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

ISBN-10: 0387115870

ISBN-13: 9780387115870

ISBN-10: 3540115870

ISBN-13: 9783540115878

ISBN-10: 3540393420

ISBN-13: 9783540393429

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Additional resources 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

Example text

Threefolds It is conjectured fibres are of a surface S to S ~ I . to A s h i k a g a [ ~ For algebraic -c~ . 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 .

1 also shows that an analytic surfaces. 2, 2) 4) is k = 0 . 1 shows that there exists an analytic : M--eA(M) M I, then the Albanese mapping is surjective. 3. with ping manifold, equivalent fibres and a general equivalent whose fibre If, moreover, If the Albanese mapping is not surjective, is a non-singular 2) Mx K = O. image of K ~ l e r is bimeromorphically over A(M) curve. a bimeromorphic and over t(M) = I, then any smooth fibre of mapping 2) ~ibre bundle to a K 3 surface. Finally we consider ([~] instanton ) the case where threefolds have negative Kodaira dimension.

7) = -] is not numerical an4 there exists ~: V - D ~ Y - ~(D) (always by ibid, Theorem 57 the following R and = ! an irreducible dim ~(D) ~ I . Moreover cases are possible: over ~(D) . Moreover we have effective . -KV) . n) and = -l ~2 (-I) . and, again using = 2 . D) = -] is a double point of throuqh e) ~(D) = Q in is I0 ~3 and the vertex"~ x PI and OD(D) ~ 0(-l,-l) and, always proceeding a "line" from above by in v , D ~ 0D(n ) ~ O p we have a quadruple In all cases [s ] c R point of ~0 ' we qet a),-,e) using this, and the results by an irreducible (-I)| .

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


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