Table Of ContentLecture Notes in Mathematics
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Continued on inside back-cover
Lecture Notes in Mathematics 1769
Editors:
J.-M. Morel, Cachan
F.Takens,Groningen
B.Teissier,Paris
Springer
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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]
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Alberich-Caffamiiiana,Maria:
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Tokyo:Springer,2002
(Lecturenotesinmathematics; 1769)
ISBN3-540-42816-X
MathematicsSubjectClassification(2000).:14EO5,14EO7
ISSN00754434
ISBN3-540-42816-XSpringer-VerlagBerfinHeidelbergNewYork
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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:
