# How can every divisor be reached by a sequence of blow-ups?

The following is a result of Zariski [cf. Lemma 2.45 of Birational Geometry of Algebraic Varieties].

$X$ : an algebraic variety over a field $k$.

$(R,m)$ : a DVR of the quotient field $K(X)$ of $X$ with $tr.deg_k(R/m)=dimX-1$.

$Y=SpecR$, $y\in Y$ the closed point and $f:Y\to X$the induced birational morphism (with $f(y)=x$).

Suppose $R\not\cong \mathcal{O}_{X,x}$.

Let $Z$ be the closure of $x$, $\pi_1:X_1\to X$ the Blowing-up of $Z$ in $X$ and $f_1:Y\to X_1$ the induced map and $f_1(y)=x_1$.

By repetitions of this procedure, $R\cong \mathcal{O}_{X_n,x_n}$ for some $n$.

I can't understand the fact that "the above result shows that every divisor can be reached by a sequence of blow-ups".

In case $\phi:Y\to X$ a birational morphism with an exceptional divisor $E$, how can I apply above result to show $E$ is obtained by a blow-ups? What are $R,y,\cdots$ in this case?

This question is probably trivial but I'm confused anyway. I appreciate very much if you give any answer.

*I posted this question at MathStack but it has not been answered.

• Take for $R$ the valuation ring of the discrete valuation on $K(X)$ given by vanishing order at $X$, and for $m$ its maximal ideal. The point $x$ is the image under $\phi$ of the generic point of $E$. What you get is a sequence of blow-ups which ultimately gives a proper birational morphism $\phi'\colon Y'\to X$ such that the birational isomorphism $(\phi')^{-1}\circ\phi$ is an isomorphism on an open subset of $Y$ which meets $Y$. – ACL Feb 6 '15 at 9:52
• math.stackexchange.com/questions/1134757/… – user26857 Feb 6 '15 at 17:11

For every divisor $$D$$ on $$X$$ we obtain a discrete valuation ring $$R$$ of $$K(X)$$ whose residue field has transcendence degree $$\mathit{dim} X -1$$ together with a morphism $$\mathit{Spec} R \to X$$, namely by setting $$R = \mathcal{O}_{X,\eta_D}$$, where $$\eta_D$$ is the generic point of $$D$$.
However, not all valuation rings as above are obtained in this way. The theorem now asserts that one can find a blowup $$Y \to X$$ (and even gives an explicit construction) such that on $$Y$$ the valuation ring $$R$$ comes from a divisor.
That every divisor can be reached does not mean every divisor on $$X$$ but every "potential divisor", i.e. every valuation as above is eventually reallised by a divisor.