6 P. A. GRIFFITH S of holomorphic differential s c o that vanis h on //—i.e., that satisf y (CJ ) D—then the numbe r of independent equation s (1.13 ) is g - /(D) . Thi s gives, finally, th e famou s Riemann-Roch theorem: (1.14) l(D) = d-g+l +i(D), derived here fro m th e existenc e theore m (1.4 ) plus elementary topologica l observations . As a corollary w e may complemen t th e uppe r boun d (1.1 ) to rea d (1.15) d-g+ 1 l(D)d + 1 . Moreover, since for an y holomorphic differentia ] c o deg(co) = 2 # - 2 , we have that i(D) = 0 in cas e de g D 2g - 1 , which whe n combine d wit h th e Riemann - Roch theore m give s the answe r (1.16) l(D) = d-g+ 1 if de g D 7g - \ to ou r proble m fo r divisor s of larg e degree . However, as is well known, in geometri c question s it i s most ofte n th e divisor s o f low degree that ar e interesting, an d a t thi s stag e al l tha t w e ca n d o easil y i s t o giv e 1(D) for generic D, a s follows : (1.17) For a generic divisor D of degree d we have ifdg-l, r " [g-d+l if 1(D) = ifgd2g-2. PROOF. Fo r a generic D w e have (cf. th e proo f o f (1.19 ) below) ! g - d if dg, 0 ifdg+l. Now use the Riemann-Roc h theore m (1.14) . (b) I n this section w e shall introduce som e furthe r algebro-geometri c terminolog y an d then begi n ou r discussio n o f specia l divisors . Two divisor s D and D' o f the sam e degre e d ar e sai d to b e linearly equivalent, writte n D ~ £' , if ther e i s a meromorphic functio n f on C whose diviso r is (f) = D'-D. The se t o f al l divisor s D' tha t ar e linearly equivalen t t o a fixed D is a projective spac e calle d the complet e linea r system an d denote d b y \D\. I n fact , usin g our previou s notation i t i s clear that th e divisor s in \D\ are in a natural one-to-on e correspondenc e wit h th e lines in the vecto r spac e L(D) so tha t \D\ = P(Lp)) .
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