Note on parametrisations of normal elliptic scrolls

Extract

1. If w(mod 2ω₁, 2ω₂) is an elliptic parameter for points of a normal elliptic curve C = ¹Cⁿ[n − 1], then it is well known that the sets of n points in which C is met by primes have a constant parameter sum k (mod 2ω_l, 2ω₂), and we may express this for convenience by saying that k is the prime parameter sum for the parametrisation of C by w. If we take the origin of w (the point for which w ≡ 0) to be one of the points of hyperosculation of C, then k ≡ 0, and we may say that w is a normal parameter for C. In the same way, if Γ is the Grassmannian image curve of the generators of a normal elliptic scroll ¹R₂ⁿ[n − 1], then a normal parametrisation of Γ defines a normal parameter w for the generators of ¹R₂ⁿ, such that n of the generators have parameter sum zero if and only if they belong to a linear line-complex not containing all the generators of ¹R₂ⁿ; or, in particular, if they all meet a space [n – 3] that is not met by every generator of the scroll. In this paper we are concerned in the first instance with the type of normal elliptic scroll ¹R₂^2m+1[2m] whose points can be represented by the unordered pairs (u₁; u₂) of values of an elliptic parameter u(mod 2ω₁, 2ω₂); and we establish a significant connection between any normal parametrisation of the generators of ¹R^2m+1 and an associated parametric representation (u₁u₂) of its points. We also add a brief note to indicate the lines along which this kind of connection can be extended to apply to a general normal elliptic scrollar variety ¹R_k^mk+1[mk] whose points can be represented by the unordered sets (u₁, …, u^k) of values of an elliptic parameter u.