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I have found a statement (Harris "Algebraic Geometry", p.269) saying that a hypersurface in $\mathbb{P}^m\times\mathbb{P}^n$ can be written as a single equation. I couldn't find the proof.

How can I prove it? Is there a good reference for the statement?

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    $\begingroup$ The equation is just given by a bi-homogeneous polynomial of bi-degree $(m, \, n)$. $\endgroup$
    – Francesco Polizzi
    Commented Jan 31, 2022 at 13:25
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    $\begingroup$ This site is for questions of math research, please consider using MSE instead. $\endgroup$
    – abx
    Commented Jan 31, 2022 at 19:17

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Let me expand my comment into a short answer. A subset $X \subset \mathbb{P}^m \times \mathbb{P}^n$ is a closed algebraic subvariety if and only if it is given by a system of polynomial equations $$G_k(x_0, \ldots, x_m; \, y_0, \ldots, y_n)=0 \quad \operatorname{for}\;k=1, \ldots, t,$$ homogeneous separately in each set of variables $x_i$ and $y_j$. In particular, if $X$ is a hypersurface one has $t=1$, for dimension reasons. See for instance [1, Theorem 1 p. 57].

References.

[1] I. G. Shafarevich: Basic Algebraic Geometry 1, Second Edition, Springer 1994.

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