Geometric Applications of the Residue Theorem on Algebraic Curves

Many classical theorems of the intersection theory of plane algebraic curves can be derived from the residue theorem on such curves. One may ask for generalizations of these results to curves in higher dimensional spaces, or to hypersurfaces, or even to arbitrary varieties, and whether they are consequences of residue theory. We describe in this survey, mainly without proofs, but with references to original articles, generalizations of some beautiful classical results, which follow from the residue theorem on projective algebraic curves. We present in particular some results of the thesis of Gerhard Quarg [Q] and relate them to previously established theorems of intersection theory. The general idea is that residues of properly chosen differentials are intersection invariants which have a geometric meaning. The residue theorem then gives global relations between these invariants.