[Paper Review] Toric Q-Gorenstein Singularities

Klaus Altmann|ArXiv.org|Mar 2, 1994

Advanced Combinatorial Mathematics4 references3 citations

TL;DR

This paper studies toric Q-Gorenstein singularities via lattice polytopes, showing that infinitesimal deformations are governed by Minkowski summands of faces of the defining polytope. The key result is that such singularities are rigid unless they are Gorenstein and three-dimensional, in which case deformations correspond to Minkowski decompositions of the polytope, with a natural Kodaira-Spencer map and connections to the Picard group of the associated projective variety.

ABSTRACT

For an affine, toric Q-Gorenstein variety Y (given by a lattice polytope Q) the vector space T^1 of infinitesimal deformations is related to the complexified vector spaces of rational Minkowski summands of faces of Q. Moreover, assuming Y to be an isolated, at least 3-dimensional singularity, Y will be rigid unless it is even Gorenstein and dim Y=3 (dim Q=2). For this particular case, so-called toric deformations of Y correspond to Minkowski decompositions of Q into a sum of lattice polygons. Their Kodaira-Spencer-map can be interpreted in a very natural way. We regard the projective variety P(Y) defined by the lattice polygon Q. Data concerning the deformation theory of Y can be interpreted as data concerning the Picard group of P(Y). Finally, we provide some examples (the cones over the toric Del Pezzo surrfaces). There is one such variety yielding Spec C[e]/e^2 as the base space of the semi-universal deformation.

Motivation & Objective

• To understand the deformation theory of affine toric Q-Gorenstein varieties using lattice polytopes.
• To determine conditions under which such singularities are rigid or admit nontrivial deformations.
• To relate the space of infinitesimal deformations T^1 to rational Minkowski summands of faces of the defining polytope Q.
• To establish a geometric interpretation of the Kodaira-Spencer map in terms of Minkowski decompositions for 3-dimensional Gorenstein toric singularities.
• To connect deformation data of the singularity to the Picard group of the projective variety P(Y) defined by the lattice polygon Q.

Proposed method

• The paper uses the combinatorial data of a lattice polytope Q to describe the affine toric variety Y, which is assumed to be Q-Gorenstein.
• It analyzes the tangent space T^1 of infinitesimal deformations using rational Minkowski summands of faces of Q.
• For isolated singularities of dimension at least 3, the rigidity condition is derived from the structure of the Minkowski summands.
• In the 3-dimensional Gorenstein case, deformations are parametrized by Minkowski decompositions of Q into lattice polygons.
• The Kodaira-Spencer map is interpreted geometrically through these decompositions.
• The Picard group of the projective variety P(Y) is used to interpret deformation-theoretic data of the singularity.

Experimental results

Research questions

• RQ1 When is a toric Q-Gorenstein singularity rigid, and what conditions allow nontrivial deformations?
• RQ2 How are the infinitesimal deformations of a toric Q-Gorenstein variety related to Minkowski summands of its defining polytope?
• RQ3 What is the geometric meaning of the Kodaira-Spencer map in the context of toric deformations of 3-dimensional Gorenstein singularities?
• RQ4 How can deformation data of the singularity be recovered from the Picard group of the associated projective variety P(Y)?
• RQ5 What role do Minkowski decompositions of the lattice polygon Q play in parametrizing deformations of the singularity?

Key findings

• The space of infinitesimal deformations T^1 of a toric Q-Gorenstein variety is isomorphic to the complexified vector space of rational Minkowski summands of faces of the defining polytope Q.
• An isolated toric Q-Gorenstein singularity of dimension ≥3 is rigid unless it is Gorenstein and 3-dimensional.
• In the 3-dimensional Gorenstein case, deformations correspond bijectively to Minkowski decompositions of the lattice polygon Q into a sum of lattice polygons.
• The Kodaira-Spencer map for such deformations has a natural geometric interpretation via the decomposition data of Q.
• The deformation theory of the singularity is equivalent to data on the Picard group of the projective variety P(Y) defined by Q.
• An example is provided where the base space of the semi-universal deformation is Spec C[e]/e^2, showing a non-trivial deformation space in the 3-dimensional Gorenstein case.