By Sydney H. Gould
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D ŒP C ŒQ C ŒR Hence ŒP C ŒQ defined by (8). k/, then the group law we get is just the translate of that given by O. Perfect base fields We now allow k to be a perfect field; for example, k could be any field of characteristic zero or any finite field. All curves over k will be assumed to be nonsingular and absolutely irreducible. f / D kŒx; y; 36 CHAPTER I. PLANE CURVES and call it the ring of regular functions on C . k/ may be empty. K/ ! K for each field K k, and kŒC can be identified with the ring of families of such functions, compatible with inclusions K L, defined by polynomials in the coordinate functions x and y.
Z/ nomials. C/ (even holomorphic) that is not rational. X; Y; Z/. As for meromorphic functions on Riemann surfaces, we try to understand the rational functions on a C in terms of their zeros and poles. k/. k/: P The degree of D is nP . C /: X X nP ŒP mP ŒP ” nP mP for all P: P In particular, nP ŒP 0 if and only if all the np are nonnegative. Let ' be a nonzero rational function on C . X;Y;Z/ of two polynomials of the same degree, say m, such that F doesn’t divide H . Because ' ¤ 0, F doesn’t divide G either (recall that kŒX; Y; Z is a unique factorization domain).
Thus P1 has genus 0. W D/ for any “canonical” divisor W ; moreover, W has degree 2g 2, and so equality holds in (6) if deg D > 2g 2. The canonical divisors are exactly the divisors of differentials of K over k. See Fulton 1969, Chap. 8, 5,6. D/ is defined to be the dimension of the space of meromorphic functions ' with div'CD 0. For the Riemann surface of a nonsingular projective curve over C, the canonical divisors in the two senses coincide, and so the genuses in the two senses also coincide.