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.


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.

  • $\begingroup$ It was actually very easy! Thanks a lot... $\endgroup$ – gio Aug 10 '15 at 18:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.