Thank you everybody!! Is it smooth also in higher rank?

Thank you, that's a cool observation. Does something similar exist in dimension 3 as well?

Thank you for your help! @Jason: If you happen to take a look at that Atlas, it would be of great help, thank you.

Thank you for your comments. @Noam: In fact, given a quartic scroll in $P^5$, I haven't computed the dimension of the space of cubics in the ideal of the scroll... is it what you meant? @Jason: in fact the question came up to me while reading Be-Do. I seem to understand that une implication is easy, but they don't seem to prove the other. Am I wrong?

Actually the two line bundles that I am considering are GIT-line bundels. That is: the map to the projective space that they induce via global sections yields a map to a GIT-compactification of $\overline{M}_{0,n}$ (non functorial compactification, due to the existence of strictily semi-stable points). Thank you for your comments!