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
A line bundle by its global h1(Ft ) = O. p : C M PicO(c)--~ Put threefolds which shows curve of genus For any point ~ICXt is generated strange hyper~lliptic There exists such that restriction of analytic First we give an example of degree . threefolds. the structure may have be a the double Then . threefolds : C--~ p1 Put of anal2tic C be qTt is over U. and we have F~ 2 analytic ramified family at the divisor (s) " 30 where is a generic s element of HO(c, L2k). ~ K C = f*L g-1+k 9 Put Then we have ~ Mt = Mt ~ C .
Then (this is the classical d = 3 , cubic surfaces in I-KsI = (or proper) transform of the ~bics of strict of fixed degree smooth projectively normal are Del Pezzo surfaces definition of Del Pezzo surfaces). e. ,ag_ d 9 In particular: ~or d = 2 the for d ~ 2 %1_KS ! the system has no base points. I-Ks1 gives a double covering of locus a quartic curve). ,a 8 in the plane), and we get a rational map to r = 1 and canonical embedding d e 3 the Del Pezzo surfaces have, using the anti- If_Ks I , lines (for r = 3 but not in the anticanonical ~!
44 The fact iv) we shall often shortly (I) V = v23g-2 c ~ g + l EXAMPLES. 1) In case V ~ ~3 (g->3) . we have g = 89 2) v ~ V(2) c P 4 ~ g = 28 ; 3) V ~ V(3) c 4) v ~ V(4) c P 4 ~ g = 3. usual embedding Let H v c ~N and Then H' v + 1 = 33 ; ~ 4 ~ g = 13 ; Also note that in Examples is not via There is the following PROPOSITION indicate by w r i t i n g I-KVI I), 2) and 3) the ! converse: 2. H' is a canonically embedded curve of genus is a Fano 3-fold of the ~rincival series embedded by means of T-~I PROOF.