Let \(k\) be an algebraically closed field.
If \(K\) is a function field of dimension \(1\) over \(k\), then every valuation ring of \(K/k\) except \(K\) itself is discrete. Thus the set of all of them is just the abstract nonsingular curve \(C_K\) of (I, §6).
If \(K/k\) is a function field of dimension two, there are several different kinds of valuations. Suppose that \(X\) is a complete nonsingular surface with function field \(K\).
If \(Y\) is an irreducible curve on \(X\) with generic point \(x_1\), then the local ring \(R = \OO_{x_1, X}\) is a discrete valuation ring of \(K/k\) with center at the nonclosed point \(x_1\) on \(X\).
If \(f: X' \to X\) is a birational morphism, and if \(Y'\) is an irreducible curve in \(X'\) whose image in \(X\) is a single closed point \(x_0\), then the local ring \(R\) of the generic point of \(Y'\) on \(X'\) is a discrete valuation ring of \(K/k\) with center at the closed point \(x_0\) on \(X\).
Let \(x_0 \in X\) be a closed point. Let \(f: X_1 \to X\) be the blowing-up of \(x_0\) (I, §4) and let \(E_1 = \inverseof{f}(x_0)\) be the exceptional curve. Choose a closed point \(x_1 \in E_1\), let \(f_2: X_2 \to X_1\) be the blowing-up of \(x_1\), and let \(E_2 = \inverseof{f_2}(x_1)\) be the exceptional curve. Repeat.
In this way we obtain a sequence of varieties \(X_i\) with closed points \(x_i\) chosen on them, and for each \(i\) the local ring \(\OO_{x_{i+1}, X_{i+1}}\) dominates \(\OO_{x_i, X_i}\). Let \(R_0 = \bigcup_{i \geq 0} \OO_{x_i, X_i}\). Then \(R_0\) is a local ring, so it is dominated by some valuation ring \(R\) of \(K/k\) by (I, 6.1A).
Show that \(R\) is a valuation ring of \(K/k\), and that it has center \(x_0\) on \(X\). When is \(R\) a discrete valuation ring?
Note: we will see later (V, Ex. 5.6) that in fact the \(R_0\) of the third kind is already a valuation ring itself, so \(R_0 = R\). Furthermore, every valuation ring of \(K/k\) except \(K\) itself is one of the three kinds just described.
(Erratum)
The second example in part (b) requires \(X'\) to be normal at the generic point of \(Y'\). An arbitrary birational morphism does not supply this hypothesis.
For a counterexample let \(X=\PP_k^2\) and blow up the zero-dimensional subscheme supported at the affine origin with ideal \((x^2,y^2)\). One affine chart of this birational model has coordinate ring \[ B=k[x,y,s]/(y^2-sx^2)\subset k(x,y),\qquad s=y^2/x^2. \] The exceptional curve has generic prime \(\mathfrak p=(x,y)\). After inverting \(k[s]\setminus\{0\}\), its local ring is \[ B_{\mathfrak p}\cong \bigl(k(s)[x,y]/(y^2-sx^2)\bigr)_{(x,y)}. \] This is a one-dimensional local domain: \(s\) is not a square in \(k(s)\), so the defining quadratic is irreducible. Its maximal ideal has two independent classes \(x,y\) modulo its square, since the defining relation has degree two. Hence this local ring is not regular and is not a DVR [@Har10a, Theorem I.6.2A]. The exceptional curve is nevertheless contracted to the closed origin of \(X\). Normality at its generic point is therefore necessary for the asserted DVR conclusion, and Theorem I.6.2A makes it sufficient.
Solution
For the third construction put \(X_0=X\), \(A_i=\OO_{X_i,x_i}\), and let \(\mathfrak m_i\) be the maximal ideal of \(A_i\). All these rings are viewed inside \(K\) through the birational maps. The second construction is treated with \(X'\) normal at the generic point of \(Y'\), as justified in the erratum.
1. In part (a), every valuation ring \(R\subsetneq K\) of \(K/k\) is the local ring of a unique closed point on the nonsingular projective model of \(K\).
Choose the nonsingular projective curve \(C\) with function field \(K\) [@Har10a, Chapter I, §6]. The valuative criterion for properness extends the generic-point map \(\Spec K\to C\) uniquely to \(\Spec R\to C\) [@Har10a, Theorem II.4.7]. If the image \(x\) of the closed point were generic, domination would give \(K=\OO_{C,x}\subseteq R\), contrary to \(R\ne K\). Thus \(x\) is closed, and \(\OO_{C,x}\) is a one-dimensional regular local ring, hence a DVR [@Har10a, Theorem I.6.2A]. A valuation ring dominating another valuation ring of the same field equals it [@Har10a, Theorem I.6.1A]. Consequently \[ R=\OO_{C,x}. \] Conversely each such local ring is a valuation ring of \(K/k\). These are precisely the points in the abstract nonsingular curve \(C_K\).
