Download Algebraic Threefolds: Proceedings of the 2nd1981 Session of by Kenji Ueno (auth.), Alberto Conte (eds.) PDF

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

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Extra 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

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H Note that due to our assumptions This step is now entirely similar . C is a smooth irreducible curve. to Step I. STEP III. The case of curves. r = 1 : Then C deg XIH I = 1 implies and hence I~! is a curve such that that the restriction gives the canonical is now the hypervlane SnH0(C,0c(1)) ~ TrcIH I = K C embedding. system on H0(C,0c(n)) of (see Lemma 5). Moreover %IH [ to C has also degree Using this embedding C ), the assertion (and hence becomes for the (now) cannonical this is true since by the Woether-Enriques-Petri 1 embedded theorem curve; C is projectively C on normal.

Is torsion free Cand hence H unique). PROOF. ~#e have seen H2(V,~) r K3-surface~ Pic(V) H2(H,~) = H2(V,~) (Corollary 4 of Lemma I); by Lefschetz and the latter is torsion free since H is Del Pezzo or 48 COROLLARY 4. TrHO ]nE! (n~l) is complete . PROOF. 13. ----+ O v ( n H ) HI(V,0v((n-I)H) = 0 ---+ 0H0(nH) n -> 1 if ---+ (Lermna 6). THE DEGREE OF A FArO VARIETY Put d = d(v) = H 3 = - ~ i r This integer is called the de~ree of V . Wow consider the rational map (2) %I HI: V ...... '~at are the possible values of deg I]H I ?

In the lower row. --~ . H0(V,nH) . 12), hence the surjectivity 52 STEP II. H Note that due to our assumptions This step is now entirely similar . C is a smooth irreducible curve. to Step I. STEP III. The case of curves. r = 1 : Then C deg XIH I = 1 implies and hence I~! is a curve such that that the restriction gives the canonical is now the hypervlane SnH0(C,0c(1)) ~ TrcIH I = K C embedding. system on H0(C,0c(n)) of (see Lemma 5). Moreover %IH [ to C has also degree Using this embedding C ), the assertion (and hence becomes for the (now) cannonical this is true since by the Woether-Enriques-Petri 1 embedded theorem curve; C is projectively C on normal.

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