SPECIAL DIVISOR S O N ALGEBRAIC CURVE S 11 In coordinat e fre e notatio n thi s is 4L:C^?{H\L)*) where, for a general poin t pGC, 0Z,(P) = {hyperplan e o f sections s with s(p) = 0}. The base locus B of the linear syste m \L\ is defined t o be the finite se t of points /?GC where s(p) = 0 for all sections s G H°(L). I n case \L\ is base-point-free w e have *2«V(D) = L, while in general the relation *i(0 p r (l)) = I(-2O is valid. Since the sections s t are linearly independent th e image curve (j L (C) C ?r i s nonde- generate—i.e., it doe s not lie in a Pr _ 1 . Consequently , there are (r + 1 ) points of C whose images span ? r , an d it follow s tha t an y general set of q r 4- 1 points on C are in general position on 0L(C). Thi s is equivalent t o the assertion in the proof o f (1.19), that an y r general point s on C impose independent condition s on H°(L). A s we shall presentl y see , of great interes t ar e the configurations o f points tha t ar e in special position (tri-chords , etc.) on Suppose we now take L = K to be the canonical line bundle. Fo r any points p, qEC the Riemann-Roc h theore m (1.25 ) gives l(p) = h°(K(-p))-g + 2, (1.31) l(p + q) = h°(K(-p-q))-g+l. Now l(p) = 2 for some point p if, and only if, C is the Riemann spher e P 1 and l(p + q) = 2 for som e pair of points p, q exactly whe n ther e is a 2-to-l ma p (1.32) C-Z+V 1 i.e., whe n C is hyperelliptic. I t follows fro m (1.14 ) and (1.31) that i f g 1 then h°(K(rP)) = g - \ , and if moreover C is nonhyperelliptic the n h°(K(-p-q))=g-2. The first o f these say s that th e canonical linear syste m \K\ is always base-point-free, an d the second that i f C is nonhyperelliptic the n the canonical mappin g is a 1-to-l embedding . I t is well known tha t fo r hyperelliptic curve s
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