The Enumerative Theory of Conics after Halphen
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1196 Eduardo Casas-Alvero Sebastian Xamb6-Descamps
The Enumerative Theory of Conics after Halphen
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Authors
Eduardo Casas-Alvero Sebastian Xamb6-Descamps Department of Geometry and Topology, University of Barcelona Gran Via 585, Barcelona 08007, Spain
Mathematics Subject Classification (1980): 14N10; 14C17; 51N15
ISBN 3-540-16495-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-16495-2 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Introduction. This
work
Halphen's on
deals
on
the
one
hand
contribution
to
the
subject
with
understanding theory
of enumerative
the
contents
and
of conics,
the other with extending his theory to conditions of any codimension.
reader interested in
the history of this subject may profit from
of
The
the beautiful
paper of Kleiman [K.2]. In
the
enumerative
approaches,
namely
theory
those
of
COnICS
associated
to
there
De
have
been
Jonquieres ,
to
basically Chasles,
three
and
to
Halphen {see the works of these authors referred to in the references, as well as
[K.2] and the references
therein}.
in
that
computations
they
correspond
to
Conceptually
the
and of the variety of complete conics, respectively. obtained
with
these
if the
data
famous
example
aproaches
the
In
problem
of this
need
under
failure
not
first
two
are similar
performed in the Chow ring of have
enumerative significance,
consideration
JP5
Unfortunately the numbers
are
general
in
even
position.
is the answer given by De Joriquieres
A
theory
to the problem of finding the number of conics that are tangent to five given in general position.
conics
factory
situation,
needless
Similarly,
Halphen gave examples of this unsatis-
to
little more involved,
say
a
for the theory of
Chasles {see [H.].], §15, or the example 14.8 in this memoir}. On the other hand, tion between proper and
the starting point
In
improper solutions {see
Halphen 's theory is the distin§
6} to an enumerative problem
and his goal is to count the number of proper solutions. The numbers produced with the such
this
theory have
data
of the
numbers
always
problem.
In addition,
solutions
and,
conversely, conditions
always
{reduced} are
all
data
proper data
significance
the sense that if
in
the
number
of
distinct
proper
solutions
then
of
the
it turns out that all nondegenerate solutions are proper
if the
with
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