PSC GALOIS EXTENSIONS OF HILBERTIAN FIELDS

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Wulf-Dieter Geyer, Erlangen University and Moshe Jarden, Tel Aviv University

Directory: \jarden\diary\totsig 26 March, 1999 Introduction The main goal of our work is to prove the following result:

Theorem A: Let K be a countable separably Hilbertian field, let S be a finite set of local primes of K, and let e be a nonnegative integer. Then, for almost all σ σ σ ∈ G(K)^{e}, the field K_{s}[σ σ σ] ∩ K_{tot,S} is PSC.

0.1 Notation in theorem A. For an arbitrary field K we denote the separable (resp., algebraic) closure of K by K_{s} (resp., K̃). Let G(K) = G(K_{s}/K) be the absolute Galois group of K. If σ_{1}, …, σ_{e} ∈ G(K), then K_{s}(σ σ σ) is the fixed field of σ_{1}, …, σ_{e} in K_{s}. The field K_{s}[σ σ σ] is the maximal Galois extension of K which is contained in K_{s}(σ σ σ). In particular, if e = 0, then K_{s}[σ σ σ] = K_{s}(σ σ σ) = K_{s}.

The absolute Galois group of K is compact with respect to the Krull topology. So, for each e, G(K)^{e} is a probability space with respect to the normalized Haar measure [FJ2, Chap. 16]. Accordingly, the clause “for almost all σ σ σ ∈ G(K)^{e}” means “for all σ σ σ ∈ G(K)^{e} but for a set of measure 0”.

Suppose now that K is a countable separably Hilbertian field [FJ2, Chap. 11]. By [JR1, Prop. 3.1], the field K_{s}(σ σ σ) is PAC over K, for almost all σ σ σ ∈ G(K)^{e}. In general, for a field M and a subset A of M, we say that M is PAC over A if for each dominant separable rational function φ: V → 𝔸^{r} of absolutely irreducible varieties of dimension r over M there exists x ∈ V (M) such that φ(x) ∈ A^{r}. A weaker possible property of M is to be PAC. This means that each absolutely irreducible variety over M has an M-rational point. In other words, M is PAC over M. By [Ja4, Lemma 1.2], K_{s}[σ σ σ] is PAC for almost all σ σ σ ∈ G(K)^{e}. However, we don’t know if K_{s}[σ σ σ] is PAC over K for almost all σ σ σ ∈ G(K)^{e}.

A local prime of K is an equivalence class 𝔭 of absolute values such that the completion K̂_{𝔭} is a local field. Thus, K̂_{𝔭} is a locally compact field with respect to the 𝔭-adic topology. The field K_{𝔭} = K_{s} ∩ K̂_{𝔭} is then well defined up to a K-isomorphism. It is the Henselian closure of K at 𝔭 if | |_{𝔭} is ultrametric, the real closure of K at 𝔭 if 𝔭 is metric and K is formally real, and the algebraic closure of K if 𝔭 is metric and K is not formally real.

Suppose that S is a finite set of local primes of K. Let

K_{tot,S} = ⋂_{𝔭∈S} ⋂_{σ∈G(K)} K_{𝔭}^{σ}

be the field of totally S-adic numbers. It is the largest Galois extension of K in which each 𝔭 ∈ S totally splits. An algebraic extension M of K is PSC if each absolutely irreducible variety V over M which has a simple M_{𝔮}-rational point for each 𝔭 ∈ S and for each extension 𝔮 of 𝔭 to M also has an M-rational point. This concludes the explanation of the terminology of Theorem A.

0.2 Historical remarks. In [Po1], Pop proves that if char(K) = 0, then K_{tot,S} is PSC. If char(K) > 0, [Po1] replaces K_{tot,S} by its maximal purely inseparable extension. In the case that K is a global field, Green, Pop, and Roquette [GPR, p. 47] sharpen this result. They fix a local prime 𝔭_{0} of K not in S. Then they prove that if V is an absolutely irreducible affine variety over K, and if for each local prime 𝔭 ≠ 𝔭_{0} of K there exists x_{𝔭} ∈ V (K_{𝔭}) such that |x|_{𝔭} ≤ 1, and, in addition, x_{𝔭} is simple if 𝔭 ∈ S, then there exists x ∈ V (K_{tot,S}) such that |x|_{𝔭} ≤ 1 for all 𝔭 ≠ 𝔭_{0}. This is Rumely’s local-global principle for K_{tot,S}.

[JR2, Thm. 1.5] proves that if K is a number field and M is an algebraic extension of K which is PAC over the ring of integers of K and S consists of ultrametric primes, then M ∩ K_{tot,S} satisfies Rumely’s local-global principle (where instead of excluding one prime 𝔭_{0}, one has to exclude all metric primes). This implies that M is PSC [JR2, Remark 8.3(a)]. In particular, K̃(σ σ σ) ∩ K_{tot,S} is PSC for almost all σ σ σ ∈ G(K)^{e}. If, however, K is a function field of one variable over a finite field, then [JR2, Thm. 1.5] proves Rumely’s local global principle, like [Po1], only for the maximal purely inseparable extension of M ∩ K_{tot,S} and not for M ∩ K_{tot,S} itself. We explain the reason for this failure below.

Since we do not know whether K_{s}[σ σ σ] is PAC over K for almost all σ σ σ ∈ G(K), we can not use the results of [JR2] in order to prove Theorem A, even in the case where K is a number field. So, we must supply a direct proof for it.

It turns out that the only property of almost all fields K_{s}(σ σ σ) that we use in order to prove Theorem A is that they are PAC over K. In other words, we prove the following result:

Theorem B: Let S be a finite set of local primes of a field K. Let M be a field which is PAC over K. Denote the maximal Galois extension of K which is contained in M ∩ K_{tot,S} by N. Then N is PSC.

0.3 Ingredients of the proof. Intersecting an arbitrary variety by a suitable hyperplane reduces the check for the PSC property of a field to that of curves. Indeed, in order to prove that a field N between K and K_{tot,S} is PSC it suffices to prove that each curve C over N is birationally equivalent over N to a curve Γ with the following property:

PSC condition: If Γ_{simp}(K_{𝔭}) ≠ ∅ for each 𝔭 ∈ S, then Γ(N) is infinite.

If the function field of C over K is conservative, then C is birationally equivalent to a smooth projective curve Γ. In this case, the proof of the PSC condition for Γ relies on a combination of the theorem about the existence of a stabilizing element for regular extensions of transcendence degree 1 [FJ1, p. 654 and GeJ, p. 336, Thm. F], and Rumely’s existence theorem with rationality conditions [Po1, Thms. 1.1 and 3.1]:

Proposition C: Let S be a finite set of local primes of a field K. Consider a smooth projective curve Γ over K and let F be the function field of Γ. For i = 1, 2 and for each 𝔭 ∈ S let U_{i,𝔭} be a nonempty 𝔭-open subset of Γ(K_{𝔭}). Write U_{i} = ⋂_{𝔭∈S} ⋂_{σ∈G(K)} U_{i,𝔭}^{σ}, i = 1, 2. Let d_{0} ∈ ℕ. Then F/K has a separating transcendental element t such that (a) all geometric zeros of t belong to U_{1} and each of them has multiplicity 1; and

(b) all geometric poles of t belong to U_{2} and each of them has multiplicity 1.

(c) Moreover, let F̂ be the Galois closure of F/K(t). Then F̂/K is a regular extension and G(F̂/K(t)) ≅ S_{d} with d > d_{0}.

Here is the version of Rumely’s existence theorem which we use in the proof of Proposition C. It incorporates a theorem about the continuity of geometric zeros of a curve.

