The Enumerative Theory of Conics after Halphen
Lecture Notes in Mathematics 1196, Springer-Verlag, 1986.
ix+130 pp. ISBN: 978-3-540-16495-1.
In the enumerative theory of conics there have been basically three approaches, namely those associated with De Jonquières, with Chasles, and with Halphen. Conceptually, the first two are similar in that they correspond to computations performed in the Chow rings of P5 and the variety of complete conics, respectively. Unfortunately, the numbers obtained with these approaches need not have enumerative significance.
Contrasting with these, Halphen's theory deals with proper solutions to
enumerative problems, i.e. solutions that move when the data of the conditions are
moved, and gives numbers that always have enumerative significance. The computations
are carried out in a graded commutative ring in which conditions are represented
by means of "characteristic numbers". These numbers encode information related to
the local and global structure of the conditions. Computations are made explicitely
by constructing a basis for each graded piece, by computing the corresponding table
of products, and by showing how to find the characteristic numbers of a given
condition.