I'm reading Mumford's book *Geometric Invariant Theory* and confused about the proof of a lemma on Page 91&92:



> 
> Lemma:
> 
> 
> Let $V\_0$ be a smooth surface over an algebraically closed field $k$
> with char$k=0$, and let $P$ be a closed point on $V\_0$. Let $x,y$ be uniformizing parameters at $P$ on $V\_0$ and $I:=(x^{r\_0}\cdot y^{s\_0},...,x^{r\_n}\cdot y^{s\_n})$ where $r\_i,s\_i$ are integers. For all positive rational number $a=p/q$ where $(p,q)=1$, define a valuation associate to $a$ such that discrete, rank 1 and centered at $P$:
> 
> 
> $v\_a(\sum a\_{i j} x^iy^j)=\min\_{a\_{i j}\neq 0}\{ip+jq\}$.
> 
> 
> Let $V$ be the normalization of blow up $Bl\_{I} V\_0$. Then the prime exceptional divisors on $V$ are exactly those prime divisors $E\_a$ corresponding to valuations $v\_a$, where the least integer in the sequence of integers $r\_ip+s\_iq, 0\leq i\leq n$ occurs at least twice.
> 
> 
> 


Q1: In the proof of Lemma, he first claimed if $v$ is a valuation of $K(V)=K(V\_0)$ centered at $P$ , then the center of $v$ has positive dimensional center if and only if $v$ has positive dimensional center on one of the following surfaces:


$Spec(\mathcal{O}\_{P, V\_0}[\frac{x^{r\_0}\cdot y^{s\_0}}{x^{r\_i}\cdot y^{s\_i}},...,\frac{x^{r\_n}\cdot y^{s\_n}}{x^{r\_i}\cdot y^{s\_i}}]), 0\leq i\leq n$


How to prove this claim?


Q2: To show every discrete valuation on $V$ is of form $v\_a$ for some positive rational number $a$, he pass to the completion $Spec(\widehat{\mathcal{O}\_{P, V\_0}})$, and claimed that since $I$ is invariant under automorphisms $(x,y)\mapsto (\alpha x, \beta y)$, $v\_a$ should also invariant under these automorphism, and he said it's easy to check only discrete valutions with required properties are $v\_a$'s. But how to check this?


For Q1, I know the center of $v\_a$ on $V$ is irreducible, hence to show center has positive dimension, we only need to show $v\_a$ centered at two distince closed points. But I don't know how to continue.


Thanks for any help.



Question about valuation and blow up (a lemma in GIT book)
do you know how we cunstruct a blow up? for your first question blow up of affine $A$ at the ideal $I$ is covered by $$Spec(A[\frac{x^{r\_0}\cdot y^{s\_0}}{x^{r\_i}\cdot y^{s\_i}},...,\frac{x^{r\_n}\cdot y^{s\_n}}{x^{r\_i}\cdot y^{s\_i}}]),0\le i\le n$$, now let $Q$ be a prime ideal defining a point with positive dimension at center of $v$ over $$Spec(\mathcal{O}\_{P, V\_0}[\frac{x^{r\_0}\cdot y^{s\_0}}{x^{r\_i}\cdot y^{s\_i}},...,\frac{x^{r\_n}\cdot y^{s\_n}}{x^{r\_i}\cdot y^{s\_i}}])$$ and because you have finitely many dominator it define a point with positive dimension of one $Spec(A[\frac{x^{r\_0}\cdot y^{s\_0}}{x^{r\_i}\cdot y^{s\_i}},...,\frac{x^{r\_n}\cdot y^{s\_n}}{x^{r\_i}\cdot y^{s\_i}}])$ where $U= Spec A$ is a neighborhood of $P$.


For your second question, I don't completely understand your statement but the idea should be something like this: let's $v(x)=p,v(y)=q$ because the valuation is discrete, after a rescaling we can assume $(p,q)=1$. now we know that
$$v(\sum a\_{i j} x^iy^j)\ge\min\_{a\_{i j}\neq 0}\{ip+jq\}$$
by third property of valuations,in fact from the third property we get that if $$v(a)\not =v(b),v(a+b)=min(v(a),v(b))$$, so we only have to handle the case that $\forall i,j,ip+jq$ is a constant. so we can assume that in $$t=\Sigma a\_{ij}x^iy^j$$ powers of $x$ are strictly increasing. we prove by induction on the number of terms of $t$ that $$v(\sum a\_{i j} x^iy^j)=ip+jq$$.assume that $$v(\sum a\_{i j} x^iy^j)>ip+jq$$ then by invariance you mentioned $$v(\sum a\_{i j} x^iy^j)=v(\sum a\_{i j} (\alpha x)^iy^j)$$ lets $n$ be the biggest power of $x$ appearing in $t$ and take $\alpha^n=-1$ (base is algebraicly closed so such an $\alpha$ exist) now $$\sum a\_{i j} x^iy^j+\sum a\_{i j} (\alpha x)^iy^j$$ has less terms than $t$ and it's valuation is bigger than $v(t)>ip+jq$ which is a contradiction by induction hypothesis.


