Projective Nullstellensatz and regular rings

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?