Main objects of the chapter. The present chapter has the main purposes of establishing the definition of that most important number called the genus of a curve, which is invariant in all birational transformations of the curve; and of obtaining the fundamental results of the theory of linear series of sets of points upon the curve. Both these are intimately related to the theory of the rational functions existing on the curve, and to the theory of the algebraic integrals belonging thereto, especially the so-called everywhere finite integrals. But it is desired that the account given shall be logically sound, shall be brief, and shall be simple; these conditions seem best satisfied by employing, together, ideas from several different modes of approach, due mainly to Abel, Riemann, Weierstrass, and Brill and Noether. A further method, developed by Kronecker and by Dedekind, in extension of the arithmetical theory of integer numbers, is explained at length in a subsequent chapter (VII). Unless the contrary is stated, the curve considered is supposed to be a plane curve.

We distinguish provisionally between what we may call the integral genus, and the arithmetical genus; the former is easy to explain in general, the latter can be computed for a curve of sufficient simplicity; it is part of our task to shew that these are the same.