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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).

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  • $\begingroup$ What is the definition of $V_p$ ? $\endgroup$
    – Jérémy Blanc
    Commented Apr 15, 2014 at 14:43
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    $\begingroup$ Why can't you take the closure $\bar V$ of V in $\mathbb P^n\times \mathbb P^m$ and apply your argument in the projective case? It seems to me that $\deg V_p\leq \deg (\bar V)_p$, so this should be OK. $\endgroup$
    – Sándor Kovács
    Commented Apr 17, 2014 at 0:01

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