Sur les vari\'et\'es X dans P^N telles que par n points passe une courbe de X de degr\'e donn\'e

TL;DR

This paper classifies certain projective varieties containing rational curves of fixed degree passing through generic points, revealing their structure and conditions for being of a specific birational form.

Contribution

It determines all varieties in the class X_{r+1,n}(q) for q ≠ 2n-3 and characterizes them via flatness of a quasi-Grassmannian structure, introducing new geometric insights.

Findings

Classified varieties X_{r+1,n}(q) for q ≠ 2n-3.

Established a birational correspondence with minimal degree varieties.

Identified conditions under which the structure is flat, characterizing the varieties.

Abstract

Given integers r>1, n>1 and q> n-2, we consider projective varieties X of dimension r+1 such that through n generic points of X passes a rational curve of degree q, contained in X. More precisely, we study the class X_{r+1,n}(q) of such varieties which moreover generate a projective space of the maximal dimension. We determine all varieties of a class X_{r+1,n}(q) when q is not equal to 2n-3. In particuliar, we show that there exists a variety X' in P^{r+n-1}, of minimal degree and a birational map F: X'---> X which sends a generic section of X' by a P^{n-1} onto a rational normal curve of degree q. Without hypothesis on q, we define a quasi-grassmannian structure on the space of the rational normal curves of degree q contained in a variety X of the class X_{r+1,n}(q). We prove that X is of the form described above if and only if this quasi-grassmannian structure is flat. We also give examples of varieties of the classes X_{r+1,3}(3) et X_{r+1,4}(5) which are not of this form.