What are the discrete valuation rings for the affine plane?

Question

Let $X = \mathbb{A}^2_k$ be the affine plane over an algebraically closed field $k$, and let $K = k(x,y)$ be the field of rational functions over $X$. How can one describe all discrete valuation rings $R$ with quotient field $K$, please?

In particular, is there a valuation associated with any maximal ideal $(x-a,y-b)$? How is it defined, please?

I am reading Hartshorne's "Algebraic Geometry", but so far in the book, the author only uses discrete valuation rings $R$ having $\dim(R) = 1$. Is that necessarily so?

Some comments from experts will be appreciated.

Edit 1: It turns out (see KReiser's answer below) that DVR must have dimension $1$. So, at least for "nice" function fields, valuation rings correspond to subvarieties of codimension $1$.

Edit 2: In the comments below KReiser's answer, I was essentially looking at the valuation on the function field of the affine plane blown-up at the origin corresponding to the exceptional divisor. I think my example is interesting. It is a discrete valuation on $k(x,y)$ whose corresponding valuation ring is not a localization of $k[x,y]$, unlike what one may conjecture (unless I am mistaken somewhere). It is related to the fact that the affine plane and its blow-up at the origin are birational, and so have isomorphic function fields.

Answer 1

Valuations correspond (under some niceness assumptions) to subvarieties of codimension one, so you will not be able to find a valuation corresponding to $(x-a,y-b)$. With some quick checking (say on wikipedia, stacksproject, any good commutative algebra book), you will find that all DVRs have Krull dimension one.

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Edit: the above answer is incorrect. I'm not sure why I didn't correct this earlier, but the full solution is that every valuation ring of $k(x,y)$ trivial on $k$ is of one of three types (per Hartshorne exercise II.4.12(b)):

• $k(x,y)$
• The generic point of some irreducible curve in a blowup of $\Bbb P^2$ (including the trivial blowup, $\Bbb P^2$)
• Take a sequence of points as follows: pick $x_0\in\Bbb P^2$, blow it up to get $X_1$, pick a point $x_1$ on the exceptional fiber over the blowup, blow that up to get $X_2$, pick a point $x_2$ on the exceptional fiber over the blowup, repeat off to infinity. Then the local ring $\mathcal{O}_{X_{i+1},x_{i+1}}$ dominates $\mathcal{O}_{X_i,x_i}$ and the union of all of these is a valuation ring.

The example Malkoun found is of the second type: it's the valuation associated to the generic point of the exceptional divisor of the blowup at the origin. As one can see here on MO, valuations can get tricky, and I'm sorry for leaving something misleading up for so long.