Abstract
Given a lattice polytope Q ⊆ ℝn, we define an affine scheme
that reflects the possibilities of splitting Q into a Minkowski sum. Denoting by Y the toric Gorenstein singularity induced by Q, we construct a flat family over
with Y as special fiber. In case Y has an isolated singularity, this family is versal.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Altmann, K.: Computation of the vector space T l for affine toric varieties. J. Pure Appl. Algebra 95 (1994), 239–259.
Altmann, K.: Minkowski sums and homogeneous deformations of toric varieties. Tôhoku Math. J. 47 (1995), 151–184.
Altmann, K.: Obstructions in the deformation theory of toric singularities. Preprint 61 “Europäisches Singularitätenprojekt”, Boston 1994; e-print alg-geom/9405008; to appear in J. Pure Appl. Algebra.
Arndt, J.: Verselle Deformationen zyklischer Quotientensingularitäten. Dissertation, Universität Hamburg, 1988.
Christophersen, J.A.: On the Components and Discriminant of the Versal Base Space of Cyclic Quotient Singularities. In: Singularity Theory and its Applications, Warwick 1989, Part I: Geometric Aspects of Singularities, pp. 81–92, Springer-Verlag Berlin Heidelberg, 1991 (LNM 1462).
Ishida, M.-N.: Torus embeddings and dualizing complexes. Tôhoku Math. Journ. 32, 111–146 (1980).
Jong, T. de, Straten, D. van: On the deformation theory of rational surface singularities with reduced fundamental cycle. J. Algebraic Geometry 3, 117–172 (1994).
Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings I. Lecture Notes in Mathematics 339, Springer-Verlag, Berlin-Heidelberg-New York, 1973.
Kollár, J., Shepherd-Barron, N.I.: Threefolds and deformations of surface singularities. Invent. math. 91, 299–338 (1988).
Matsumura, H.: Commutative Algebra. W.A. Benjamin, Inc., New York 1970.
Oda, T.: Convex bodies and algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3/15), Springer-Verlag, 1988.
Riemenschneider, O.: Deformationen von Quotientensingularitäten (nach zyklischen Gruppen). Math. Ann. 209 (1974), 211–248.
Stevens, J.: On the versal deformation of cyclic quotient singularities. In: Singularity Theory and its Applications, Warwick 1989, Part I: Geometric Aspects of Singularities, pp. 302–319, Springer-Verlag Berlin Heidelberg, 1991 (LNM 1462).
Author information
Authors and Affiliations
Corresponding author
Additional information
Oblatum 9-V-1996 ⇐p; 30-IX-1996
This paper was written at M.I.T. and supported by a DAAD-fellowship
This article was processed by the author using the LATEX style file pljour 1m from Springer-Verlag.
Rights and permissions
About this article
Cite this article
Altmann, K. The versal deformation of an isolated toric Gorenstein singularity. Invent. math. 128, 443–479 (1997). https://doi.org/10.1007/s002220050148
Issue Date:
DOI: https://doi.org/10.1007/s002220050148