Contact conditions on conic varieties and Halphen's \(2^{\text{nd}}\) formula (Q1200937)

\textit{G. H. Halphen} [``Sur les charactéristiques des systèmes de coniques et des surfaces de second ordre''; ``Sur le nombre des coniques qui, dans un plan, satisfont à cinq conditions projectives et indepéndentes entre elles'', Oeuvre II, 1-57 and 275-289 (Paris 1918)] developed an enumerative calculus for conics quite different from Chasles' calculus, whose computations are made in the intersection ring of the variety of complete conics. Halphen's calculus computes proper solutions of enumerative problems and has the advantage of its results to be always significant: the true number of different conics that properly satisfy given conditions (with general data) is always given by Halphen's calculus regardless problems of intersections or multiplicities. Halphen's work has been updated by the reviewer and \textit{S. Xambó- Descamps} [``The enumerative theory of conics after Halphen'', Lect. Notes Math. 1196 (1986; Zbl 0626.14037)] by providing modern proofs and extending the theory to cycles (conditions) of intermediate dimensions. The paper under review is devoted to Halphen's second formula (theorem 14.6 in the cited book), a formula that gives the number of conics properly satisfying five conditions of a very particular type that Halphen called elementary conditions. This formula is important because of the fact (well known by Halphen but from which the authors seem not to be aware), that elementary conditions are a \(\mathbb{Z}\)-basis of the module of all conditions of order (=codimension) one, modulo strict numerical equivalence. The result is thus not new, and even the proof provided may hardly be considered as new: it is obtained after blowing-up all improper solutions [along the line of \textit{C. De Concini}, and \textit{C. Procesi} in Algebraic groups and related topics, Proc. Symp., Kyoto and Nagoya/Jap. 1983, Adv. Stud. Pure Math. 6, 481-513 (1985; Zbl 0596.14041)] and it is rather more complicated than Halphen's original proof [see the reviewer and \textit{Xambó} (loc. cit.)], due to the way the authors are dealing with infinitely near varieties of double lines.