By E. Ballico, Ph. Ellia (auth.), Ciro Ciliberto, Franco Ghione, Ferruccio Orecchia (eds.)
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Extra info for Algebraic Geometry — Open Problems: Proceedings of the Conference Held in Ravello, May 31 – June 5, 1982
Ainsi projectif. dans de c e n t r e courbe d~finie 7~ £ c o e f f i c i e n t s une ~ ~4) f : Q£(X)~g dans une quadrique restric- de q u a d r i q u e s des de ~ x G ~ que la ~6. r~seau (isomorphe la ; unirationnelle. Choisissons sous-vari~t~ est obtenue en c onique s . de £ X est Notons W le que vari~t~ 5 . imm~diatement de s o r t e la droite 2-plans form~e fait des de Q ~ ( X ) sous-vari~t~ gne par <~x> plan singuli~re si C est l'ensemble des par l'annulation lin~aires, d'un et par obtenue en ~ c l a t a n t ~ d a n s X~ e t des couples (q~x) que <~x>cq engendr~ par £ et tels x ; ce s ym bole S la a un s e n s On a u n d i a g r a m m e avec a(q,x)= (q~<~,x>) Q~(x) II est imm~diat que P2 e t a sous- (on d~si- S X£ de q est 7.
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. i,T. is d i v i s i b l e , y ~ w*AI(v) an element all x = ~*x' ~ A 1 (T).
Of the Our p u r p o s e A2(X) in terms is to of the • AI(s) . 10) = ~(f,~l) and Therefore nonsingular divisor system f I~ We say that be seen of L look at the e m b e d d i n g -i TX Tx we can find an i r r e d u c i b l e belonging (i)~ ~ X ' to the c o m p l e t e linear by use of B e r t i n i ' s is a t a u t o l o g i c a l in the sequel X ÷ ~(f,~xl~i) our a r g u m e n t s divisor of X do not d e p e n d Theorem. 4). 1. Y = f-l(c) U = X\Y Let in this i be the d i s c r i m i n a n t be the ruled We d e n o t e U , by C by surface of the s i n g u l a r s e c t i o n we w a n t T to d e s c r i b e the r e s t r i c t i o n the i n c l u s i o n T + U curve of of Tx and we set X and let fibres.
Algebraic Geometry — Open Problems: Proceedings of the Conference Held in Ravello, May 31 – June 5, 1982 by E. Ballico, Ph. Ellia (auth.), Ciro Ciliberto, Franco Ghione, Ferruccio Orecchia (eds.)