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Let $R=\mathbb{C}[x_0,x_1,...,x_n]$ and $I$ be a homogeneous $J$-primary ideal, where $J=\sqrt{I}$ and $J$ is a homogeneous prime ideal. Assume $V(J)$ is a $d$-dim projective subvariety inside $\mathbb{P}^n$.

I have a few questions concerning the Hilbert polynomials of $I$ and $J$ which I will call them $h_I$ and $h_J$.

The degree of polynomial $h_I$ and $h_J$ should be $d$.

Let the leading coefficient of $h_I$ and $h_J$ be $\frac{a_I}{d!}$ and $\frac{a_J}{d!}$.

How to show that $a_J$ divides $a_I$ or is there any counterexample that this is not true?

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  • $\begingroup$ I assume $h_I$ is the Hilbert polynomial of $R/I$, not of the $R$-module $I$? $\endgroup$ – Jesko Hüttenhain Feb 10 '18 at 10:23
  • $\begingroup$ @Jesko Hüttenhain, you are right. $\endgroup$ – Ben Feb 10 '18 at 15:51
  • $\begingroup$ Take $I = (x_0)^2$ and $J = (x_0)$. $\endgroup$ – Youngsu Feb 10 '18 at 16:50
  • $\begingroup$ @Youngsu, degree($I$)=2, degree($J$)=1. $\endgroup$ – Ben Feb 11 '18 at 0:14
  • $\begingroup$ Oh right I should have meant a_J divides a_I. I edited my question. $\endgroup$ – Ben Feb 11 '18 at 0:15

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