On the hypersurfaces in toric varieties

Abstract

Theory of toric varieties provides fruitful interactions between algebraic geom etry and combinatorics. It is remarkably fertile in terms of connections with many areas of mathematics and has plentiful applications to other disciplines as well. We introduce and study toric varieties and their hypersurfaces in the realm of algebraic geometry with a focus on quasismooth hypersurfaces. This is because quasismooth hy persurfaces are general enough to contain many examples of elements in some special families (e.g. regular hypersurfaces and Calabi-Yau hypersurfaces) that have a fre quent appearance in mirror symmetry, complex and differential geometry, and physics; interesting enough to have special roles in some areas of research such as toric GIT and moduli problems; and easy to characterize using combinatorial tools agreeably to the spirit of toric geometry.