Sequences of Enumerative Geometry: Congruences and Asymptotics

We study the integer sequence vn of numbers of lines in hypersurfaces of degree 2n−3 of Pn, n > 1. We prove a number of congruence properties of these numbers of several different types. Furthermore, the asymptotics of the vn are described (in an appendix by Don Zagier). An attempt is made at a similar analysis of two other enumerative sequences: the numbers of rational plane curves and the numbers of instantons in the quintic threefold. We study the sequence of numbers of lines in a hypersurface of degree D = 2n − 3 of P, n > 1. The sequence is defined by (see e.g. [8]) vn := ∫ G(2,n+1) c2n−2(Sym Q), (1) where G(2, n + 1) is the Grassmannian of C subspaces of C (i.e. projective lines in P) of dimension 2(n + 1 − 2) = 2n − 2, Q is the bundle of linear forms on the line (of rank r = 2, corresponding to a particular point of the Grassmannian), and Sym is its Dth symmetric product – of rank ( D+r−1 r−1 ) = D−1 = 2n−2. The top Chern class (Euler class) c2n−2 is the class dual to the 0-chain (i.e. points) corresponding to the zeros of the bundle Sym(Q), i.e. to the vanishing of a degree D equation in P; this is the geometric requirement that the lines lie in a hypersurface. The integral (1) can actually be written as a sum: