Mori fiber space contractions of the blow-up of a projective space

Question

Let $\mathbb P(V)$ be a projective space containing $Y$ as a subvariety. Let $Z$ be the blow-up of $\mathbb P(V)$ along $Y$.

Clearly there exists a divisorial contraction given by $b: Z \to \mathbb P(V)$. On the other hand the Picard number of $Z$ is two, hence we have another contraction. I want to understand more about this second contraction. I have read that it is a Mori fiber space contraction. Can one say something else in the general contest?

Moreover, if we suppose that $V=T_{X,x}$ is the tangent space of a smooth unruled projective variety $X$ at a point $x \in X$ and that $Y=C_X$ is the variety of minimal rational tangent, can one say something more about this Mori fiber space contraction?

Answer 1

You don't necessarily have another Mori contraction, the Kleiman-Mori cone might stray outside the half space $-K_{Z}.C >0$, so one cannot guarantee the existence of two distinct extremal rays. Sometimes you have it, for example if the blow-up is Fano, but not always.