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?