1994/03/02 by Klaus Altmann, Altmann, Klaus · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.alg-geom/9403003
openalex publication_date 1994/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For an affine, toric Q-Gorenstein variety Y (given by a lattice polytope Q) the vector space T1 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]/e2 as the base space of the semi-universal deformation.