Projective normality of cones over projectively normal varieties

Question

Let $X\subseteq\mathbb{P}^n$ be a smooth subvariety, with homogeneus ideal $I\subseteq k[x_0,\ldots,x_n]$. Let $C(X)\subseteq\mathbb{P}^{n+1}$ be the projective cone over $X$, so that $C(X)$ is defined by the ideal generated by $I$ in $k[x_0\ldots,x_{n+1}]$.

Question: Is true that, if $X$ is projectively normal, then $C(X)$ is also so? Equivalently, is true that, if $k[x_0,\ldots,x_n]/I$ is a normal ring, then also $k[x_0,\ldots,x_{n+1}]/I$ is a normal ring?

Thanks in advance for any eventual suggestion.

Answer 1

If we let $R=k[x_0,…,x_n]/I$, then $k[x_0,…,x_{n+1}]/I\cong R[x_{n+1}]$. If $R$ is normal, then $R[x_{n+1}]$ is certainly normal.