# How to Construct a ''Nice'' Birational Model in Characteristic $p>0$?

Let $X={\rm Spec} A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ is a prime Weil-divisor on $X$.

Now, I need to construct a birational model $g:Y\to X$ such that $Y$ and the birational transform $S'$ of $S$ are both normal and $g$ is a projective (or proper) morphism .

Since we don't have the Resolution of Singularities in characteristic $p>0$, I don't really know a good (or standard) way of doing this in higher dimension ($n>3$).

Any help will be greatly appreciated.

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What do you mean by $S'$ normal? I don't think one can in general ask for $S'$ a normal variety. For $n=2$, if $X= \mathbb P^2$ and $S$ is the union of two lines, then in any proper birational $Y \to X$, $S'$ won't be normal. It shouldn't get easier if $n>2$. – Will Sawin Mar 23 '13 at 0:30
Hi Will, by $S'$ normal I mean every local ring of $S'$ is integrally closed domain, I don't need $S'$ to be irreducible, even though in the situation stated above it will be irreducible since $S$ is. Your example doesn't quite fit to my problem, your $S$ is union of two lines but mine is an irreducible closed subset of co-dimesion $1$. However even in your example it is possible to make $S'$ normal, since resolution of singularity exists in dimension $2$, therefore we can choose a log resolution $g:Y\to X$ such that support $S'=f^{-1}_*S$ is disjoint union of smooth irreducible components. – Omprokash Mar 23 '13 at 1:55
Then $S'$ is normal – Omprokash Mar 23 '13 at 1:55
Let $B=A/I$ be the ring of functions on $S$. $y/x$ be a fraction in the normalization of $B$. Lift $y$ and $x$ arbitrarily to $A$, call them $y'$ and $x'$. Blow up $X$ at the ideal $(x',y')$. We certainly have a natural immersion from $\operatorname{Spec} B [y/x]$ to this blow-up. Since $\operatorname{Spec} B[y/x]$ is proper over $A$, and the map is over $A$, the map is closed, so a closed immersion. Since the image is birational to $S$, this is the strict transform of $S$. Repeat until the strict transform of $S$ is the spectrum of the integral closure of $B$. Is $Y$ then normal? – Will Sawin Mar 23 '13 at 3:51
Hi Will, thanks for your reply. I did something very similar as you. I took directly the the normalization of $B$ and then similar calculation as you but without the language of blow up, however I liked your arguments over mine. I have some questions though, you said there is a natural immersion from $Spec\ B[y/x]$ to the blow up, say $\tilde{X}$ and then you argued that since $Spec\ B[y/x]$ is proper over $A$, so it's a closed immersion. I have two questions here: (1) How do you say $Spec\ B[y/x]$ is proper over $A$ ? (2) I find it more natural to say directly that – Omprokash Mar 23 '13 at 6:14