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Let $X \subseteq \mathbb P^n, Y \subseteq \mathbb P^m$ be varieties and $h_X,h_Y$ be the hilbert polynomial. then, I know that $$h_{X \times Y}=h_X \cdot h_Y$$ But, I can't prove. In special case $X \times Y= \mathbb P^n \times \mathbb P^m$, I prove using the Segre embedding. But above case, I can't easily apply.

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    I assume, since you mention it, that we are embedding $X\times Y$ into $\mathbb{P}^N$ via the Segre embedding? The Hilbert polynomial, $h_{X\times Y}$, depends on the embedding in general.2012-09-25
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    yes, Matt. I assume that $X \times Y$ is embedded in $\mathbb P^N$ via the Segre embedding.2012-09-26
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    http://math.stackexchange.com/a/139323/32172012-09-26

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