Scrolls containing binary curves

Abstract

We study families of scrolls containing a given rational curve and families of rational curves contained in a fixed scroll via a stratification in terms of the degree of the induced map onto $$\mathbb P^1$$ and we prove that there is no rational normal scroll of minimal degree and of dimension $$\le \frac{n}{2}$$ containing a general binary curve in $$\mathbb P^n$$.