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Let $k$ be an algebraically closed field of characteristic zero, $X, Y$ be projective $k$-schemes. Fix closed immersions $i:X \hookrightarrow \mathbb{P}^n$ and $j:Y \hookrightarrow \mathbb{P}^m$ for some positive integers $n$ and $m$. This induces a natural closed immersion $i \times j:X \times_k Y \hookrightarrow \mathbb{P}^n \times_k \mathbb{P}^m \to \mathbb{P}^{nm+n+m}$, where the last morphism is the Segre embedding. Let $F$ be a coherent sheaf on $X \times_k Y$ flat over $X$. Denote by $p:X \times Y \to X$ be the natural projection map. Denote by $P(F)$ (resp. $P(p_*F)$) the Hilbert polynomial of $F$ (resp. $p_*F$) with respect to the ample line bundle arising from the closed immersion $i \times j$ (resp. $i$). Is there a formula relating the Hilbert polynomials $P(F)$ and $P(p_*F)$?

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  • $\begingroup$ The Hilbert polynomial depends on a choice of an ample line bundle. What are the line bundles in your question? $\endgroup$ – Sasha Jan 19 '17 at 4:39
  • $\begingroup$ @Sasha I have edited the question to fix the choice of ample line bundles. $\endgroup$ – Naga Venkata Jan 19 '17 at 12:29
  • $\begingroup$ Do you mean, relating $P(F,(i \times j)^*O(1))$ with $P(p*F,i^*O(1))$? If this is what you mean, then the answer is negative. $\endgroup$ – Sasha Jan 19 '17 at 16:59

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