Let $\mathfrak{m} = (x,y) \subset k[x,y]$. Then the valuation $v$ of $k(x,y)$ associated to the exceptional divisor of the blowup should be defined by $$v(f) = \mathrm{sup}(n|f \in \mathfrak{m}^n), f\in \mathfrak{m}$$
How does one extend $v$ to all of $k(x,y)$?
The valuation ring $\mathcal{O}_v$ will be the elements with nonnegative valuation. What is good way to represent the elements of $\mathcal{O}_v$?
How to calculate the residue field $k(v) = \mathcal{O}_v/\mathfrak{m}$?
How would these calculations differ for blowups at other ideals supported at $\mathfrak{m}$, like $(x^2,y), (x^3,xy,y^2)$?
Thank you.
The local calculation on one affine chart of the blow-up is $O_E = k[x,y/x]_{(xk[x,y/x])}$. On this chart, the exceptional divisor is generated by $x$. (Since $y = x \cdot y/x$, $mk[x,y/x] = xk[x,y/x]$.)
Any elements $z$ in $k(x,y)$ has a presentation of $f(x,y)/g(x,y)$, where $f,g \in k[x,y]$. Such element $z$ is in $O_E$ if and only if $v(f/g)= v(f) - v(g) \ge 0$. To calculate $v(f), v(g)$, you can extend $f,g$ to $O_E$, or one can use the valuation you defined in the post.