I have some questions about Exercise 8.4 (b) of Hartshorne:
Let $Y$ be a closed subscheme of $\mathbf{P}_k^n$. If $Y$ is a complete intersection of codimension $r$ in $\mathbf{P}_k^n$ (i.e. the homogeneous ideal $I_Y$ of $Y$ in $S=k[x_0,\cdots,x_n]$ can be generated by $r$ elements), and if $Y$ is normal, then $Y$ is projectively normal.
I probably know how to solve this question, but there are some details I cannot solve:
Step 1: $Y$ is projectively normal $\Leftrightarrow$ $S(Y)=S/I_Y$ is an integrally closed domain $\stackrel{(*)}\Leftrightarrow$ $\forall p\in \operatorname{Spec}S(Y)$, $S(Y)_p$ is an integrally closed domain $\Leftrightarrow$ $\operatorname{Spec}S(Y)$ is normal.
Step 2: Since the affine cone $C(Y)=\operatorname{Spec}S(Y)$ is a locally complete intersection subscheme of $\mathbf{A}_k^{n+1}$, proposition 8.23 shows that $C(Y)$ is normal if and only it is regular in codimension 1. The normality of $Y$ implies that it is regular in codimension 1, (**) which implies that $C(Y)$ is regular in codimension 1.
My questions:
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(*) holds when $S(Y)$ is an integral domain, but in our case, the homogeneous ideal $I$ of $Y$ may not be a prime ideal. Can we show that it is an integral domain by the assumption that $\operatorname{Proj} S(Y)$ is normal?
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I do not know how to prove (**) since there are some inhomogeneous prime ideals of $S(Y)$.
Best Answer
I'd do it slightly differently, bypassing some of your trouble spots.
From (a), $I_Y$ is generated by $r$ elements, thus $I_{C(Y)}\subset k[\Bbb A^{n+1}]$ is also generated by $r$ elements and so $C(Y)$ is a complete intersection. As $Y$ is a complete intersection and normal, it's singular set is codimension at least 2 by proposition II.8.23(b). But by the Jacobian criteria, the singular set of $C(Y)$ is the cone on the singular set of $Y$ (potentially with the origin thrown in), so the singular set of $C(Y)$ is of codimension at least 2 and by another application of proposition II.8.23(b), $C(Y)$ is normal and hence $Y$ is projectively normal.