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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.

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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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    $\begingroup$ I have a stupid question. Suppose we define $v(g)$, where $g$ is a non-zero element of $k[x,y]$, to be the largest $n$ such that $v(g) \in (x-a,y-b)^n$, and then $v(f/h) = v(f) - v(h)$ for a non-zero element $f/h \in k(x,y)$, where $f,h \in k[x,y]$ and both are non-zero. Why is this not a discrete valuation on $K = k(x,y)$? $\endgroup$ – Malkoun Jun 19 '18 at 22:17
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    $\begingroup$ I may be wrong but, if you consider the blow-up map $p$ from the affine plane blown-up at the origin, denoted by $\hat{X}$, to the affine plane $X$, then isn't my valuation $v$ applied to a rational function $a/f$ on the plane essentially the order of the proper transform (I think this is what it is called) of $a/f$ to $\hat{X}$ at the generic point of the exceptional divisor? $\endgroup$ – Malkoun Jun 20 '18 at 10:11
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    $\begingroup$ 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. My example corresponds to the exceptional divisor in the blow-up. $\endgroup$ – Malkoun Jun 20 '18 at 13:58
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    $\begingroup$ KReiser, do you agree with my last paragraph please? After all, the properties of a valuation seem to be satisfied. $\endgroup$ – Malkoun Jun 20 '18 at 13:59
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    $\begingroup$ Yes, you've found something here. It's known (see Hartshorne exercises II.4.12, V.5.6) that there's some chicanery with valuations in higher dimensions - this is one of the reasons that their use changes after we leave curves for higher-dimensional settings. Emerton's post at mathoverflow.net/questions/12717/points-and-dvrs may help with understanding. $\endgroup$ – KReiser Jun 20 '18 at 16:21

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