Degree of a variety vs degree of its blow-up

Question

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 $D$?

Answer 1

First, the ample divisors on $\mathbf P^n \times \mathbf P^{n-1}$ are of the form $aH_1+bH_2$ for $a,b>0$, which means that every ample divisor is of the form $H_1+H_2+N$ for some nef divisor $N$. This shows that $\operatorname{deg}_{H_1+H_2}\widetilde{V} \leq \operatorname{deg}_D\widetilde{V}$ for every other ample divisor $D$.

Next, the restriction of $H_1+H_2$ to the blowup $\operatorname{Bl}_p\mathbf P^n \subset \mathbf P^n \times \mathbf P^{n-1}$ is represented by the strict transform $\widetilde{Q}$ of any quadric $Q$ smooth at $p$.

For a variety $V \subset \mathbf P^n$ of dimension $d$, we can find $d$ such quadrics $Q_1,\ldots,Q_d$ such that the projectivised tangent cones of $V$ and $Q_1 \cap \cdots \cap Q_d$ at $p$ are disjoint, and away from $p$ the intersection $V \cap Q_1 \cap \cdots \cap Q_d$ is a finite set.

Hence $$\operatorname{deg}_D \widetilde{V} = \widetilde{V} \cdot \widetilde{Q_1} \cdots \widetilde{Q_d}= V \cdot Q_1 \cdots Q_d -\operatorname{mult}_pV =2^d \operatorname{deg} V -\operatorname{mult}_p V.$$

In particular taking $V=\mathbf P^n$ as in Remy van Dobben de Bruyn's comment, we get

$$\operatorname{deg}_D \widetilde{V} = (2^n-1)\operatorname{deg} V$$

while for any other $V$ we have

$$\operatorname{deg}_D \widetilde{V} \leq 2^d \operatorname{deg} V<(2^n-1) \operatorname{deg} V.$$

So $c_n=2^n-1$.