2. The first two surface constructions give DVRs with the stated centers, under the corrected hypothesis in the second construction.
For the first construction, the generic point \(x_1\) of \(Y\) has codimension one in the nonsingular surface \(X\). Thus \(\OO_{X,x_1}\) is a one-dimensional regular local domain with fraction field \(K\), so is a DVR [@Har10a, Theorem I.6.2A]. It contains \(k\) and dominates itself, giving center \(x_1\).
For the second construction, write \(\eta'\) for the generic point of \(Y'\). The local ring \(\OO_{X',\eta'}\) is a one-dimensional noetherian local domain with fraction field \(K\). Normality at \(\eta'\) makes it integrally closed, hence a DVR by the same theorem. Since \(f(\eta')=x_0\), the induced inclusion \(\OO_{X,x_0}\hookrightarrow\OO_{X',\eta'}\) is local. This is domination, so its center on \(X\) is \(x_0\). In both cases \(k^\times\) consists of units, so the valuation is trivial on \(k\).
3. The union \(R_0\) is a local domain with maximal ideal \(\mathfrak m_\infty=\bigcup_i\mathfrak m_i\) and residue field \(k\).Every valuation ring \(R\) of \(K\) dominating \(R_0\) is a valuation ring of \(K/k\) with center \(x_0\).
A blowup of a nonsingular surface at a closed point is nonsingular [@Har10a, Proposition V.3.1]. Hence every \(A_i\) is a two-dimensional regular local domain with fraction field \(K\) and residue field \(k\). The maps \(A_i\hookrightarrow A_{i+1}\) are local, so \(\mathfrak m_{i+1}\cap A_i=\mathfrak m_i\). An element outside \(\mathfrak m_i\) is already a unit in \(A_i\); an element in \(\mathfrak m_i\) remains a nonunit in every later ring. Thus \(R_0\) is local with the stated maximal ideal. For every \(a\in A_i\) there is a unique \(c\in k\) with \(a-c\in\mathfrak m_i\), so \(R_0/\mathfrak m_\infty\cong k\).
The domination theorem supplies a valuation ring \(R\subset K\) dominating \(R_0\) [@Har10a, Theorem I.6.1A]. It contains \(k\), and every nonzero constant is a unit. Moreover \[ \mathfrak m_R\cap A_0=\mathfrak m_\infty\cap A_0=\mathfrak m_0. \] Thus \(R\) dominates \(\OO_{X,x_0}\) and has center \(x_0\). It is not a field, since \(\mathfrak m_0\ne0\).
4. If a DVR \(V\) of \(K\) dominates every \(A_i\), then \(V=R_0\).
Normalize its valuation as \(v:K^\times\to\ZZ\). For a nonzero \(z\in V\), write \(z=a_0/b_0\) with nonzero \(a_0,b_0\in A_0\). Suppose \(z=a_i/b_i\) with \(a_i,b_i\in A_i\). If \(b_i\) is a unit, then \(z\in A_i\). Otherwise \(v(b_i)>0\), and \(v(a_i)\ge v(b_i)\) implies \(a_i,b_i\in\mathfrak m_i\).
Choose regular parameters \(p_i,q_i\) in \(A_i\). The blowup has charts \(A_i[q_i/p_i]\) and \(A_i[p_i/q_i]\) [@Har10a, Chapter II, §7]. On either chart the extended ideal \((p_i,q_i)\) is generated by its denominator. Thus \(\mathfrak m_i A_{i+1}=(e_i)\) for a local equation \(e_i\) of the exceptional curve. Since \(x_{i+1}\) lies on this curve, \(e_i\in\mathfrak m_{i+1}\). Set \(a_{i+1}=a_i/e_i\) and \(b_{i+1}=b_i/e_i\), which belong to \(A_{i+1}\) and still have quotient \(z\). Then \[ 0\le v(b_{i+1})=v(b_i)-v(e_i)<v(b_i). \] These nonnegative integers cannot decrease indefinitely. Eventually \(v(b_N)=0\), so domination makes \(b_N\) a unit of \(A_N\) and \(z\in A_N\). This proves \(V\subseteq R_0\); the reverse inclusion is assumed.
5. The ring \(R\) in the third construction is discrete precisely when there is a \(k\)-embedding \(\iota:K\hookrightarrow k((t))\) such that, for every \(i\),\[ \iota(A_i)\subseteq k[[t]], \qquad \iota(\mathfrak m_i)\subseteq t k[[t]]. \]For any such embedding,\[ R=R_0=\boxed{\iota^{-1}(k[[t]])}. \]Geometrically, this says that the prescribed sequence is realized by the successive lifts of a formal arc \(\Spec k[[t]]\to X\) whose generic point maps to the generic point of \(X\).
