Download e-book for iPad: Algebraic Threefolds: Proceedings of the 2nd1981 Session of by Kenji Ueno (auth.), Alberto Conte (eds.)

By Kenji Ueno (auth.), Alberto Conte (eds.)

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Additional info for Algebraic Threefolds: Proceedings of the 2nd1981 Session of the Centro Internazionale Matematico Estivo (C.I.M.E.), Held at Varenna, Italy, June 15–23, 1981

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O. 9 we have the same argument~we for a general g1(V) is not ~urjective <(v w) t ~(v) z fibration. dim W~__ 2, that and curves, W is biration- From the universal we infer that A(V). W is mapping is birationally This shows that ~ has con- g1(V) ~-~ 2. Now let us consider the case g1(V)~ 3. As ~ is surjective, m Pg(V) = I. First assume We let i~__i niEi that eodim~(Ei)~ (Ei)), i=I A(V/ A(V)' = W - the covering ~ groups ~ : V' : V--~ A(V). there is a natural a bimeromorphic 9 o an abelian of A(V) ~ A(V)' V mapping.

10). LE}@~A 5. F d~f TrF]_~71 b) the genus = Kr . g(F) = -~(~T) 3 + ! (g=g(F) is called the genus o~ the Fano variety V). PROOF. By the adjunction formula we have K F = F XI. F . 43 Using the exact sequence o the fact that --- q(Y) = 0 Ox hence the anticanonieal on Ox(r ) --+ r~(r)i --~ 0 , and Lemma 4 we get H0(v,-r~7) KF --* --+> system ~~ I~7 T --+> H~ cuts out the complete canonical system F ; this ~ives a). As to b) this follows now by Lemma 2 since we have (as follows easily from the above used exact sequences) g(r) = dim H~ that = dim HO(V,0v(-K V) - 2 = - 89 + I.

Using the theory infer that we can find a finite ramified is birationally k= that the Kodaira dimension in this case there is a finite ramified A that V ~ W is b i r a t i o n a l l y equivalent W ruled. If dim W =I, then general fibres ruledsurfaces. with is uniruled. 1. we consider ~ = -~ Let C Then we have K C = L g-1. be the line bundle Ft = F~[t~ Ft 2m - g + I, covering. 0 on C p,~ is the natural of the origin phic of ~ ~t = ~ p-1(t ), zero on II , ~ C considered for each point = 0 I ' t ~ U .

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