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Rational singularity of Spec, Proj and Spec of localization of a standard graded $2$-dimensional ring

If $X$ is a two dimensional Noetherian reduced excellent scheme, then we know by a Theorem of Lipman that $X$ has a desingularization, i.e., there exists a regular scheme $Y$ and a proper birational map $f: Yto X$. A Noetherian reduced excellent scheme $X$ of dimension $2$ is said to have rational singularity if there exists a regular scheme $Y$ and a proper birational map $f: Yto X$ such that $R^i f_*mathcal O_Y=0,forall i>0$.

Now let $k$ be a perfect field and $R=k[x_1,…,x_n]/I$ be a standard graded ring of dimension $2$, where $I$ is a homogeneous radical ideal (hence $R$ is reduced) of $k[x_1,…,x_n]$. Let $mathfrak m$ be the unique homogeneous maximal ideal of $R$. Consider the following statements:

(1) $operatorname{Spec}(R)$ has rational singularities

(2) $operatorname{Proj}(R)$ has rational singularities

(3) $operatorname{Spec}(R_{mathfrak m})$ has rational singularities

(4) $operatorname{Spec}(R_{P})$ has rational singularities for every maximal ideal $P$ of $R$

My question is: What is the relationship between these statements (1), (2), (3) and (4)? Is there any references where I can find implications between these statements?

(If needed, I’m willing to assume $R$ is a normal ring.)

Mathematics Asked on November 14, 2021

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