Lattice-ordered Abelian groups and Toric varieties
by
Vincenzo Marra
University of Milan, Milan, Italy, and Freie Universitaet, Berlin, Germany

This talk is an attempt to explain how one relates questions about lattice-ordered Abelian groups to questions in algebraic geometry. The celebrated dictionary between toric varieties and integral geometry in the style of Minkowski and Hadwiger affords a complete translation of algebro-geometric concepts into the discrete geometry of rational polyhedra, and vice versa. While barely thirty years old, this surprising bridge between apparently distant mathematical worlds has already proved its worth beyond any reasonable doubt. In a parallel and hitherto independent development whose full import is just beginning to be appreciated, K. R. Baker and W. M. Beynon brought to light in the 1970s an intimate connection between finitely generated projective Abelian l-groups and supports of rational polyhedral sets. We offer some preliminary considerations on the relationships between these two intriguing duality phenomena. We show that a finitely generated projective Abelian l-group with a distinguished generating set of elements satisfying a certain condition completely encodes the absolute geometry (i.e. the geometry which is independent of the ground field, or, indeed, integral domain) of a smooth toric variety. As an example, we translate the notorious Birational Factorisation Conjecture of algebraic geometry, whose absolute toric version is also known as the Strong Oda Conjecture, into an equivalent purely l-group-theoretical statement. We then show how to obtain a version of the same conjecture over the real numbers, that is, for Archimedean finitely generated o-groups. We thus arrive at a completely elementary arithmetic version of the conjecture expressing a confluence property of slow multi-dimensional continued fraction expansions. We close with a brief discussion of further research.

Date received: December 28, 2002