On completeness of varieties in enumerative geometry (Q583319)

An enumerative problem is solved by intersecting suitable subvarieties \(V_ 0,...,V_ r\) (corresponding to conditions) of a given parameter variety X (whose points parametrize the objects to be counted). In addition to the formal intersection product of the \(V_ i\), in the intersection ring of X, we also need to know that the intersection of the \(V_ i\) is proper (or better, transverse) in order to be sure that the formal product counts solutions to the given problem. Usually, this second step is accomplished by translating all but one of the \(V_ i\) under a given group of motions G (for example a projective group) to obtain a proper intersection. When G is transitive, we can always do this (and in characteristic 0 obtain a reduced intersection as well), by the well-known results of \textit{S. L. Kleiman}, Compos. Math. 28, 287-297 (1974; Zbl 0288.14014)]. In general, however, X will have more than one orbit, possibly infinitely many. Here, if each \(V_ i\) but one meets all the orbits properly (respectively transversely, in characteristic 0), we again obtain a proper (respectively transverse) intersection. This follows from results of the reviewer in Algebraic Geometry, Proc. Conf., Sundance/Utah 1986, Lect. Notes Math. 1311, 235-252 (1988; Zbl 0683.14003); theorem 1.3, p. 240, a general statement about fiber products over a smooth variety with an algebraic group action, which follows almost immediately from some basic facts about flat and smooth morphisms. The article under review, by contrast, establishes some special cases, by longer arguments, more or less in a classical style. When a subvariety of X, for example one of the \(V_ i\), does not meet an orbit properly, it is easy to describe what happens to the orbits; the authors discuss this possibility also, in a similar style.