Winfried Bruns, Joseph Gubeladze
Polytopal linear groups
The paper is published: J. Algebra 218, 715 - 737 (1999)

MSC:

14M25 Toric varieties, Newton polyhedra

20G99 None of the above but in this section

Abstract: There are many natural reasons justifying the assignment of the general
linear group GL_n(k) (k a field) to a simplex with n vertices. What is
the most natural way to generalize this assignment to arbitrary convex
lattice polytopes? We give an answer to this question by describing the
group of graded automorphisms of normal semigroup rings. The arguments
are based on elementary properties of lattice polytopes and they are
rather transparent.

The objects immediately associated to the graded automorphisms of
affine semigroup rings are automorphisms of projective toric
varieties. Our results also yield a description of the automorphism
group of a projective toric variety (over any algebraically closed
field) without use of linear algebraic groups.

The description of the automorphism group of a complete (not necessarily
projective) smooth toric C-variety in terms of roots goes back to
M. Demazure in his fundamental work [Sous-groupes algebriques de rang
maximum du groupe de Cremona, Ann. Sci. Ecole Norm. Sup. (1970)],
initiating the theory of toric geometry. The analogous description for
all quasi-smooth complete toric C-varieties has recently been obtained
by D. Cox (1995). The general case has been completed by D. Buehler
using Cox' approach. In our approach Demazure's roots are replaced by
"column structures" of lattice polytopes. The descriptions provided by
the two approaches are essentially the same on the subclass of
projective toric varieties.

In a subsequent paper we will generalize our results to graded rings
defined by (not necessarily Euclidean) lattice polyhedral complexes.





Keywords: affine semigroup ring, projective toric variety, automorphism group