Correspondences with unequal valencies

Extract

In an earlier note (3) we have shown by examples that it is possible to have an irreducible non-degenerate point-point correspondence on an algebraic surface such that the correspondence and its inverse have arbitrary unequal negative valencies. We have also remarked that, by multiplying such a correspondence by one with equal positive valencies, we can find correspondences with unequal positive valencies. In (4) we have given reasons, falling short of complete proof, which suggest that, on any surface whose intersection group is cyclic, correspondences with unequal Albanese valencies exist. But the only proofs of the existence of such correspondences are the examples due to the author, Severi and Todd in (3) which all refer to ruled surfaces. Severi has rightly pointed out that these examples are not entirely satisfactory, since, as ruled surfaces have zero geometric genus, there is a confusion between valency correspondences in Severi's and in Albanese's senses, whereas the theoretical arguments of (4) apply only to Albanese valency correspondences. In this note we remark that on hyperelliptic surfaces there exist correspondences with valencies γ and γ₃ in the two directions for any integer γ, and that unless |γ| = 1, in which case the two valencies are the same, these correspondences cannot have a Severi valency in either direction. This result is an immediate consequence of some classical results on Abelian varieties (c.f.(2)), but the application seems of sufficient interest to note explicitly. We remark that we are still not able to find examples of correspondences which are irreducible and non-degenerate and whose valencies in the two directions are of opposite sign; it seems a plausible conjecture that such correspondences do not exist.