La variété des triplets complets. (The variety of complete triplets) (Q1262922)

The author generalizes a construction of a variety \(W^*\) of ``complete'' point triplets in \({\mathbb{P}}^ 2\) due to \textit{J. G. Semple} [Mathematika 1, 80-88 (1954; Zbl 0057.371)] and which has been studied by J. Roberts and R. Speiser in a series of articles, to a variety \(\hat H^ 3(V)\) of point triplets of a non-singular variety V. The main result is that \(\hat H^ 3(V)\) is a non-singular subvariety of \(V\times V\times V\times Hilb^ 2V\times Hilb^ 2V\times Hilb^ 2V\times Hilb^ 3V\). The technique to prove it is an intelligent use of local parameters in the several kinds of special points of \(\hat H^ 3(V)\). By the way some nice divisors on \(\hat H^ 3(V)\) occur in the discussions which could be helpful in the computation of the intersection ring of \(\hat H^ 3(V).\) The reviewer asks of what use this generalization is for any question in the enumerative geometry.