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By Smirnov V. A.

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University Contributions automorphisms 17 ( 1 9 8 1 ) , K. TIMMERSCHEIDT J. Oxford I. threefolds, On t h e J. rational [Ty] : Curven, (1974). Birational V. algebrica, rationale Berlin-Heidelberg-New IS] [Sg] varieties, New Y o r k Springer-Verlag, ~ber 163-165. Annalen of 258 threefold, in geometria 149-186. threedimensional (1982), Proc. 267-275. Cambridge Phil. CONIC BUNDLES ON N O N - R A T I O N A L SURFACES by M. B e l t r a m e t t i and P. F r a n c i a (*) Contents Introduction §i. Notations, definitions, and p r e l i m i n a r y §2.

2. conic fibres. that Xs f -I e(s) (s), As to exists = f*2a + f * 2 b - i , T x = f * 2 a = O. Hence, as b e f o r e = 2 a . f , i , T x = 2 a - ~ , T x = 4a, proved in [R] the g r o u p the discriminant surface a canonical is t h e s i n g u l a r for the b l o w i n g - u p divisor Then a = O we f i n d = f * 2 a - i , T x. X a n d the r u l e d There such E ker~ A2(S)4 AI(s) 2 to A I b ( S ) 4 • P i c ° ( S ) 2 , so it is f i n i t e . We return bundle f,f (a,b) contained so that O = i*f*2a is i s o m o r p h i c Since + f * b ' i , T x) is a ~ A 2 (S) 4.

On c h e r c h e r a I1 faut On e s t ne serait-ce maintenant description n'est [A-M]. lieu Nous a l l o n s une cette un description ~ cette que une Pour donc pour loin que cela dis- de s a v o i r pour classe prou- a montrer les vari6- de v a r i 6 t 6 s pour existe. t 6. FI BRES EN CONIQUES. D~finition : existe une fibres sont I1 tion On d i t surface des est coni~ues facile rationnelle sur (ou, let de c o u r b e s un qui fibr~ Sous qu'il - - si en les - sip est pest On d i t cependant le Proposition surface est Soit rationnelle la tout r~sultat la une s'il dont une les une une applica- courbe rationnelle birationnellement radu ~quivalente classe de vari~t~s rationnelles.

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