Sur les variétés x ⊂ ℙ^N telles que par n points passe une courbe de x de degré donné

For given integers r ≥ 1, n ≥ 2 and q ≥ n - 1, we introduce the class χ_r=1,n(q) of (r + 1)-dimensional subvarieties X of a projective space, such that: - any generic set of n points of X is contained in a rational normal curve on X, of degree q; - X spans a projective space the dimension of which is the biggest possible, considering the first property. Our main result is the following. Theorem. - If q ≠ 2n - 3 and X ∈ χ_r+1,n(q), there exists a variety X₀ in ℙ^r+n-1, of dimension r + 1 and minimal degree n - 1, and a birational map X₀ → X, such that a section of X₀ by a generic ℙ^n-1 is mapped onto a rational normal curve of degree q. Without any assumption on q, we say that a variety X ∈ χ_r+1,n(q) is standard if it satisfies the conclusion of the preceding theorem. Building upon the classification of varieties of minimal degree, which is well-known, we give a complete classification of standard varieties in each class χ_r+1,n(q). The existence and classification of non-standard varieties X ∈ χ_r+1,n(2n - 3), for r ≥ 2 and n ≥ 3, remains an open problem. However, though the condition q ≠ 2n - 3 in the theorem above may not be sharp, we give examples of non-standard varieties in/χ_r+1,3(3) and in χ_r+1,4(5). In the general case, if X ∈ χ _r+1,n(q), we show that the space of rational normal curves of degree q on X carries a natural quasi-grassmannian structure. Our second main result is: Theorem. - A variety X ∈ χ_r+1,n(q) is standard if and only if the associated quasi-grassmannian structure is integrable, that is locally isomorphic to the natural stucture of the grassmannian of (n - 1)-planes in ℙ^r+n-1. In a forthcoming paper we shall apply our results to the so-called Problem of alge-braization of webs of maximal rank, giving in most cases a solution to a question first raised, in this generality, by Chern and Griffiths.