Algebraic Geometry Bucharest 1982: Proceedings of the by Lucian BĂdescu (auth.), Lucian Bădescu, Dorin Popescu (eds.)

By Lucian BĂdescu (auth.), Lucian Bădescu, Dorin Popescu (eds.)

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Additional info for Algebraic Geometry Bucharest 1982: Proceedings of the International Conference held in Bucharest, Romania, August 2–7, 1982

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3) the surfaces d=n and [9]. to see that surfaces surfaces (see [i0] or our Lemmas p r o b l e m of e n n u m e r a t i n g surface It is well A=d-n+l is classical, m u s t De ruled surfaces space pn. 6)). for w h i c h This suggests the n u m b e r d-2n+2 the is small. In 52 of this p a p e r we e n n u m e r a t e ces with d=2n-l, d=2n and d=2n+l whose hyperplane section these surfaces surfaces. dy exis t n o n - r u l e d irregular gular surface (any a b e l i a n In §3 we prove (where n~4, is a regular are regular all types surfaces It turns out that all that for d=2n+2 whose surfa- n~5 and n~6 respectively) divisor.

B) X=BL2(Y) pn+2 (II 4) a) X is minimal elliptic with ~=i, pg=2, q=0. The elliptic fibration is given by the canonical map fK:X--~ I. The fibres of fK have degree 4 and are not multiple. b) X=BLI(Y) where Y is of type 13 in ~n+l , 56 (II6) x is Castelnuovo. Proof. 5) we have HK even ~6 and Pa(X)=HK/2. Suppose HK=0. with M=0 and Pa=0, Suppose HK=2. Since tl2Kl~0 we get 12K~0. So X must be minimal hence X is Enriques. Since Pa(X)=l there exists an effective canonical divisor K. There are two possibilities: l) K is a smooth conic.

Let Y~ = Y). O >pn(s) be the closed embedding given by the complete li- near system IOy(s)~ (n(1) - n). Then the cone X = C(Y,O. (s)) lles in pn(s)+l. x n(s)+l Step I. Every small (embedded) deformation of X in P is again a cone of type C(¥t,OYt(s)) , where Yt is a complete intersection in pn of type (dl,. • • ,dr). ~oo~ o~ step 1. ~et {Xt~ t ~ ~ be an ( a l ~ b r a i o , subeohemes of pn(a)+l parametrized by B, such that X point o 6 B . embedded) f~ily of c l o s e d = X for a k-rational o Let H - pn(S) be the hyperplane at infinity of pn(s)+l.

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