If I am not mistaken, and according to the projective Nullstellensatz, we have: $\mathbb{P}_{\mathbb{C}}^n = \mathrm{Proj} (\mathbb{C} [X_0 , \dots, X_n])$, by the correspondence: $ A \to \mathrm{Proj} (A) $ with $ A $ a graded ring.

By this correspondence, is $ \mathrm{Proj} (A) $ non-singular (i.e: smooth) if and only if $ A $ is a regular graded ring?

We can notice that: $ \mathbb{P}_{\mathbb {C}}^n $ is non-singular, and $ \mathbb{C} [X_0, \dots, X_n] $ is a regular ring.

If yes, and if $ I $ is a homogeneous ideal of the ring of homogeneous polynomials $ \mathbb {C} [X_0, \dots, X_n] $ , under which conditions, on the ideal $ I $, 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.


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