Introduction. We can state the problem of the moduli of algebraic curves in the following way. Consider all irreducible algebraic curves, defined over a ground field k, of a fixed genus g. Can we, in some natural fashion, associate to each birational class of these curves a point of an algebraic variety N, defined over k, in such a way that for distinct birational classes we obtain distinct points? We impose the further condition that those points of N which do not represent any of the above birational classes form a proper algebraic subvariety of N, defined over k. We shall consider this problem only for the special case of hyperelliptic curves, van der Waerden [3] has attacked the general problem. There is, however, an important gap in the proof of Theorem 7 [p. 698, loc. cit.j. We shall use the notation of van der Waerden in pointing this out. I, is an irreducible algebraic system of regular plane curves of genus g [1J4, p. 698, loc. cit.]. The algebraic correspondence 93, transforms 7, onto itself. The difficulty arises from the fact that, as far as we know, an irreducible regular plane curve V of genus g may be fundamental for the correspondence 93,. This situation has not been ruled out since V may correspond, under 93,., to curves in I, which do not belong to ^Jr. These curves could not be irreducible regular curves of genus g. Such "limiting curves" can be ignored when considering I, as the domain of 93" but not when considering it as the range of this correspondence. If T is fundamental for 93, then the birational class of r would be represented by infinitely many points on the moduli-variety My. In the special case of hyperelliptic curves the objects involved are much simpler, hence the corresponding difficulty can be handled easily. We shall assume the basic concepts and definitions of the theory of algebraic varieties. Our terminology conforms to that of Zariski [6]. We suppose that a universal domain ft has been fixed, once and for all. All quantities which occur will lie in ft, and any ground field k that we choose will be such that ft has infinite transcendence degree over it.
