I am interested in proving the following result:

Let $k$ be an algebraically closed field, $X$ a normal integral variety over $k$ and $x\in X$ a closed point.

Write $\mathfrak{m}_x$ for the maximal ideal of the local ring $\mathcal{O}_{X,x}$ and consider the function $$\begin{align}\text{ord}_x:\mathcal{O}_{X,x}&\rightarrow \mathbb{Z}_{\geq 0} \\ f&\mapsto \max\{j\geq 0\mid f\in \mathfrak{m}^j\}\end{align}$$ Then if $x$ is a smooth point $\text{ord}_x$ is a valuation and it correspond to the valuation given by the exceptional divisor of the blow-up of $X$ at $x$.

I am not sure about how to approach this but this is what I have notice until now:

As the problem is local we can assume $X=\text{Spec}(A)$ and hence $x$ correspond to a maximal ideal $\mathfrak{m}$ of $A$ such that $\mathfrak{m}A_\mathfrak{m}=\mathfrak{m}_x$ is the maximal ideal of $\mathcal{O}_{X,x}$.

Now consider the blow-up $$\text{Bl}_x(X)=\text{Proj}(\oplus_{j\geq 0} \mathfrak{m}^j)\rightarrow X$$ the exceptional divisor is given by the fiber product $$\begin{align}\text{Bl}_x(X)\times_X \text{Spec}(k(x))=&\text{Proj}_X(\oplus_{j\geq 0} \mathfrak{m}^j)\times_{\text{Spec}(A)}\text{Spec}(A/\mathfrak{m})\\ =& \text{Proj}((\oplus_{j\geq 0} \mathfrak{m}^j)\otimes_A A/\mathfrak{m})\\ =& \text{Proj}(\oplus_{j\geq 0} \mathfrak{m}^j/\mathfrak{m}^{j+1})\end{align}$$

Now as we have the following equalities

  • $\mathfrak{m}^j/\mathfrak{m}^{j+1}\cong\mathfrak{m}_x^j/\mathfrak{m}_x^{j+1}$
  • $\mathfrak{m}_x^j/\mathfrak{m}_x^{j+1}\cong \hat{\mathfrak{m}}_x^j/\hat{\mathfrak{m}}_x^{j+1}$ (the hat is the completion in $\mathcal{O}_{X,x}$ wrt $\mathfrak{m}$)
  • $\widehat{A_\mathfrak{m}}\cong k[[x_1,\dots,x_d]]$ (because $x$ is smooth)
  • $\displaystyle \frac{(x_1,\dots,x_d)^{j}k[[x_1,\dots,x_d]]}{(x_1,\dots x_d)^{j+1}k[[x_1,\dots,x_d]]}\cong \frac{(x_1,\dots,x_d)^{j}k[x_1,\dots,x_d]}{(x_1,\dots x_d)^{j+1}k[x_1,\dots,x_d]}$
  • $\displaystyle \frac{(x_1,\dots,x_d)^jk[x_1,\dots,x_d]}{(x_1,\dots x_d)^{j+1}k[x_1,\dots,x_d]}\cong k[x_1,\dots,x_d]_j$

we conclude that the exceptional divisor is $$E=\text{Proj}(\oplus_{j\geq 0} \mathfrak{m}^j/\mathfrak{m}^{j+1})\cong \text{Proj}(k[x_1,\dots,x_d])=\mathbb{P}^d$$ In particular it is integral.

Now some questions that I have are:

  • Something that should be obvious: Why the generic point of $E$ is normal?
  • Is there a concrete way to see the generic point $\eta$ of $E$ inside $\text{Bl}_x(X)$? having it concretely is the only way I have in mind to prove this.

I would be very thankful if someone have a proof. Maybe it can be with other approach (like working with a presentation of the ring $A$). A good reference would be good as well.

Remark: This result appears for granted at the paper A refinement of Izumi's Theorem and understanding that paper is the original motivation.


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