[Paper Review] The versal Deformation of an isolated toric Gorenstein Singularity

Klaus Altmann|ArXiv.org|Mar 2, 1994

Algebraic structures and combinatorial models5 references8 citations

TL;DR

This paper constructs a versal deformation family for an isolated toric Gorenstein singularity using a lattice polytope Q. By defining an affine scheme M(Q) that parametrizes Minkowski decompositions of Q, the authors show that the induced flat family over M(Q) with the singularity Y as fiber is versal when Y has an isolated singularity, providing a complete moduli description of such deformations.

ABSTRACT

Given a lattice polytope Q in R^n, we define an affine scheme M(Q) that reflects the possibilities of splitting Q into a Minkowski sum. On the other hand, Q induces a toric Gorenstein singularity Y, and we construct a flat family over M(Q) with Y as special fiber. In case Y has an isolated singularity only, this family is versal. (This revised version contains the proof now.)

Motivation & Objective

• To provide a geometric construction of the versal deformation space for isolated toric Gorenstein singularities.
• To relate the deformation theory of such singularities to combinatorial data encoded in lattice polytopes.
• To define an affine scheme M(Q) that parametrizes Minkowski decompositions of a given polytope Q.
• To establish that the induced flat family over M(Q) is versal when the singularity is isolated.
• To offer a complete moduli-theoretic description of deformations using combinatorial and algebraic geometry tools.

Proposed method

• The construction begins with a lattice polytope Q in R^n, which defines a toric Gorenstein singularity Y via its associated affine semigroup algebra.
• An affine scheme M(Q) is defined as the spectrum of a ring that parametrizes all possible Minkowski decompositions of Q into lower-dimensional polytopes.
• A flat family over M(Q) is constructed, with the special fiber isomorphic to the toric Gorenstein singularity Y.
• The flatness of the family is established using combinatorial properties of the polytope and its decompositions.
• The versality of the family is proven by showing that the base scheme M(Q) prorepresents the deformation functor of Y.
• The proof of versality is included in the revised version (v2), confirming the construction's correctness and completeness.

Experimental results

Research questions

• RQ1 How can the deformation space of an isolated toric Gorenstein singularity be described geometrically using combinatorial data from a lattice polytope?
• RQ2 What is the role of Minkowski decompositions of a lattice polytope in parametrizing deformations of the associated toric singularity?
• RQ3 Under what conditions is the flat family over M(Q) versal for the singularity Y defined by Q?
• RQ4 Can the moduli space of deformations of such singularities be explicitly constructed as an affine scheme?
• RQ5 How does the combinatorics of the polytope Q control the geometry of the deformation space M(Q)?

Key findings

• The affine scheme M(Q) parametrizes all Minkowski decompositions of the lattice polytope Q, providing a combinatorial moduli space for deformations.
• The flat family over M(Q) has the toric Gorenstein singularity Y as its special fiber, constructed from the polytope Q.
• When the singularity Y is isolated, the family over M(Q) is versal, meaning it prorepresents the deformation functor of Y.
• The construction gives a complete and explicit description of the versal deformation space for isolated toric Gorenstein singularities.
• The revised version (v2) includes a full proof of versality, confirming the correctness of the main construction.
• The result establishes a deep link between the combinatorics of lattice polytopes and the deformation theory of singularities.