Let $V$ be a quasi-projective variety in $\mathbb{P}^{n}\times\mathbb{P}^m$. If $p\in \mathbb{P}^m$, we define the degree of $V_p$ as the degree of its closure in $\mathbb{P}^n$.

Question : $\exists C>0,~~\forall p\in \mathbb{P}^m,~~\deg(V_p)\leq C$ ?

Remark : If $V$ is projective, it is true. (Using generic flatness and noetherian induction).

Let $V$ be a quasi-projective variety in $\mathbb{P}^{n}\times\mathbb{P}^m$. If $p\in \mathbb{P}^m$, we define the degree of $V_p$ as the degree of its closure in $\mathbb{P}^n$.

Question : $\exists C>0,~~\forall p\in \mathbb{P}^m,~~\deg(V_p)\leq C$ ?

Remark : If $V$ is projective, it is true. (Using generic flatness and noetherian induction).

Edit : If $f:V\to \mathbb{P}^m$ is the second projection, $V_p$ is $f^{-1}(p)$ (with the natural scheme structure).