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I have a stupid question: assume $X \subset P:=\mathbb{P}^{N_1}\times \mathbb{P}^{N_2} $ closed, irreducible, Cohen-Macaulay, not a product of two varieties and non degenerate, meaning that it is not contained in any hyperplane $H\times K$ where $H$ is an hyperplane in the first factor and $K$ an hyperplane in the second. Is there a lower bound depending on $N_1$ and $N_2$ for the degree of $X$ in terms of the dimension of $P$?

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    What do you mean by hyperplane of that product?2012-05-07
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    @Mariano Suárez Alvarez Sorry I edited2012-05-07
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    If $H$ and $K$ are hyperplanes, then $H \times K$ is not a hyperplane in the Segre embedding of $P^{N_1} \times P^{N_2}$, nor is it necessarily contained in one.2012-05-07

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