On multiple curves. I

Extract

Let γ be an irreducible variety of dimension d, and let I_n denote an involution of order n on Γ. The image of I_n is a variety C, also irreducible and of dimension d, and Γ is said to be a multiple of C. Among the possible multiples of a given variety C there is a class of particular interest; this arises when the involution I_n possesses the following properties:

(1) it is unramified, that is, in every set of the involution the n points are all distinct;

(2) it is generated by an Abelian group G of order n of birational transformations of Γ into itself. Given any point P of Γ the n transformations of G transform P into n points P₁ = P, P₂, …, P_n which constitute a set of I_n. The two conditions together imply that the function field κ of Γ is an unramified Abelian Galois extension of the function field k of C, and this shows that the study of the particular class of multiple varieties—which we may call normal multiple varieties when it is necessary to give a name to them—is analogous to the study of a well-known branch of the theory of numbers.