12 P.A.GRIFFITH S * = /*(O pl te-i)), and th e canonica l ma p factor s a s n \ / \ ^ A P1 P * " 1 7 r ° A where t// : P 1 P g_1 i s the standar d rationa l norma l curv e o f degre e g - 1 . In the remainde r o f thi s sectio n an d i n § 2 w e shall assume tha t C is nonhyperelliptic, and shal l identify th e curv e wit h it s canonica l image. Explicitly , choosing a basi s cjj,... , OJ for theholomorphi c differential s o n C, the homogeneous coordinates of a point p is written PK(P)= I w iO0 •••* 0 ? ) ] . If Pj, ... , P j ar e point s o n C , we shall denot e b y Pj pd th e linea r spa n o f thes e point s in the spac e P^ - 1 o f th e canonica l curve . I t i s clear what thi s mean s if th e p( ar e distinct , and if , e.g. , p1 = = p k = p, the n w e take the spa n o f pk + 1 , ..., p d togethe r wit h th e fcth osculating spac e to th e canonica l curv e at p. Equivalently , if D = Pl + '-' + p d , then th e linear spa n is the intersection o f al l hyperplanes {a = 0}wher e c o €H°(K(-D)). Fro m thi s it follow s that dim D = g - 1 - i(D) . On the othe r han d th e Riemann-Roc h theore m (1.18 ) gives r(D) = d~g + i(D). Eliminating the inde x o f specialit y betwee n thes e tw o equation s w e arrive a t th e Geometric form of the Riemann-Roch theorem. (1.33) dim. D = d - l -r(D). In words, On a nonhyperelliptic curve C a divisor D moves in a linear system whose dimension is exactly proportional to the failure of the points of D to be in general position on the canonical curve. Using (1.33) w e may, following th e classica l paper [3 ] o f Bril l and Noether , give the expected dimensio n coun t fo r th e varietie s o f specia l divisors . Fo r thi s we denote b y C d the se t o f al l effectiv e divisor s D = px + + p d o f degre e d. I t i s well known tha t thi s
Previous Page Next Page