Let $V\subset \mathbf{P}^n$ be a variety. Let $f:\mathbb{P}^n\times \mathbb{P}^{n-1}\to \mathbb{P}^n$ be a blow-up of a point $P$ on $V$.

It feels as if it would make sense for there to be an ample divisor $D$ on $\mathbb{P}^n\times \mathbb{P}^{n-1}$ such that, for $\deg_D$ the associated degree, $$\deg_D f^{-1}(V) = \deg V$$ holds. Is that the case? Is it the same divisor $D$ for any $V$? Does the Segre embedding induce such a divisor/degree function?