5.1. Discreteness implies the existence of this embedding.
3. Choose a uniformizer \(\pi\in R\).The completion map \(R\to\widehat R\) is injective, since a nonzero element has finite valuation and cannot belong to every \((\pi^n)\).Every element of \(\widehat R\) has a unique expansion \(\sum_{n\ge0}c_n\pi^n\) with \(c_n\in k\): subtract its residue, divide by \(\pi\), and repeat, using completeness.Thus \(\widehat R\cong k[[t]]\) with \(\pi\) corresponding to \(t\).Passing to fraction fields embeds \(K\) into \(k((t))\).The inclusions of the \(A_i\) into \(R\) are local, giving the required conditions.
3.2. Such an embedding implies discreteness.
Let \(V=\iota^{-1}(k[[t]])\) and restrict the \(t\)-adic valuation to \(K\). Its value group is a nonzero subgroup of \(\ZZ\): every nonzero element of \(\mathfrak m_0\) has positive value. A nonzero subgroup of \(\ZZ\) is infinite cyclic, so \(V\) is a DVR. For every \(i\), units of \(A_i\) map to units of \(k[[t]]\), since both the unit and its inverse lie in \(A_i\). The stated condition on \(\mathfrak m_i\) therefore says that \(V\) dominates \(A_i\). By step 4, \(V=R_0\). Since \(R\) dominates the valuation ring \(V\) inside the same field, \(R=V\) [@Har10a, Theorem I.6.1A].
3.3. Q.E.D.
Steps 1 and 2 prove the equivalence and identify the ring. The local maps \(A_i\to k[[t]]\) give compatible morphisms \(\Spec k[[t]]\to X_i\) centered at \(x_i\); their maps on function fields are \(\iota\). Conversely such lifts induce these local maps.
6. Discrete valuations occur in the third construction, even though every center \(x_i\) is closed.
Take \(X=\PP_k^2\), with affine coordinates \(x,y\), and put \[ \phi(t)=\sum_{n\ge1}t^{n!}\in t k[[t]]. \] This series is transcendental over \(k(t)\). Indeed, let \(P(T,Z)\in k[T,Z]\) be nonzero and suppose \(P(t,\phi(t))=0\). For \(\phi_N(t)=\sum_{n=1}^N t^{n!}\), the polynomial \(P(t,\phi_N(t))\) is divisible by \(t^{(N+1)!}\), whereas its degree is at most \(\deg_T P+(\deg_Z P)N!\). For all sufficiently large \(N\) this degree is smaller than \((N+1)!\), so \(P(t,\phi_N(t))=0\). This gives infinitely many distinct roots \(\phi_N\in k(t)\) of the nonzero polynomial \(P(t,Z)\), a contradiction.
Consequently \(x\mapsto t\), \(y\mapsto\phi(t)\) defines a \(k\)-embedding \(k(x,y)\hookrightarrow k((t))\). The restricted valuation has value group \(\ZZ\), since \(v(x)=1\), and residue field \(k\). It defines a DVR \(V\) centered at the origin. At each successive blowup, properness gives a center of \(V\) over the preceding center. Its residue field embeds into \(V/\mathfrak m_V=k\), so this center is a closed \(k\)-point. This constructs an infinite sequence of the required form, all of whose local rings are dominated by \(V\). Step 4 identifies their union with \(V\).
7. Nondiscrete valuations also occur in the third construction.
Again take \(X=\PP_k^2\) and \(K=k(x,y)\). For a nonzero polynomial \(P=\sum c_{ab}x^a y^b\), define \[ v(P)=\min\{(a,b):c_{ab}\ne0\}\in\ZZ^2_{\mathrm{lex}}, \] and extend by \(v(P/Q)=v(P)-v(Q)\). The least monomial of a product is the product of the least monomials, so this is well-defined and multiplicative; the inequality for addition follows from the minimum. Thus it is a valuation, trivial on \(k\), with value group \(\ZZ^2_{\mathrm{lex}}\).
At stage \(i\) use coordinates \(y,u_i=x/y^i\), starting with \(u_0=x\). Their values are \((0,1)\) and \((1,-i)\), both positive in lexicographic order. The center is therefore the closed point \(y=u_i=0\). Blowing up that point and taking the chart \(u_{i+1}=u_i/y\) repeats this construction, with the next center on the exceptional curve \(y=0\). The valuation ring dominates every \(A_i\), hence their local union. Its value group is not infinite cyclic, so it is not a DVR.
8. Q.E.D.
Step 1 proves part (a). Step 2 proves the first two surface examples with the stated correction, and step 3 proves the domination and center assertions for the third. Steps 4 and 5 characterize when the third construction is discrete; steps 6 and 7 show that both possibilities occur.