Proposition D: Let S be a finite set of local primes of a field K. Consider a smooth projective curve Γ over K and let F be the function field of Γ. Let 𝔞_{0} be a positive divisor of F/K. For each 𝔭 ∈ S let U_{𝔭} be a nonempty 𝔭-open subset of Γ(K_{𝔭}) and let U = ⋂_{𝔭∈S} ⋂_{σ∈G(K)} U_{𝔭}^{σ}.

Then there exists a positive integer k_{0} such that for each multiple k of k_{0}, for 𝔞 = k𝔞_{0}, and for each basis y_{0}, y_{1}, …, y_{n} of L_{K}(𝔞) there exists a nonempty intersection A ⊆ K^{n+1} of 𝔭-open sets, 𝔭 ∈ S, such that for each a ∈ A, t = ∑_{i=0}^{n} a_{i}y_{i} is a Rumely element with respect to S, 𝔞, U. This means that div(t) = p_{1} + ⋯ + p_{d} − 𝔞, where p_{1}, …, p_{d} are distinct points of Γ(K̃) which belong to U ∖ Supp(𝔞).

To achieve t of Proposition C, suppose that Γ is embedded in ℙ^{n}. We project Γ from a point o_{n} ∈ ℙ^{n}(K) down to ℙ^{n−1}, and then from a point o_{n−1} ∈ ℙ^{n−1} down to ℙ^{n−2} until we reach a plane node curve, which we further project to ℙ^{1}. This is done by choosing a rational function t = t_{1}/t_{2} such that t_{i} is a Rumely element with respect to S, 𝔞, U_{i}, i = 1, 2. The poles of t_{1} and t_{2} cancel each other and we have div(t) = div_{0}(t_{1}) − div_{0}(t_{2}).

The proof of Proposition D takes advantage of the Jacobian variety J of the smooth curve Γ and the compactness of J(K̂_{𝔭}).

In addition to Proposition D, we use the following classical result:

Proposition E: Let Γ be a smooth projective curve over K and let F be the function field of Γ over K. Then there exists a positive integer d_{0} such that each divisor 𝔞 of F/K of degree at least d_{0} is very ample. That is, each basis of L(𝔞) defines an embedding of Γ into the appropriate projective space.

If the function field F of C is not conservative (so, char(K) > 0), then C is not birationally equivalent to a smooth curve over K any more. This forces us to use heavier tools in order to sharpen and generalize Propositions C, D, and E.

First of all we extend the classical proof of Proposition E to an arbitrary projective curve Γ over K. The main ingredient in this generalization is the Riemann-Roch theorem for a semilocal ring O of the function field F of Γ. The striking phenomena here is that, unlike the genus of Γ which may decrease under purely inseparable extensions of the base field K, the genus of O does not change under arbitrary change of the base field K.

The generalized Riemann-Roch theorem is due to Rosenlicht [Ro1] who uses it in [Ro2] to construct a generalized Jacobian variety J of Γ. This variety is not an abelian variety, as it is in the case where Γ is smooth. It is rather an extension of Γ by a linear algebraic group Λ. In particular, for 𝔭 ∈ S, J(K̂_{𝔭}) need not be compact. This presents a severe obstruction to the proof of Rumely’s existence theorem. Nevertheless, in the case where Γ is K-normal and K̂_{𝔭}/K is separable, Green, Pop, and Roquette [GPR] overcome this difficulty by replacing K̂_{𝔭} with a purely inseparable extension L̂ and observing that the group Λ(L̂) is annihilated by a power of char(K). As a result, one may find a Rumely element as in Proposition D. Then it is possible to find an element t that satisfies Conditions (a) and (b) of Proposition C.

This is not good enough, since t has to satisfies Condition (c) of Proposition C as well. Fortunately, Neumann [Neu] establishes this condition in the case where, in our terminology, Γ is a special cusp curve. It is a lucky coincidence that the generalized Jacobian variety of such a curve is an extension of an abelian variety by a linear group Λ such that Λ(K̂_{𝔭}) is annihilated by a power of char(K). So, we are able to use the method of [GPR] to prove the analog of Proposition C for special cusp curves.

Acknowledgement: The authors are indebted to Aharon Razon for critical reading of earlier versions of this work.

1. Pseudo S-adically closed fields Consider a field K and an equivalence class 𝔭 of absolute values of K (see [CaF, Chap. II] or [Art, Chap. 1] for the definition and the basic properties of absolute values). Choose a completion, K̂_{𝔭}, of K at 𝔭. Choose a representative | |_{𝔭} for 𝔭 and extend it to K̂_{𝔭} in the unique possible way. We call the induced topology on K and on K̂_{𝔭} the 𝔭-adic topology. A basic open neighborhood of a ∈ K̂_{𝔭} in the 𝔭-adic topology is the set {x ∈ K̂_{𝔭} | |x − a|_{𝔭} < ε} for some positive real number ε.

We say that 𝔭 is a local prime of K if K̂_{𝔭} is locally compact in the 𝔭-adic topology. If | |_{𝔭} is ultra-metric (i.e., |x + y|_{𝔭} ≤ max(|x|_{𝔭}, |y|_{𝔭}) for all x, y ∈ K), then K̂_{𝔭} is either a finite extension of ℚ_{𝔭} for a certain prime number p, or K̂_{𝔭} is a formal power series 𝔽_{q}((t)) for some power q of a prime number [Kow]. If | |_{𝔭} is metric (i.e., can be chosen to satisfy |x + y|_{𝔭} ≤ |x|_{𝔭} + |y|_{𝔭} for all x, y ∈ K and |n| = n for all n ∈ ℕ), then K̂_{𝔭} = ℝ (in this case we call 𝔭 real) or K̂_{𝔭} = ℂ (in which case we call 𝔭 complex).

We assume that all completions of K as well as their algebraic closures are embedded in a universal field Ω which is algebraically closed. We call K_{𝔭} = K_{s} ∩ K̂_{𝔭} the 𝔭-closure of K at 𝔭. In the ultra-metric case K_{𝔭} is the Henselization of K with respect to 𝔭. In the metric case K_{𝔭} is either a real closure of K with respect to the 𝔭-ordering or K_{𝔭} = K̃. In each case K_{𝔭} depends on the embeddings of K̃ and K̂_{𝔭} in Ω only up to K-conjugation. So, we define

K_{t𝔭} = ⋂_{σ∈G(K)} K_{𝔭}^{σ}

and call it the field of totally 𝔭-adic elements of K̃. It is a Galois extension of K which is independent of the above embeddings.

Let S be a set of local primes of K. Then

K_{tot,S} = ⋂_{𝔭∈S} K_{t𝔭}

is the field of totally S-adic elements of K̃. By definition, K_{tot,S} is the maximal Galois extension of K in which for each 𝔭 ∈ S and for each prime 𝔓 of K_{tot,S} above 𝔭, we have

({ \atop K_{tot,S}})_{𝔓} = K̂_{𝔭}.

Remark 1.1: The S-topology. For each algebraic extension L of K we denote the set of all extensions of the primes in S to L by S_{L}. A basic S-open subset of K̃ is a set of the form

U = ⋂_{𝔭∈S_{L}} ⋂_{𝔭̃|𝔭}{x ∈ K̃ | |x − a_{𝔭}|𝔭̃ < ε_{𝔭}},

where L is a finite extension of K, for each 𝔭 ∈ S_{L} the element a_{𝔭} belongs to L, ε_{𝔭} > 0, and 𝔭̃ ranges over all extensions of 𝔭 to K̃. By the weak approximation theorem [CaF, p. 48], U contains elements of L, in particular U is nonempty.

Each nonempty finite intersection of basic S-open subsets of K̃ contains a basic S-open subset of K̃. Hence, the unions of basic S-open subsets of K̃ form a topology on K̃ which we call the S-topology.

For each positive integer n, equip K̃^{n} with the product topology. It is a Hausdorff topology. Each polynomial in K̃[X_{1}, …, X_{n}] induces an S-continuous map K̃^{n} → K̃. So, if A is an affine K̃-algebraic set, then A(K̃) is closed. Hence, the S-topology naturally extends to a topology of the sets A(K̃) for each quasi-projective set A over K̃.

For each algebraic extension L of K the S-topology of A(K̃) induces a topology on A(L). Thus, we call a subset of A(L̃), S-open, if it is the intersection of an S-open subset of A(K̃) with A(L).

If L ⊆ K_{tot,S}, 𝔭 ∈ S, and 𝔮 is an extension of 𝔭 to L, then each extension of 𝔮 to K_{tot,S} is conjugate over K to the prime of K_{tot,S} which is induced by K_{𝔭}. It follows that 𝔮 is a local prime of L and each closure L_{𝔮} of L at 𝔮 is isomorphic to K_{𝔭} over K. Example 1.2: Local primes.

(a) If K is a number field, then each absolute value of K defines a local prime 𝔭 of K. In case 𝔭 is a metric prime, one also says that 𝔭 is infinite or archimedean. If 𝔭 is a finite prime (in our terminology, ultra-metric prime) and p is the rational prime under 𝔭, then K̂_{𝔭} is a finite extension of the field ℚ̂_{p} of p-adic numbers.

(b) If K is a finite extension of 𝔽_{p}(t), then each absolute value of K defines an ultra-metric prime 𝔭. The residue field K̄_{𝔭} is 𝔽_{p^{m}}, for some positive integer m, and K̂_{𝔭} = 𝔽_{p^{m}}((x)), where x is a prime element of K at 𝔭 [Che., p. 46].

(c) Let S be a set of rational primes. For each p ∈ S the field ℚ̂_{p} has cardinality 2^{ℵ_{0}} and hence also transcendence degree 2^{ℵ_{0}} over ℚ. Choose a transcendence base X_{p} for ℚ̂_{p}/ℚ. Let T be a set of cardinality ≤ 2^{ℵ_{0}} and let K = ℚ(T). Then we may embed T into X_{p} and extend this embedding to an embedding of K into ℚ̂_{p}. It induces a local prime 𝔭 on K such that K̂_{𝔭} = ℚ̂_{p}. Another embedding into ℚ̂_{p} may induce another local prime on K. In this way we get sets of local primes of K which may have large cardinalities.

(d) If K is as in (b), then K̂_{𝔭} is a separable extension of K [Ja3, Lemma 2.2]. We give an example where the latter conclusion does not hold.

Consider the field 𝔽_{p}((t)) of formal power series in t over 𝔽_{p}. There are 2^{ℵ_{0}} elements of the form ∑_{n=0}^{∞} a_{n}t^{pn} with a_{n} ∈ 𝔽_{p}. Since the algebraic closure of 𝔽_{p}(t) is countable, we may choose x of that form which is not algebraic over 𝔽_{p}(t). Let K = 𝔽_{p}(t, x) and let 𝔭 be the local prime of K that the t-adic valuation of 𝔽_{p}((t)) induces. Then K̂_{𝔭} = 𝔽_{p}((t)).

Now observe that x^{1/p} = ∑_{n=0}^{∞} a_{n}t^{n} is in K̂_{𝔭} while K(x^{1/p}) is a purely inseparable extension of K(x) of degree p. Conclude that K̂_{𝔭}/K is not a separable extension.

Definition 1.3: PSC fields. Let S be a set of local primes of a field K and let N be an algebraic extension of K. We say that N is pseudo S-adically closed, and abbreviate it by PSC, if every variety V over N satisfies the following local global principle for N:

(1) If V_{simp}(N_{𝔮}) ≠ ∅ for each 𝔮 ∈ S_{N}, then V (N) ≠ ∅.

Here, and in the sequel, whenever we speak about a variety over K, we mean a scheme V of finite type over K such that V ×_{K} K̃ is reduced and irreducible. In particular, V is nonempty. In the terminology of Weil’s Foundation [Wei], V is a nonvoid abstract variety defined over K (in particular V is absolutely irreducible). Likewise a curve over K is a variety over K of dimension 1. As usual, V_{simp} denotes the Zariskiopen subset of V of all simple points of V. Also, V (N) is the set of all N-rational points of V. Usually, we take V as embedded in 𝔸^{n} or in ℙ^{n} for some positive integer n. Then we view each N-rational point of V as an n-tuple of elements of N or as an equivalence class of nonzero (n + 1)-tuples of N modulo multiplication by elements of N.

Remark 1.4: Comparison with other definitions. Consider a field K, a set S of local primes of K, and an algebraic extension N of K.

(a) By definition, N is PSC if and only if N is PS_{N}C.

(b) Suppose that V is a variety over K with a simple K_{𝔭}-rational point for each 𝔭 ∈ S. Then it has a simple K_{𝔭}^{σ}-rational point for each 𝔭 ∈ S and each σ ∈ G(K).

(c) Let K be a family of field extensions of K. We say that K is PKC if every variety V over K with a simple K̄-rational point for each K̄ ∈ K has a K-rational point [Ja2, §7]. In particular, N is PSC if and only if N is P N C with N being the set {N_{𝔭}^{σ} | 𝔭 ∈ S_{N}, σ ∈ G(N)}.

(d) In particular, suppose that S is the empty set. Then for N to be PSC means that every variety V over N has an N-rational point. So, N is PSC if and only if N is PAC.

(e) Let S_{0} be the set obtained from S by removing all complex local primes. Since each variety has simple K̃-rational points, N is PSC if and only if N is PS_{0}C.

(f) One says that N is PRC if N is P N C with respect to the family N of all real closures of N. Suppose that S consists of all real metric absolute values of K. Then, the corresponding completions are ℝ. Hence, if in this case N is PSC, then it is also PRC.

The converse is not true. Indeed, take K = N to be a real closed field which can not be embedded in ℝ (e.g., take K to be a nonprincipal ultrapower of ℝ). Then K is PRC but not PAC [FJ2, Thm. 10.12]. Since S is empty, this means, by (d), that K is not PSC.

(g) Consider a rational prime p. Take now S to be the set of all local primes 𝔭 of a field K such that K̂_{𝔭} = ℚ̂_{p}. For such 𝔭, char(K) = 0, the residue field, K̄_{𝔭}, of K_{𝔭} is 𝔽_{p}, and v_{𝔭}(p) is the smallest positive integer of v_{𝔭}(K_{𝔭}^{×}). In other words v_{𝔭} is a p-adic valuation of K_{𝔭}. Also, [E : K_{𝔭}] = [Ē_{𝔭} : K̄_{𝔭}][v_{𝔭}(E^{×}) : v_{𝔭}(K_{𝔭}^{×})] for each finite extension E of K_{𝔭} [Ja3, Lemma 2.2(a)]. Thus if [E : K_{𝔭}] > 1, then either the residue field of E is larger than 𝔽_{p}, or v_{𝔭}(p) is not the smallest positive element of v_{𝔭}(E^{×}). So, the unique extension of v_{p} to E is not a p-adic valuation. Conclude that K_{𝔭} is a p-adic closure of K.

It follows that if N is a subextension of K_{tot,S}/K and N is PSC, then N is also PpC. That is, N is P N C with respect to the set N of all p-adic closures of N.

The converse is not true. Suppose, for example, that K is a p-adically closed field such that the value group of the unique p-adic valuation v of K [HJ1, Prop. 6.3] is not isomorphic to ℤ (e.g., K is a nonprincipal ultrapower of ℚ̂_{p}). Then K is PpC and G(K) ≅ G(ℚ̂_{p}) [HJ1, Cor. 6.6]. In particular cd(G(K)) = 2 [Rib, p. 281] and therefore G(K) is not projective. Hence, K is not PAC [FJ2, Thm. 10.17]. Since S is empty, this implies that K is not PSC.

(h) Let 𝔭 be a local prime of K. If K̂_{𝔭}/K_{𝔭} is a separable extension, then it is also a regular extension. It follows from [Ja3, Lemma 2.3] that K_{𝔭} is existentially closed in K̂_{𝔭}. In particular, if a variety V over K_{𝔭} has a simple K̂_{𝔭}-rational point, then V also has a simple K_{𝔭}-rational point. If K̂_{𝔭}/K_{𝔭} is not separable, then char(K) > 0. Nevertheless, as Luck Darnière (oral communication) pointed out, K_{𝔭} is Henselian. So, the implication “V_{simp}(K̂_{𝔭}) ≠ ∅ ⟹ V_{simp}(K_{𝔭}) ≠ ∅” is still valid.

Indeed, the function field F of V over K_{𝔭} is a regular extension of K_{𝔭}. Let t = (t_{1}, …, t_{r}) be a separating transcendence basis for F/K_{𝔭}, let y be a primitive element for F/K_{𝔭}(t), and let f ∈ K_{𝔭}[t, Y] be a monic irreducible polynomial such that f(t, y) = 0. Then \frac{∂f}{∂Y} (t, y) ≠ 0. The equation f(T, Y) = 0 defines a variety W over K_{𝔭} which is birationally equivalent to V. Thus V_{simp} has a nonempty Zariski-open subset V_{0}, W has a nonempty Zariski-open subset W_{0} on which \frac{∂f}{∂Y} does not vanish, and there exists an isomorphism φ: V_{0} → W_{0} which is defined over K_{𝔭}. As V_{simp}(K̂_{𝔭}) ≠ 0, the density theorem 8.2(b) gives a point p̂ ∈ V_{0}(K̂_{𝔭}). Then (â, b̂) = φ(p̂) ∈ W_{0}(K̂_{𝔭}). In particular, f(â, b̂) = 0 and \frac{∂f}{∂Y} (â, b̂) ≠ 0. Since K_{𝔭} is 𝔭-dense in K̂_{𝔭}, we may approximate â by a ∈ K_{𝔭}^{r}. Then, we may use the Henselian property of K̂_{𝔭} in order to find b ∈ K̂_{𝔭} such that f(a, b) = 0 and \frac{∂f}{∂Y} (a, b) ≠ 0. Thus, b is separable over K_{𝔭} and therefore belongs to K_{𝔭}. It follows that (a, b) ∈ W_{0}(K_{𝔭}). The point p = φ^{−1}(a, b) belongs then to V_{simp}(K_{𝔭}).

Note that indeed, the assumption “K̂_{𝔭}/K is separable” does not appear in any of the theorems of this work.

(i) Suppose that S is a finite set of local primes of K, let N = K_{tot,S}, and let N_{ins} be the maximal purely inseparable extension of N. The main theorem of [Po1] states:

(2) If V is a K̃-normal variety over N_{ins} (i.e., V ×_{K} K̃ is normal) and V has a simple N̂_{ins,𝔮}-rational point for each 𝔮 ∈ S_{N_{ins}}, then V has an N_{ins}-rational point. One knows that every variety V over N_{ins} is birationally equivalent over N_{ins} to

a K̃-normal variety. Moreover, under this equivalence, simple N̂_{ins,𝔮}-rational points are mapped onto simple N̂_{ins,𝔮}-rational points. Hence, by (2), N_{ins} is PSC.

(j) In [GPR, §1.4], Green, Pop, and Roquette consider a global field K and a finite set of local primes S of K. Let N = K_{tot,S} and let O_{N} be the ring of all elements a ∈ N which are 𝔮-integral for each 𝔮 ∈ S_{N}. For each 𝔭 ∈ S let Ô_{𝔭} be the ring of integers of K̂_{𝔭}. The Main Theorem of [GPR] implies that each affine variety W over K satisfies the following local-global principle:

(3) If W_{simp}(Ô_{𝔭}) ≠ ∅ for each 𝔭 ∈ S, then W (O_{N}) ≠ ∅.

We claim that this result implies that N is PSC. Indeed, suppose that V is a variety over N with a point a_{𝔮} ∈ V_{simp}(N_{𝔮}), for each 𝔮 ∈ S_{N}. Then V is already defined over a finite subextension K′ of N/K. For each 𝔭 ∈ S_{K′}, let O_{𝔭} be the ring of 𝔭-integers of K′_{𝔭}. Further, choose 𝔮 ∈ S_{N} over 𝔭 and let a_{𝔭} = a_{𝔮}. By Proposition 8.2(b), we may assume that V is affine and is embedded in 𝔸^{n}. Next choose c ∈ K^{×} such that ca_{𝔭} ∈ O_{𝔭}^{n} for each 𝔭 ∈ S_{K′}. Multiplication with c gives an automorphism of 𝔸^{n} which maps V onto an affine variety W over K′ and a_{𝔭} onto the point ca_{𝔭} ∈ W_{simp}(O_{𝔭}). By (3), applied to K′ instead of to K, W (O_{N}) ≠ ∅. Hence, applying c^{−1}, we get that V (N) ≠ ∅. Conclude that N is PSC.

(k) Suppose again that K is a global field and that S is a finite set of ultra-metric local primes of K. [JR2, Thm. 1.5 and Cor. 1.9] proves that for almost all σ σ σ ∈ G(K)^{e}, each affine variety W which is defined over N = (K_{s}(σ σ σ) ∩ K_{tot,S})_{ins} satisfies (3). As in (j), this implies that N is PSC.

Once a variety V over a PSC field N has a simple N-rational point, then it has many N-rational points:

Proposition 1.5: Let S be a set of local primes of a field K and let N be a subextension of K_{tot,S}/K. Suppose that N is PSC. Let V be a variety over K such that V_{simp}(K_{𝔭}) ≠ ∅ for each 𝔭 ∈ S. Then V (N) is Zariski-dense in V. In particular, V_{simp}(N) is not empty.

Proof: Let W be a nonempty Zariski-open subset of V. In particular, W itself is a variety over K. By Proposition 8.2(b), W_{simp}(K_{𝔭}) ≠ ∅ for each 𝔭 ∈ S. Hence, W (N) ≠ ∅.

The following lemma reduces the check for PSC to curves, indeed to only one curve out of each birationally equivalence class.

Lemma 1.6: Let S be a finite set of local primes of a field K and let N be a subextension of K_{tot,S}/K. Suppose that each curve C over K is birationally equivalent to a curve Γ over K that satisfies the PSC-Condition: Γ_{simp}(K_{𝔭}) ≠ ∅ for each 𝔭 ∈ S ⟹ Γ(N) is Zariski-dense in Γ. Then N is PSC.

Proof: The proof splits into three parts.

Part A: Varieties over K. We prove that if W is an arbitrary affine variety over K and W_{simp}(K_{𝔭}) ≠ ∅ for each 𝔭 ∈ S, then W (N) ≠ ∅.

Indeed, for each 𝔭 ∈ S choose a_{𝔭} ∈ W_{simp}(K_{𝔭}). Since all K_{𝔭} are separable over K, there exists an affine curve C on W over K which goes through each of the points a_{𝔭} [JR2, Lemma 10.1]. By assumption, there exists a curve Γ which is birationally equivalent to C over K and which satisfies the PSC-Condition. Then there exist nonempty affine Zariski-open subsets Γ_{0} and C_{0} of Γ_{simp} and C_{simp} respectively, and there exists an isomorphism Γ_{0} → C_{0} over K. By the density theorem 8.2(b), we may assume that for each 𝔭 ∈ S we have a_{𝔭} ∈ C_{0}(K_{𝔭}). Hence, Γ_{0}(K_{𝔭}) ≠ ∅. So, by the PSC-condition, Γ_{0}(N) ≠ ∅. Hence C(N) ≠ ∅. So, W (N) ≠ ∅, as claimed.

Part B: Descent. Let U be an affine variety over N with a simple N_{𝔮}-rational point for each 𝔮 ∈ S_{N}. We prove that U(N) ≠ ∅.

Indeed, U is already defined over a subextension K′ of N/K of a finite degree d. Choose σ_{1}, …, σ_{d} ∈ G(K) whose restrictions to K′ are the d K-embeddings of K′ into K_{s}. Weil’s descent gives an affine variety W over K and a K′-isomorphism φ: W → ∏_{i=1}^{d} σ_{i}U [FJ2, Prop. 9.34]. Let 𝔭 ∈ S and let 1 ≤ i ≤ d. Then N ⊆ K_{tot,S} ⊆ σ_{i}^{−1}K_{𝔭}, and therefore σ_{i}^{−1}K_{𝔭} is N_{𝔮} for some 𝔮 ∈ S_{N}. Choose a_{i} ∈ U_{simp}(σ_{i}^{−1}K_{𝔭}). Then σ_{i}a_{i} ∈ (σ_{i}U)_{simp}(K_{𝔭}). Hence, φ^{−1}(σ_{1}a_{1}, …, σ_{d}a_{d}) ∈ W_{simp}(K_{𝔭}). By Part A, W (N) ≠ ∅. Hence, U(N) ≠ ∅.

Part C: Conclusion of the proof. Consider an arbitrary variety V over N such that V_{simp}(N_{𝔮}) ≠ ∅ for each 𝔮 ∈ S_{N}. Let U be an open affine subset of V. By Proposition 8.2(b), U_{simp}(N_{𝔮}) ≠ ∅ for each 𝔮 ∈ S_{N}. Hence, by Part B, U(N) ≠ ∅ and therefore V (N) ≠ ∅. Conclude that N is PSC.

The local-global theorem for varieties that a PKC field satisfies gives a local-global theorem for the corresponding Brauer groups.

Proposition 1.7: Let K be a field and let K be a family of separable algebraic field extensions of K such that K is PKC. Then

(a) the natural map ι: Br(K) → ∏_{K̄∈K} Br(K̄) is injective.

(b) If G(K̄) is projective for each K̄ ∈ K, then G(K) is projective.

Proof of (a): Starting from a central simple K-algebra A, one may use the properties of the Brauer-Severi variety associated with A (as stated in [Jac, §§3.5–3.8]) to prove (a). We prefer however to use the notion of the reduced norm in order to give a more elementary proof of (a).

We are using Br(K) to denote the Brauer group of K. Each nontrivial element of Br(K) may be represented by a central division algebra D over K such that dim_{K} D = n^{2} and n > 1. Assume that for each K̄ ∈ K the algebra D ⊗_{K} K̄ represents the trivial element of Br(K̄). This means that there exists a K̄-algebra isomorphism α: D ⊗_{K} K̄ → M_{n}(K̄). Choose a K-basis {e_{ij} | 1 ≤ i, j ≤ n} for D and let E_{ij} = α(e_{ij}). Then {E_{ij} | 1 ≤ i, j ≤ n} is a K̄-basis for M_{n}(K̄). Write each a ∈ D as ∑_{i,j=1}^{n} a_{ij}e_{ij} with a_{ij} ∈ K. The matrix a = (a_{ij})_{1≤i,j≤n} ∈ M_{n}(K) then satisfies α(a) = ∑_{i,j=1}^{n} a_{ij}E_{ij} = (λ_{kl}(a))_{1≤k,l≤n}, where λ_{kl} are linearly independent linear forms over K̄ in the n^{2} variables X_{ij}. Let X = (X_{ij})_{1≤i,j≤n}. By a theorem of Skolem and Noether [Deu, p. 43, Satz 5], each automorphism of M_{n}(K̄) is inner. This implies that p(X) = det(λ_{kl}(X)) is a polynomial of degree n over K which is independent of K̄ and α. The element red.norm(a) = p(a) = det(α(a)) = det(λ_{kl}(a)) is the reduced norm of a. It satisfies the formula red.norm(ab) = red.norm(a) · red.norm(b) for all b ∈ D. In particular, if a ∈ D and a ≠ 0, then there exists a′ ∈ D such that aa′ = 1. So, red.norm(a) ≠ 0.

The change of variables Y_{kl} = λ_{kl}(X) maps p(X) onto det(Y), which is an absolutely irreducible polynomial. Hence, p(X) is also absolutely irreducible. Let b be the matrix in M_{n}(K̄) which has a zero entries everywhere except in the first n − 1 places along the diagonal matrix. Then b is a simple zero of det(Y) = 0. It follows that also p(X) has a simple K̄-rational zero. Since K is PKC, there exists a ∈ M_{n}(K) which is a simple nontrivial zero of p(X) = 0. Then a = ∑_{i,j=1}^{n} a_{ij}e_{ij} is a nonzero element of D whose reduced norm is 0. This contradiction proves that the map ι is injective.

Proof of (b): Let K′ be a finite separable extension of K. Let K′ = {K̄K′ | K̄ ∈ K}. By [Ja2, Lemma 7.2], K′ is PK′C. Since G(K̄) is projective, Br(K̄K′) = 0 [Rib, p. 261, Cor. 3.7]. Hence, by (a), Br(K′) = 0. It follows from [Rib, p. 261] that G(K) is projective.

Remark 1.8: Part (b) of Proposition 1.7 generalizes the second part of Theorem 3.2 of [Po2], which is proved by different methods.

2. Ample divisors of curves The theory of very ample divisors for smooth curves over algebraically closed fields is well documented in various text books. Here we prove the existence of very ample divisors for singular curves over arbitrary fields.

2.1 Points on curves versus prime divisors of function fields. Let K be a field. Consider the m-dimensional projective space ℙ^{m} = ℙ_{K}^{m} over a field K. For each field extension L of K the points of ℙ^{m}(L) are the equivalence classes (p_{0}:p_{1}: ⋯:p_{m}) of (m + 1)-tuples of elements of L, not all zero, modulo multiplication by elements of L^{×}. We write K(p_{0}: ⋯:p_{m}) for the field K(\frac{p_{0}}{p_{j}} , \frac{p_{1}}{p_{j}} , …, \frac{p_{m}}{p_{j}}), where j is chosen such that p_{j} ≠ 0.

Consider a projective curve Γ ⊂ ℙ^{m} over K (which is, by our convention, absolutely irreducible). Let F be the function field of Γ over K. Denote the genus of F/K by g. Denote the set of all prime divisors of F/K by PrimDiv(F/K). For each P ∈ PrimDiv(F/K) let v_{P} be the normalized discrete valuation associated with P. Also, let O_{P} (resp., M_{P}) be the valuation ring (resp., its maximal ideal) of P. Then choose a place φ_{P}: F → K̃ ∪ {∞} that represents P. Note that φ_{P} is determined by P only up to conjugation over K. For each f ∈ O_{P} write φ_{P} (f) also as f(P).

Now choose a generic point x = (x_{0}:x_{1}: ⋯:x_{m}) for Γ over K with homogeneous coordinates x_{0}, x_{1}, …, x_{m} in F such that K(x_{0}: ⋯:x_{m}) = F. For each prime divisor P of F/K choose y ∈ F^{×} such that \frac{x_{i}}{y} ∈ O_{P} for each i and \frac{x_{j}}{y} ∉ M_{P} for at least one j (e.g., y = x_{j} with v_{P} (x_{j}) = min(v_{P} (x_{0}), …, v_{P} (x_{m})). Then the prime ideal K[\frac{x_{0}}{y} , …, \frac{x_{m}}{y}] ∩ M_{P} of K[\frac{x_{0}}{y} , …, \frac{x_{m}}{y}] gives a closed point p of Γ called the center of P at Γ. The point (\frac{x_{0}}{y} (P): ⋯:\frac{x_{m}}{y} (P)) of Γ(K̃), with homogeneous coordinates \frac{x_{i}}{y} (P), lies over p and we sometimes abuse notation and call it also the center of P at Γ (although it is determined by P only up to K-conjugation). We will also say that P lies over p.

Conversely, let p be a closed point of Γ. Denote the local ring of p in F by O_{Γ,p} and let M_{Γ,p} be its maximal ideal. There are finitely many K-homomorphisms φ_{p}: O_{Γ,p} → K̃ with kernel M_{Γ,p}, all conjugate over K. Nevertheless, we abuse notation and for f ∈ O_{Γ,p} we write φ_{p}(f) also as f(p) while we remember that f(p) is determined by p only up to K-conjugation. Next, extend φ_{p} to a K-place φ: F → K̃ ∪ {∞}. Let P be the corresponding prime divisor of F/K that φ determines. Then P lies over p. That is, O_{Γ,p} ⊆ O_{P} and O_{Γ,p} ∩ M_{P} = M_{Γ,p}.

There are only finitely many prime divisors P of F/K that lie over each point p of Γ. If p is simple^{∗}, then O_{Γ,p} is a valuation ring and P is unique^{∗∗}. So, we identify p in this case with P and write v_{p} instead of v_{P}.

2.2 Semilocal rings. A semilocal ring O of F/K is the intersection of finitely many local rings O_{1}, …, O_{n} of F that contain K such that F is the quotient field of O. Denote the maximal ideal of O_{i} by M_{i}, i = 1, …, n. Like every ring between K and F, the ring O is Noetherian. Indeed, let A be a nonzero ideal of O and choose b ∈ A, b ≠ 0. Then dim_{K} O/Ob < ∞ [Ro1, Thm. 1]. As A/Ob is a subspace of O/Ob, it has a finite basis, say a_{1} + Ob, …, a_{m} + Ob. It follows that A = Oa_{1} + ⋯ + Oa_{m} + Ob is finitely generated. Conclude that O is Noetherian. Since the transcendence degree of F/K is 1, the dimension of O is at most 1. That is, each nonzero prime ideal of O is maximal.

For each i between 1 and n consider the prime ideal P_{i} = M_{i} ∩ O of O, which, by what we have just said, is maximal (but it is possible that P_{i} = P_{j} for i ≠ j). Then, O ⊆ ⋂_{i=1}^{n} O_{P_{i}} ⊆ ⋂_{i=1}^{n} O_{i} = O. So, O = ⋂_{i∈I} O_{P_{i}}. Hence, by [Bou, p. 93, Cor.], the P_{i}’s are all maximal ideals of O.

Conversely, let R be a subring of F which contains K such that F is the quotient field of R. Suppose that R has only finitely many maximal ideals Q_{1}, …, Q_{k}. Then R = ⋂_{j=1}^{k} R_{Q_{j}} and therefore R is a semilocal ring of F/K.

Let S̄ be the set of all P ∈ PrimDiv(F/K) that lie over O, that is O ⊆ O_{P}. By [Ro1, Thm. 3], S̄ is also the set of all prime divisors of F/K that lie over at least one of the rings O_{i}. Let Ō = ⋂_{P ∈S̄} O_{P}. By [Lan, p. 12, Prop. 4], Ō is the integral closure of O in F. It is a Dedekind domain with finitely many maximal ideals (hence a principal ideal domain), namely M̄_{P} = Ō ∩ M_{P}, with P ∈ S̄ [FJ2, Prop. 2.12]. By [Ro1, p. 170, Thm. 1] dim_{K} Ō/O < ∞. Let C = {a ∈ O | aŌ ⊆ O} be the conductor of O in Ō. It is the largest ideal of O which is also an ideal of Ō. In particular, C = ∏_{P ∈S̄} M̄_{P}^{k_{P}} for some nonnegative integers k_{P} (see also [Ro1, p. 171, Cor. 2]). It follows that

* Following Weil [Wei, p. 99] and Lang [Lan, p. 198] we call p simple if it satisfies the

Jacobian criterion. That is , we take an affine open neighborhood Γ_{0} ⊂ 𝔸^{m} for p, let

f_{1}, …, f_{r} be generators for the ideal of all polynomials in K[X_{1}, …, X_{m}] that vanish on

Γ_{0} and demand that rank( \frac{∂f_{i}}{∂X_{j}} (p)) = m − 1. If p is not simple, we call it singular. We

denote the set of all singular points of Γ by Γ_{sing}.

** Moreover, if Γ̃ = Γ ×_{K} K̃ and p̃ is a point of Γ̃ that lies over p, then O_{Γ̃,p̃} is still a

valuation ring. So, one may also say that p is geometrically simple.

(1) if a ∈ O, x ∈ F, and v_{P} (x − a) ≥ k_{P} for each P ∈ S̄, then x ∈ O.

By [ZaS, p. 269, Cor.],

(2) k_{P} > 0 if and only if the local ring of O at O ∩ M_{P} is not integrally closed. We call 𝔠 = ∑_{P ∈S̄} k_{P} P the conductor divisor of O.

For each divisor 𝔞 of F/K we consider the K-vector space

L(𝔞) = L_{F/K}(𝔞) = {x ∈ F | div(x) + 𝔞 ≥ 0}.

A divisor 𝔞 = ∑_{i=1}^{m} a_{i}P_{i} of F/K is O-smooth if none of the P_{i}’s is in S̄. In this case we also consider the subspace L_{O}(𝔞) = L_{O/K}(𝔞) = L(𝔞) ∩ O. By (1) (with a = 0),

(3)  L(𝔞 − 𝔠) ⊆ L_{O}(𝔞).

Indeed, if x ∈ L(𝔞 − 𝔠) and P ∈ S̄, then v_{P} (𝔞) = 0 and so v_{P} (x) = v_{P} (x) + v_{P} (𝔞) ≥ k_{P}.

The Riemann-Roch theorem for Γ controls the dimension of L_{O}(𝔞):

(4) There exists a nonnegative integer genus(O/K) and for each O-smooth divisor 𝔞 of

F/K there exists a nonnegative integer i(𝔞) such that (4a) dim(L_{O}(𝔞)) = deg(𝔞) + 1 − genus(O/K) + i(𝔞) [Ro1, Thm. 7]; (4b) if deg(𝔞) > 2g − 2 + deg(𝔠), then dim(L_{O}(𝔞)) = deg(𝔞) + 1 − genus(O/K)

[Ro1, Thm. 9].

