m at h . A C ] 2 5 Ju l 2 00 1 PROBLEMS AND ALGORITHMS FOR AFFINE SEMIGROUPS

Affine semigroups – discrete analogues of convex polyhedral cones – mark the crossroads of algebraic geometry, commutative algebra and integer programming. They constitute the combinatorial background for the theory of toric varieties, which is their main link to algebraic geometry. Initiated by the work of Demazure [De] and Kempf, Knudsen, Mumford and Saint-Donat [KKMS] in the early 70s, toric geometry is still a very active area of research. However, the last decade has clearly witnessed the extensive study of affine semigroups from the other two perspectives. No doubt, this is due to the tremendously increased computational power in algebraic geometry, implemented through the theory of Gröbner bases, and, of course, to the modern computers. In this article we overview those aspects of this development that have been relevant for our own research, and pose several open problems. Answers to these problems would contribute substantially to the theory. The paper treats two main topics: (1) affine semigroups and several covering properties for them and (2) algebraic properties for the corresponding rings (Koszul, CohenMacaulay, different “sizes” of the defining binomial ideals). We emphasize the special case when the initial data are encoded into lattice polytopes. The related objects – polytopal semigroups and algebras – provide a link with the classical theme of triangulations into unimodular simplices. We have also included an algorithm for checking the semigroup covering property in the most general setting (Section 4). Our counterexample to certain covering conjectures (Section 3) was found by the application of a small part of this algorithm. The general algorithm could be used for a deeper study of affine semigroups.