TL;DR
Given a smooth projective variety $X\subseteq \mathbb{P}^m$ (I guess this means that the homogeneous coordinate ring $S(X)$ is regular whenever localizing at a non-maximal graded prime ideal). Is there an another projective variety $Y\subseteq \mathbb{P}^n$ isomorphic to $X$ such that the homogeneous coordinate ring $S(Y)$ is regular?
Any examples or comments are welcome!
Recently in noncommutative projective geometry, a notion of resolution of singularities has been introduced [1]. Roughly it sounds like this: Let $A$ be an $\mathbb{N}$-graded algebra which is "Gorenstein". If $B$ is an $\mathbb{N}$-graded algebra which is "regular" such that there is an equivalence of category $A{-}\text{qgr}\to B{-}\text{qgr}$, then $B$ is called a "resolution" of $A$. In particular, the case that $A$ has an isolated singularity (i.e. $A{-}\text{qgr}$ has finite global dimension) has been examined.
I want to justify the notion in commutative case. Recall that the celebrated Serre's theorem states that for a finitely generated commutative graded algebra $A$ generated in degree 1, there is an equivalence between the quotient category $A{-}\text{qgr}$ and the category of coherent sheaves $\text{coh}(X)$ over the associated projective scheme $X$ of $A$. Then asking $A$ to have an isolated singularity (in noncommutative sense) is requiring $\text{coh}(X)$ to have finite global dimension, which I guess can yield smoothness of $X$. Hence translated into commutative case, I think what the authors actually have done is to find a good coordinate ring ($B$ instead of $A$) of a smooth projective variety (maybe I should really say projective schemes but I know little about schemes) and whether there are examples or researches on this topic in algebraic geometry is what I want to ask.
Apologize for some gaps in my statements as I don't know much of algebraic geometry (I've only taken a one-semester course of algebraic geometry), and thanks!
[1] He, Ji-Wei; Ye, Yu. Preresolutions of noncommutative isolated singularities. Pacific J. Math.316(2022), no.2, 367–394. MR4404004