2.3 The map associated with a Γ-smooth divisor. Let S be the (finite) set of singular points of Γ. Let S̄ be the set of all prime divisors of F/K that lie over S. Let

O = ⋂_{p∈S} O_{Γ,p},  Ō = ⋂_{P ∈S̄} O_{P} .

We call O the semilocal ring of singularities of Γ. If Γ is smooth, then O = F and genus(F/K) = g is the usual genus of Γ (or of F/K) [Ro1, Thm. 8].

A Γ-smooth divisor is a formal sum 𝔞 = ∑_{i=1}^{n} a_{i}p_{i}, with p_{i} simple points of Γ. By our identification, it is also an O-smooth divisor. Suppose that n = dim(L_{O}(𝔞))−1 ≥ 2. Choose a K-basis f_{0}, …, f_{n} for L_{O}(𝔞). Then K(f_{0} : ⋯: f_{n}) ⊆ F and therefore f = (f_{0} : ⋯: f_{n}) generates an absolutely irreducible projective curve Δ in ℙ^{n} over K (Δ is the closure of f in ℙ^{n}). Moreover, the map x ↦ f defines a rational map ψ: Γ → Δ. The map ψ is defined at a point p of Γ if there exists y ∈ F^{×} such that each \frac{f_{i}}{y} belongs to O_{Γ,p} and not all of them are in M_{Γ,p}. In this case \frac{f}{y} (p) = (\frac{f_{0}}{y} (p): ⋯:\frac{f_{n}}{y} (p)) is a well defined point of Δ(K̃) which does not depend on the choice of y but depends on the choice of the place φ over p up to K-conjugation. So, the point q of Δ which lies below \frac{f}{y} (p) is well defined and ψ(p) = q. In particular, O_{Δ,q} ⊆ O_{Γ,p}. We abuse notation and write in this case also ψ(p) = \frac{f}{y} (p). We add the subscript 𝔞 to ψ whenever we want to emphasize the dependence of ψ on 𝔞.

Another choice of the basis changes Δ and ψ by a linear isomorphism of ℙ^{n}. We say that the map ψ: Γ → Δ is associated with 𝔞. Let p be a point of Γ. We say that the divisor 𝔞 is very ample at p if ψ is biregular at p, i.e., O_{Δ,q} = O_{Γ,p}. The divisor 𝔞 is very ample on Γ if it is very ample at each p ∈ Γ. That is, ψ is an isomorphism.

Lemma 2.4: Suppose that K is algebraically closed. Let 𝔞 be a Γ-smooth divisor and let p ∈ Γ(K). Then 𝔞 is very ample at p in each of the following two cases: (A) p is simple and deg(𝔞) ≥ 2g + 1 + deg(𝔠). (B) p ∈ S and deg(𝔞) ≥ 2g − 1 + 2 deg(𝔠).

Proof: Let ψ: Γ → Δ be the rational map associated with 𝔞.

Case A: In this case O_{Γ,p} is a valuation ring of F/K. We proceed along classical lines.

Claim A1: ψ is defined at p and O_{Γ,p} is the integral closure of O_{Δ,ψ(p)}. Indeed, let P′ be a prime divisor of F/K which is not p. Since K is algebraically closed, deg(p) = deg(P′) = 1. Hence, by assumption, deg(𝔞 − 𝔠 − p − P′) ≥ 2g − 1. By Riemann-Roch, L(𝔞 − 𝔠 − p − P′) is properly contained in both L(𝔞 − 𝔠 − p) and L(𝔞 − 𝔠 − P′). By definition, L(𝔞 − 𝔠 − p) ∩ L(𝔞 − 𝔠 − P′) = L(𝔞 − 𝔠 − p − P′). As both L(𝔞 − 𝔠 − p) and L(𝔞 − 𝔠 − P′) are contained in L(𝔞 − 𝔠), (3) implies that they are contained in L_{O}(𝔞). Moreover, both dim(L(𝔞 − 𝔠 − p)) and dim(L(𝔞 − 𝔠 − P′)) are greater than dim(L(𝔞 − 𝔠 − p − P′)) by 1.

So, we may choose a K-basis g_{0}, …, g_{n} for L_{O}(𝔞) such that g_{l} ∈ L(𝔞 − 𝔠 − P′) ∖ L(𝔞 − 𝔠 − p − P′) and g_{l+1} ∈ L(𝔞 − 𝔠 − p) ∖ L(𝔞 − 𝔠 − p − P′) for some l between 0 and n − 1. In particular v_{p}(g_{i}) ≥ −v_{p}(𝔞) for i = 0, …, n, v_{p}(g_{l}) = −v_{p}(𝔞), and v_{p}(g_{l+1}) ≥ −v_{p}(𝔞) + 1. Similarly, v_{P′}(g_{i}) ≥ −v_{P′}(𝔞), for i = 0, …, n, v_{P′}(g_{l}) ≥ −v_{P′}(𝔞 − 𝔠) + 1, and v_{P′}(g_{l+1}) = −v_{P′}(𝔞 − 𝔠).

It follows that \frac{g_{i}}{g_{l}} ∈ O_{Γ,p}, i = 0, …, n. Also, \frac{g_{l}}{g_{l}} = 1 ∉ M_{Γ,p}. Hence, ψ can be defined at p as ψ(p) = \frac{g}{g_{l}} (p). Let q = ψ(p). Since v_{P′}(\frac{g_{l+1}}{g_{l}}) ≤ −1, we have \frac{g_{l+1}}{g_{l}} ∉ O_{P′}. On the other hand, \frac{g_{0}}{g_{l}} , …, \frac{g_{n}}{g_{l}} generate the coordinate ring of the affine open subset of Δ where the lth coordinate is nonzero. Since q belongs to this set, \frac{g_{l+1}}{g_{l}} ∈ O_{Δ,q}. Hence, O_{Δ,q} ⊈ O_{P′}.

Thus, O_{Γ,p} is the only valuation ring of F that contains O_{Δ,q}. Conclude that O_{Γ,p} is the integral closure of O_{Δ,q}.

Claim A2: ψ is biregular at p. By Claim A1, O_{Γ,p} is finitely generated as an O_{Δ,q}- module [GeJ, Lemma 9.3]. Since K is algebraically closed, we have K = O_{Δ,q}/M_{Δ,q} ⊆ O_{Γ,p}/M_{Γ,p} = K. Hence, O_{Γ,p} = O_{Δ,q} + M_{Γ,p}. If we find a prime element t for M_{Γ,p} in O_{Δ,q}, then t will belong to M_{Δ,q}. Hence, M_{Γ,p} = tO_{Γ,p} = M_{Δ,q}O_{Γ,p} and therefore O_{Γ,p} = O_{Δ,q} + M_{Δ,q}O_{Γ,p}. By Nakayama’s lemma, this will imply that O_{Γ,p} = O_{Δ,q} and we will be done.

To find t, observe that deg(𝔞 − 𝔠) > deg(𝔞 − 𝔠 − p) > deg(𝔞 − 𝔠 − 2p) ≥ 2g − 1. Hence, by Riemann-Roch, dim(L(𝔞−𝔠)) = dim(L(𝔞−𝔠−p))+1 and dim(L(𝔞−𝔠−p)) = dim(L(𝔞 − 𝔠 − 2p)) + 1. So, by (3), L_{O}(𝔞) ⊇ L(𝔞 − 𝔠) ⊃ L(𝔞 − 𝔠 − p) ⊃ L(𝔞 − 𝔠 − 2p).

