# Under which conditions on the homogeneous ideal $I$, the quotient ring $\mathbb{C} [X_0, \dots, X_n]/I$ is a regular ring?

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?

$\mathbb{C}[x_1,\dotsc,x_n]/I$ 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.
• Please, is $I$ homogeneous, generated by linear forms if and only if : $I = I^{ \mathrm{lin} } = \{ \ \text{ the ideal generated by the linear terms } f^{ \mathrm{lin} } \text{ of all } f \in I \ \}$ such that : $f^{ \mathrm{lin} }$ is the linear term of $f$ as the degree one homogeneous polynomial in its expression as a sum of homogeneous polynomials in the variables $x_i$ 's ? Thank you . – YoYo Sep 9 '18 at 20:33
• @YoYo: Yes. If $I$ is generated by linear forms $\ell_i$ then generators of $I^{\text{lin}}$ include $\ell_i^{\text{lin}} = \ell_i$. Conversely $I^{\text{lin}}$ is generated by linear forms by definition, so if $I = I^{\text{lin}}$ then $I$ is generated by linear forms, too. – Zach Teitler Oct 23 '18 at 7:47