher: Elsevier
• ISSN: 0001-8708
• COBISS.SI-ID: 24891904

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.