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Published online by Cambridge University Press: 24 October 2008
A curve Γ has been defined as a normal multiple of a curve C if it contains an involution In, of order n, which has the properties: (1) the sets of In are in (1–1) correspondence with the points of C; (2) each set In consists of n points which are distinct in the birational sense; (3) In is generated by an Abelian group G of order n of birational transformations of Γ into itself. We shall denote the field of complex numbers by k, the function field of C by k(C), and the function field of Γ by k(Γ). Conditions (1) and (3) imply that k(Γ) is a commutative normal (i.e. Galois) extension of k(C). The object of this note is to show how, given the curve C and the Abelian group G of order n, we can construct a curve Γ with the properties (1), (2), (3).
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