Severi’s Results on Correspondences

Severi developed a theory of correspondences in a series of papers which appeared in 1933, introducing the notions of valences and indices. One of the results achieved by Severi is a formula for the virtual number of fixed points of a correspondence on a smooth projective surface X. These papers are part of Severi’s attempt to develop a theory of the series of equivalences on a surface. In fact Severi encountered (sometimes without being completely aware of what was going on!) the problem of not having a rigorous definition for the different equivalence relations among cycles, which are now known as rational, algebraic, homological and numerical equivalence. However as W. Fulton writes in [?, p.26]: It would be unfortunate if Severi’s pioneering works in this area were forgotten; and if incompleteness or the presence of errors are grounds for ignoring Severi’s work, few of the subsequent papers on rational equivalence would survive. The above considerations indicate that Severi was often wrong and certainly too bold in making conjectures. However Severi was somehow able to perceive the motivic content of the matter, by considering correspondences and their action both on Chow groups and cohomology groups. In fact he was the first to relate the action of a correspondence Γ ⊂ X × X on the Chow group of 0-cycles on a smooth projective surface X to the cohomology class of Γ in H(X × X,C). In [?] (see also [?, 3.3]) he made a claim that in its original form is not correct but can be easily restated as what is now known as Bloch’s conjecture.