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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**

Hence, for explicit reference, we have: COY OL~~A~9/ 2. H I has no base points and This is in narticular the case for kiwi is birational then r > 2 and IHI is very ample. r = 2 , d > 2 . COROLLARY 3. Let the assumptions be as in Lerm~a 11. Then ~rojectiVely normal. In particular: if then v is projectively normal. HI: v ~ V' = IPN with is embedded by means of v' IHI 54 PROOF. ~e have now normal; the projective Y~v I. ]7. EQI~TIONS For use later normality ~r' is smooth and hence certainly follows now from (5).

Nowever reduced . nO) 0D (dO'D) = -I D , then description ~ence 0D(D) ~ 0 I . surface Take for Y D . -~7) in point on g e 4 from (7). Hence . dim W > ! , proving Lerm~a ]3 and comnletin ~ Step 2. ST~P 3: I-~TI has no fixed components irreducible and the general D e I-Kvl is and has at most double points. For the proof we refer now to the original Here we give only some indications. I-~TI = ID0+DI irreducible. 400-403. l i up centers, and also (~ strict transform, ni->2) with a cur,re or (~) a point).

7. EQI~TIONS For use later normality ~r' is smooth and hence certainly follows now from (5). H O Let 2, hence 12. LE~IA Let by Corollary v ~ w, sm-I~0(Y/)x(H)) IHI . Let H 0 . Assume: ---+> H0(X,0x((m-I)H)) I(X) , resp. I(Y) , be the ideals defined by and let F ~ I (Y) m F*m ~ Im(X) X (m~2) and Y . resmectively, (the homogeneous part of degree m ). Then there exists m such that F*m O H0 = Fm . Moreover in case is uniquely d~tMrmined by ~E~ARK. dim X ~ 2 . Let Note that m = 2 the F*m F m (*) certainly is fulfilled if X is projectively normal.