Let El(n; p1, . . . , pm) be the family of elliptic curves of degree n + 1 in Pn containing a fixed set of m distinct points p1, . . . , pm . In modern language, El(n; p1, . . . , pm) is a Zariski-open subset of the fibre of the evaluation map ev : M1,m(P, n + 1) → (Pn)m , where M1,m(P, n + 1) is the space of regular maps of elliptic curves equipped with an ordered set of m points to a curve of degree n + 1 in Pn . Its general fibre, if not empty, is an irreducible variety of dimension (n + 1)2 − m(n − 1). The largest possible m for which the map ev is dominant is equal to 9(n = 2, 5), 8(n = 3, 4), n + 3(n ≥ 6). In the last case we show that the general fibre is isomorphic to an open subset of a complete intersection of n − 2 diagonal quadrics in Pn+2. In particular, birationally, it is a Fano variety if n ≤ 6, a Calabi–Yau if n = 7, and of general type if n ≥ 8. The group Gn = (Z/2Z)n+2 acts naturally in Pn+2 by multiplying the projective coordinates with ±1. The corresponding action of a subgroup of index 2 of Gn is induced by a certain group of Cremona transformations in Pn which we will describe explicitly. There are three cases when El(n; p1, . . . , pm) is of expected dimension 0. They are (n,m) = (2, 9), (3, 8), (5, 9). It is well known that in the first two cases El(n; p1, . . . , pm) consists of one point. Less known is the fact that the same is true in the case (5, 9). D. Babbage [1] attributes this result to T. G. Room. Apparently it was proven much earlier by A. Coble [2]. We reproduce Coble’s proof in the paper. This result implies the existence of a rational elliptic fibration f : P5− → El(5, p1, . . . , p8) which is an analog of the well-known rational elliptic fibrations