Correspondence (algebraic geometry)

In algebraic geometry, a correspondence between algebraic varieties "V" and "W" is a subset "R" of "V"×"W", that is closed in the Zariski topology. In set theory, a subset of a Cartesian product of two sets is called a binary relation or correspondence; thus, a correspondence here is a relation that is defined by algebraic equations. There are some important examples, even when "V" and "W" are algebraic curves: for example the Hecke operators of modular form theory may be considered as correspondences of modular curves.
However, the definition of a correspondence in algebraic geometry is not completely standard. For instance, Fulton, in his book on intersection theory, uses the definition above. In literature, however, a correspondence from a variety "X" to a variety "Y" is often taken to be a subset "Z" of "X"×"Y" such that "Z" is finite and surjective over each component of "X". Note the asymmetry in this latter definition; which talks about a correspondence from "X" to "Y" rather than a correspondence between "X" and "Y". The typical example of the latter kind of correspondence is the graph of a function "f":"X"→"Y". Correspondences also play an important role in the construction of motives (cf. presheaf with transfers).