If $ I $ is a homogeneous ideal of the ring of homogeneous polynomials $ \mathbb {C} [X\_0, \dots, X\_n] $ , under which conditions on the homogeneous ideal $ I $, and particularly on $ I\_m $, the $m$ -th graded piece of $I$ for every $m$, the quotient ring $ \mathbb{C} [X\_0, \dots, X\_n]/I $ is a regular ring?


According to the following wiki link: <https://fr.wikipedia.org/wiki/Anneau_local_r%C3%A9gulier> , we find that:


If $ A $ is a regular Noetherian local ring, and if: $ I $ is an ideal of $ A $. Then $ A/I $ is regular if and only if $ I $ is generated by a part of a regular parameter system of $ A $.


But here, for our case, $ \mathbb {C} [X\_0, \dots, X\_n] $ is not a local ring, which implies that we can not apply this proposition to our case of the quotient ring: $ \mathbb{C} [X\_0, \dots, X\_n] / I $. What is the solution ? How to answer to my questions in this case?


Thanks in advance for your help.



Under which conditions on the homogeneous ideal $ I $, the quotient ring $ \mathbb{C} [X_0, \dots, X_n]/I $ is a regular ring?
$\mathbb{C}[x\_1,\dotsc,x\_n]/I$ is regular if and only if the affine variety $V(I)$ is smooth. When $I$ is homogeneous,a $V(I)$ is a cone (with vertex at the origin). The only way for a cone to be smooth is if it's a linear subspace. So, for homogeneous $I$, the ring is regular if and only if $I$ is generated by linear forms.


