Let $X$ be a projective non-normal scheme, say over the complex numbers, endowed with an ample line bundle $L$. Let $\nu \colon \hat{X} \to X$ be its normalization. Is it possible that $(X,L)$ and $(\hat{X},\nu^\*L)$ have the same Hilbert polynomial?


(In the case of curves the answer is negative.)



Hilbert polynomial of the normalization
No. One has an exact sequence $0\to\mathcal{O}\_X\to \nu\_\*\mathcal{O}\_{\widehat{X}}\to F\to 0$ and $F\neq 0$ if $X\neq \widehat{X}$. If the two Hilbert polynomials are equal, it follows that $\chi(F\otimes L^n)=0$ for all $n$ and thus for large $n$, we must have $h^0(F\otimes L^n)=0$, which is impossible since $F\neq 0$.


