Geometry of the Plane Cremona Maps PDF
Preview Geometry of the Plane Cremona Maps
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Morel, Cachan F.Takens,Groningen B.Teissier,Paris Springer Berlin Heidelberg New York Barcelona HongKong London Milan Paris Tokyo Maria Alberich-Carramifiana Geometry of the Plane Cremona Maps 4 10, Springer Author MariaAlberich-Carramifiana Departamentod'AlgebraiGeometria GranViadelegCortsCatalanes,585 08007Barcelona,Spain e-mail:[email protected] Cataloging-in-PublicationDataappliedfor. DieDeutscheBibliothek-CIP-Einheitsaufhahme Alberich-Caffamiiiana,Maria: Geometryof,theplanecremonamap/MariaAlberich-Carramifiana.-Berlin; Heidelberg;NewYork;Barcelona;HongKong;London;Milan;Paris; Tokyo:Springer,2002 (Lecturenotesinmathematics; 1769) ISBN3-540-42816-X MathematicsSubjectClassification(2000).:14EO5,14EO7 ISSN00754434 ISBN3-540-42816-XSpringer-VerlagBerfinHeidelbergNewYork Thisworkissubjectto copyright.Allrights arereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstoragein,databanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965, initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer-Verlag.Violationsare liableforprosecutionundertheGermanCopyrightLaw. Springer-VerlagBerlinHeidelbergNewYorkamemberofBertelsmannSpringer Science+BusinessMediaGmbH http://www.springer.de 0Springer-VerlagBerlinHeidelberg2002 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:Camera-readyTEXoutputbytheauthor SPIN:10856615 41/3143/LK-543210-Printedonacid-freepaper To Albert and Antoni Preface The basic theory of plane birational maps was first stated by L. Cremona in his two memoirs [14] (1863), [15] (1865), and henceforth plane birational maps are known as plane Cremona maps. Other geometers soon brought substantial additions. Historical r6sum6s can be found in [34] XVII and [12]. Tostartwith,letusexplaintheconnectionbetweenplaneCremonamaps, linear systems and clusters of points. To a given plane Cremona map 4i ]?12 _-., ]?22 we associate the linear system (net) W in p21 without fixed part whichistheinverseimageby!PofthenetoflinesofP'. ThenetW,determines 2 the map 0 up to a projectivity of]?22: there is aprojectivitYU : ]?22 So that uoit is equal to the map ]?22 --4 V, with V the projective space dual to W, which sends x E ]?2 to the hyperplane in W consisting ofthe divisors passing through x. Observe that a point x E ]?2 is fundamental for 0 (i.e., x belongstotheclosedsubset where!Pcannotbedefinedasamorphism) ifand only if x is a base point ofthe net X Consider the set ofbase points ofX not onlythefundamentalpoints of0, which areproperpointsin ]?2 but also , the infinitely near base points, which are proper points in a suitable surface obtained from p2 by successive blowing-ups. They form a weighted cluster IC = (K,p), where K axe the base points ofW and p assigns to each p E K the multiplicity at p of generic curves in X The pair IC will be called the weighted cluster of base points of 0. This book studies the plane Cremona maps from the viewpoint ofthe geometry of their weighted clusters of base points. Accounts of the classical development'of the theory of plane Cremona maps axe given by Hudson [34], Godeaux [28], [29], Coble [11], Enriques- Chisini [26], Semple-Rqth [47], Coolidge [13]. All ofthem deal systematically only with plane Cremona maps whose base points are all proper; infinitely near base points do appear just in examples. By contrast, infinitely near points already appear in the easiest examples of plane Cremona maps, as for instance the quadratic ones. Besides the well-known ordinary quadratic transformationthere aretwoother types (special quadratic transformations) which have, respectively, one and two infinitely near base points (see section 2.8). Thestructureofthegroup ofplane Cremonatransformations (plane Cre- mona group) has received a good deal of attention by modem literature:
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