# The valuation attached to a smooth point of an algebraic variety

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.