Sequences of Enumerative Geometry: Congruences and Asymptotics

Abstract

We study the integer sequence v n of numbers of lines in hypersurfaces of degree 2n − 3 of P n , n > 1. We prove a number of congruence properties of these numbers of several different types. Furthermore, the asymptotics of the v n 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 , n > 1. The sequence is defined by (see e.g.where G(2, n + 1) is the Grassmannian of C 2 subspaces of C n+1 (i.e. projective lines in P n ) 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 D is its Dth symmetric product -of rank D+r−1 r−1 = D − 1 = 2n − 2. The top Chern class (Euler class) c 2n−2 is the class dual to the 0-chain (i.e. points) corresponding to the zeros of the bundle Sym D (Q), i.e. to the vanishing of a degree D equation in P n ; this is the geometric requirement that the lines lie in a hypersurface.The integral (1) can actually be written as a sum:v n = 0≤i<j≤n

Exact results for perturbative Chern–Simons theory with complex gauge group

“…-44 -Using formula (4.8) for N of the order of 5000 and the numerical interpolation method explained in [56] and [21], we computed the values of s n for 0 ≤ n ≤ 27 to very high precision, finding that (4.4) holds with s 0 = 1 2π 2 i D e πi/3 , s 1 = − 1 4 log 3 , (4.10)…”

Cotangent sums, quantum modular forms, and the generalized Riemann hypothesis

“…Also, since H is even, we can replace the term H(1/x) in (8) by H(−1/x). But the two matrices S = 0 −1 1 0 and T 2 = 1 2 0 1 generate the subgroup Γ ϑ of index 3 in the full modular group SL(2, Z) = S, T consisting of matrices congruent to 1 0 0 1 or 0 1 1 0 modulo 2 (this is the so-called "theta group," under which the Jacobi theta function ϑ(z) = n e πin 2 z transforms like a modular form of weight 1/2), so equations (8) and (9) can be combined to the following statement:…”

“…Computing the values of C 1± 1 n for 1 ≤ n ≤ 1000 and using a numerical interpolation technique that is explained elsewhere (see e.g. [9]), we find empirically an expansion of the form…”

The Erdős–Moser equation $1^{k}+2^{k}+\dots+(m-1)^{k}=m^{k}$ revisited using continued fractions

“…Thus, one arrives at a conjecture that hopefully can be turned into a proof. For more details and some "victories" achieved by the asymp k method, see Grünberg and Moree [13]. D. Zagier has applied this trick to the sequence of k = k(m) obtained from (1) by letting m run through the first thousand values.…”