Toric Mathematics from Semigroup Viewpoint

Abstract

Toric geometry is a subject of increasing activity. Toric varieties are objects on which one usually can check explicitly properties and compute invariants from algebraic geometry. This happens for the so-called normal toric varieties, i.e. algebraic varieties which are constructed from rational fans in an euclidean space. In the last 10 years the theory of non normal toric varieties has also been developed providing a very different and new scope as well as interesting and beautiful new applications. Normal toric geometry mainly uses techniques from convex geometry, as it is technically founded on the concepts of fan and cone. Fans are sets of polyhedral cones in such a way that each cone provides an affine chart of the toric variety. Namely, those charts have, as coordinate algebra, the algebra of the semigroup of lattice points lying inside the dual cone of the corresponding cone of the fan. To study non normal toric geometry one needs to be more precise than considers only cones. In fact, what one needs is to consider affine charts where coordinate algebras are semigroup ones for more general classes of semigroups. Thus, convex geometry should be used only as a tool by taking into account that nice semigroup generate concrete polyhedral cones. The purpose of this paper is to show how mathematics in toric geometry can be understood as the theory of appropriate classes of commutative semigroups with given generators. This viewpoint involves the description of various kinds of derived objects as abelian groups and lattices, algebras and binomial ideals, cones and fans, affine and projective algebraic varieties, simplicial and cellular complexes, polytopes, and arithmetics. Our approach consists in showing the mathematical relations among above objects and clarifying their possibilities for future developments in the area. For that purpose, we will survey some recent results and concrete applications.