# 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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