The enumerative theory of conics after Halphen
This book gives a detailed and self-contained exposition of the enumerative theory of conics. The authors survey the classical approaches to the subject and provide modern proofs (several are new), thus tying up with the most recent research. They give a very good introduction to this ``oldest'' part of enumerative geometry. In particular they explain the classical terminology and concepts, and thus facilitate the reading of older papers. The first part, {\S}{\S} 1-14, reworks Halphen's theory about ``proper solutions'', and leads to what are called Halphen's first and second formula. The first of these gives the proper number of members of conics in a 1-dimensional system satisfying a given condition, in terms of local and global characteristic pairs of the condition and the system, and of the degenerations of the system. The second one gives the number of conics properly satisfying \(5\quad given\quad conditions,\) in terms of characteristic numbers of the conditions. The second part of the book gives a general treatment for conditions of any codimension. In particular, the appropriate intersection ring, \(Hal^{\bullet}(W)\), is constructed - this is the abstract ring of De Concini-Procesi [cf. \textit{C. De Concini} and \textit{C. Procesi}, ``Complete symmetric varieties. II'', in Algebraic groups and related topics, Proc. Symp., Kyoto and Nagoya 1983, Adv. Stud. Pure Math. 6, 481-513 (1985; Zbl 0596.14041)] in the case of conics.
scientific article; zbMATH DE number 4017091