Question about valuation and blow up (a lemma in GIT book)

Question

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.

Answer 1

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.