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By Chern S., Osserman R. (eds.)

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Is s u r j e c t i v e (a,b,c) c ker(¢+p) , so t h a t Moreover ker2~x is a f i n i t e i) , t h e n we o b t a i n 2~ E k e r { ~ j ( C~) -T*) follow The m o r p h i s m zero map, iii) by ii). implies group Take ¢(a,b),p(c)ep(ker2¢x). of o r d e r d (d <_ 4) by therefore ¢(da,db) = p(dc) Since ker # c A 2 ( S ) 4 which is a f i n i t e =Alb(S)tors c e Pd2 Thus @ AI(s)2 group proved = O we in v i e w in [R], ker(~+p) find (a,b) of the ~ A 2 (S)4d @ A 1 (S)2d isomorphism while p(dc) is a f i n i t e group.

Automorphisms Soviet MANIN to the Math. : of three-dimensional 13 ( 1 9 8 0 ) , 815-868. Three-dimensional Lflroth problem, Math. quartics and USSR S b o r n i k 141-166. vt ELI J. LUROTH Annalen : Beweis 9 (1876), eines Satzes [M1] D. MUMFORD : Abelian EM2] D. MUMFORD : Prym varieties [R] Academic Press~ L. ROTH : Algebraic SARKISOV Mat. P. SERRE ET] : London Math. B. SEGRE : TYRELL Soc. Ergebnisse 34 ( 1 9 5 9 ) , Variazione Ann. Mat. : manifolds, : 57 ( 1 9 6 1 ) , Math. (1970). analysis, Appl.

We c a n w r i t e = %, f ~ + s,~, Then, The morphism E A 2 (V) @ A 1 (V). Take ~*y = f*a + f * b . i , T . 2 ). 1. f,n y = 8. we find so we get an e m b e d d i n g S s,~* A 1 (V). 2 (multiplication by N ~*A is c o m m u t a t i v e (') A2(U) with (U) exact W* > A 2 (U) + O n~ row and g i v e s = ~,f~, A 2 ( T ) ~ ,~ s , , A 1 (V) an i s o m o r p h i s m = f*~,A2(T)~i,~*AI(v). 4B Now we compute cycle i,~*AI(v). take the i*f*n,x ~,~* = "2 o n e has i,(2x) The group element Since = n*w,w*x' = i,(i*f*w,x) shown so we t o be o f t h e that x E w*AI(v) for such i'f* = 2x.

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