Normality and Covering Properties of Affine Semigroups
by
Winfried Bruns
Universität Osnabrück, Osnabrück, Germany

Affine semigroups are finitely generated subsemigroups of a lattice Zⁿ containing 0. Such a semigroup S is called normal if x in S for every element x in \operatornamegp(S) such that mx in S for some m in Z, m > 0. We say that S is positive if 0 is the only invertible element in S.

Affine semigroups S, and in particular normal ones appear in various mathematical contexts:

1. The algebraic varieties corresponding to the semigroup algebras K[S] (where K is an algebraically closed field) are the building blocks of toric varieties.
2. The rings of invariants of polynomial rings under actions of algebraic tori or finite abelian groups are of type K[S].
3. They occur naturally in integral optimization since the non-negative integral solutions to a system of homogeneous linear equations form a normal semigroup.
4. Last, but not least, they are interesting algebraic objects by themselves.

In our lecture we discuss conjectures of A. Sebö about the combinatorial structure of the Hilbert bases of normal affine semigroups that have been of great interest for integer optimization. These conjectures (proved by Sebö in dimesnions <= 3) have now been disproved in dimesnions >= 6 by a counter-example.

The Hilbert basis \operatornameHilb(S) of a positive affine semigroup is its set of irreducible elements. It is the unique minimal system of generators of S. The counter-example shows that Sebös Unimodular Covering Conjecture (UCC) and the so-called Integral Caratheodory Property (ICP) do not hold in general. (UCC) is the assertion that the cone generated by a normal positive affine semigroup S (in S\otimes_ZR) is the union of unimodular simplicial cones generated by elements of \operatornameHilb(S) whereas the weaker property (ICP) means that every element of S can be represented by \operatornamerank(S) elements of \operatornameHilb(S).

The counterexample is of rank 6 and has 10 elements in its Hilbert basis. It has several remarkable properties. It can be realized as the semigroup associated with a 0-1-polytope in R⁵, and its surprisingly large symmetry group acts transitively on \operatornameHilb(S). As a counterexample to (UCC) it was found in joint work with J. Gubeladze, and consecutive joint work with J. Gubeladze, M. Henk, A. Martin and R. Weismantel showed that it also violates (ICP).

The results to be discussed have been published in J. Reine Angew. Mathh. 510 (1999), 161-178 and 179-185.

Date received: June 5, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caec-42.