# 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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TERMINOLOGY A surface complex field. HI(~(D))=0. curve. will always A divisor mean D on a surface A geometrically A ruled a smooth p r o j e c t i v e surface m e a n s ruled surface over is said to be regular surface means a surface w h i c h a pl-bundle is b i r a t i o n a l l y the if over a isomor- 48 phic to a geometrically ruled surface. If a surface X is obtained a surface Y by blowing up m distinct points, we write X=BLm(Y). ly normal surfaces Xcp n will always be supposed nondegenerate contained in any hyperplane).

The bundle such that is, again, tautological p~(F'~T~-I)=o. Consequently, there is a section msH° (X×M (d,r), F'~TS-I) with non-zero image through the identification: -i P*(F'oT2-1)t/mtP*(F'~T'~I)t=H°(F'~T2-1)/mt(F'~T ~ ))=C- The corresponding map TS-F" is injective modulo m t for each tsM(d,r). Then T~-F is and injective and its cokernel C is flat over M(d,r). We have the exact sequences: O-L2-F'/mtF'-C/mtC-O 0-L2-Et-Iy~L1-0, where q(t)=z=(Li,L2,Y), Therefore moreover Ll=0x(-Co-fo)~n'(Ll) C/mtC=Iy®iI.

M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966) 129-136. 3. L. B~descu, On ample divisors, Nagoya Math. J. 86 (1982) 155-171. 4. L. B~descu, On ample divlsors:II, Proceedings of the Week of Algebraic Geometry,Bucharest 198o , Teubner-Texte Math. Band 40, Leipzig 1981. 5. L. B~descu, The projective plane blown up at a point as an ample divisor, Atti Accad. Ligure Scienze Lettere, 38 (1981) 3-7. 6. C. B&nic~, Sum lee fibres infinitesimalles d'un morphisme propre d'espaces complexes, S@minaire F.