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?

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 $V$. Write $\widetilde{V}$ for the strict transform of $V$ under $f$, i.e., the Zariski closure of $f^{-1}(V\setminus \{P\})$.

It is tempting to guess that there ought to be an ample divisor $D$ on $\mathbb{P}^n\times \mathbb{P}^{n-1}$ such that, for $\deg_D$ the associated degree, $$\deg_D \widetilde{V} = \deg V.$$ However, as is shown in a comment, this need not be the case. What is the minimal $c_n$ such that, for every $V$, $$\deg_D \widetilde{V} \leq c_n \deg V$$ for some divisor $D$?

Let $V\subset \mathbf{P}^n$ be a variety. Let $f:\mathbb{P}^n\to \mathbb{P}^n\times \mathbb{P}^{n-1}$ 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, $$\deg_D f(V) = \deg V$$ holds. Is that the case? Is it the same divisor $D$ for any $V$? Does the Segre embedding induce such a divisor/degree function?

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?