Choose a K-basis h_{0}, …, h_{n} for L_{O}(𝔞) such that h_{r} ∈ L(𝔞 − 𝔠 − p) ∖ L(𝔞 − 𝔠 − 2p) and h_{r+1} ∈ L(𝔞−𝔠) ∖ L(𝔞−𝔠−p) for some r between 0 and n−1. Then v_{p}(h_{i}) ≥ −v_{p}(𝔞) for each i, v_{p}(h_{r}) = −v_{p}(𝔞) + 1, and v_{p}(h_{r+1}) = −v_{p}(𝔞).

The (n + 1)-tuple (\frac{h_{0}}{h_{r+1}} , …, \frac{h_{n}}{h_{r+1}} ) generates the coordinate ring of an affine open neighborhood of ψ(p). In particular, t = \frac{h_{r}}{h_{r+1}} belongs to O_{Δ,q}. By our choice, v_{p}(t) = 1, as desired.

Case B: As p is singular, we have to use special considerations.

Part B1: Lifting a basis of O/C^{2} to L_{O}(𝔞). The ideal C^{2} of O satisfies

dim_{K} O/C^{2} ≤ dim_{K} Ō/C^{2} = deg(2𝔠) < ∞.

Choose y_{0}, …, y_{l} ∈ O such that y_{0}, …, y_{k} modulo C^{2} form a K-basis for C/C^{2} and y_{0}, …, y_{l} modulo C^{2} form a K-basis for O/C^{2}.

Next we change the y_{i}’s such that they will belong to L_{O}(𝔞). To do this let 𝔸 be the ring of adèles of F/K. Let Λ(𝔞 − 2𝔠) be the vector space over K consisting of all α ∈ 𝔸 such that v_{P} (α_{P}) + v_{P} (𝔞 − 2𝔠) ≥ 0 for all P ∈ PrimDiv(F/K). Since deg(𝔞 − 2𝔠) ≥ 2g − 1, the Riemann-Roch theorem implies that 𝔸 = Λ(𝔞 − 2𝔠) + F [FJ2, p. 20]. For each i let α_{i} be the adèle defined by α_{i,P} = y_{i} if P ∈ S̄ and α_{i,P} = 0 else. Then there exist β_{i} ∈ Λ(𝔞 − 2𝔠) and z_{i} ∈ F such that α_{i} = β_{i} + z_{i}. If P ∈ S̄, then v_{P} (𝔞) = 0 and v_{P} (y_{i} − z_{i}) = v_{P} (β_{i}) ≥ v_{P} (2𝔠) = 2k_{P}. Since y_{i} ∈ O, (1) implies that z_{i} ∈ O. In particular, v_{P} (z_{i}) + v_{P} (𝔞) ≥ 0. Also,

(5)  z_{i} ≡ y_{i} mod C^{2}.

If P ∉ S̄, then z_{i} = −β_{i,P} and v_{P} (𝔠) = 0. Hence, v_{P} (z_{i})+v_{P} (𝔞) = v_{P} (β_{i})+v_{P} (𝔞−2𝔠) ≥ 0. Conclude that div(z_{i}) + 𝔞 ≥ 0 and that therefore z_{i} ∈ L_{O}(𝔞).

Claim B2: In the notation of §2.2, k_{P} > 0 for each P ∈ S̄. Indeed, by assumption, each p ∈ S is singular. Since K is algebraically closed, O_{Γ,p} is not integrally closed. As the local ring of O at O ∩ M_{Γ,P} is contained in O_{Γ,P}, it is also not integrally closed. So, by (2), k_{P} > 0.

Part B3: Another application of Nakayama’s lemma. Observe that C, as an ideal of the Noetherian ring O, is a finitely generated O-module. Moreover, by Claim B2, C is contained in ⋂_{q∈S}(O ∩ M_{Γ,q}), which is the Jacobson radical of O. By (5) and by the choice of y_{0}, …, y_{k}, we have C = ∑_{i=0}^{k} Kz_{i} + C^{2} = ∑_{i=0}^{k} Oz_{i} + C^{2}. Hence, by Nakayama’s lemma [Mat, p. 11],

(6)  C = ∑_{i=0}^{k} Oz_{i}.

The choice of y_{0}, …, y_{l} and (5) implies that

(7)  O = ∑_{i=0}^{l} Kz_{i} + C^{2}.

Let R = K[z_{0}, …, z_{l}]. Then, by (7), O = R + C^{2} and therefore O = R + C. Suppose by induction that O = R + C^{k} for some k ≥ 2. Then, by (6), C = ∑_{i=0}^{k}(R + C^{k})z_{i} ⊆ R + C^{k+1}. Hence, O = R + C ⊆ R + R + C^{k+1} = R + C^{k+1} ⊆ O and therefore O = R + C^{k+1}. Thus,

(8)  O = R + C^{r}

for each r ≥ 2.

Consider the multiplicative subset U = ⋂_{p′∈S}(R ∖ M_{Γ,p′}) of R. The maximal ideals of the quotient ring U^{−1}R belong to the set {U^{−1}R ∩ M_{Γ,p′} | p′ ∈ S} [Bou, p. 93, Prop. 17]. In particular there are only finitely many of them. Hence, U^{−1}R is a semilocal ring (by §2.2). By [Ro1, Thm. 2], there exists r ≥ 2 such that U^{−1}R contains all y ∈ F such that v_{P} (y) ≥ r for all P ∈ S̄. Since k_{P} > 0 for each P ∈ S̄ (Claim B2), C^{r} ⊆ U^{−1}R. Since U ⊆ O^{×}, (8) implies that O = U^{−1}R. It follows that R_{R∩M_{Γ,p}} = O_{Γ,p}.

Part B4: Conclusion of the proof. By (5) and by the choice of y_{0}, …, y_{l}, the elements z_{0}, …, z_{l} of L_{O}(𝔞) are linearly independent over K modulo C^{2}. Extend z_{0}, …, z_{l} to a K-basis z_{0}, …, z_{n} of L_{O}(𝔞). Then assume without loss that z_{0}, …, z_{n} define ψ.

Next observe by Claim B2 that f(p) = 0 for each f ∈ C. Hence, by (7), there exists j between k + 1 and l such that z_{j}(p) ≠ 0. Also, for each i, z_{i} ∈ O ⊆ O_{Γ,p}. Hence, ψ is defined at p and q = ψ(p) = (z_{0}(p) : ⋯: z_{n}(p)). Let R′ = K[z_{0}, …, z_{n}]. By Part B3, O_{Δ,q} = R′_{R′∩M_{Γ,p}} = O_{Γ,p}. Conclude that ψ is biregular at p.

The next result will enable us to descend from K̃ to K.

Lemma 2.5: Let K′ be a field extension of K which is linearly disjoint from F, let F′ = F K′, and let Γ′ = Γ ×_{K} K′. Consider a finite subset T of Γ and let T′ be the set of all points of Γ′ that lie over T. Let R = ⋂_{p∈T} O_{Γ,p} and let R′ = ⋂_{p′∈T′} O_{Γ′,p′}. Then R′ = RK′ and R = F ∩ R′. Moreover, for each p′ ∈ T′, O_{Γ′,p′} is the local ring of R′ at R′ ∩ M_{Γ′,p′}.

Proof: If p ∈ T, p′ ∈ T′, and p′ lies over p, then O_{Γ,p} ⊆ O_{Γ′,p′}. Hence, with A′ = RK′, we have A′ ⊆ R′. Since Γ is projective, T is contained in an affine open subset Γ_{0} of Γ. Let A_{0} be the coordinate ring of Γ_{0}. Then A_{0} ⊆ ⋂_{p∈T} O_{Γ,p} = R. Moreover, A′_{0} = A_{0}K′ is the coordinate ring of Γ′_{0} = Γ_{0} ×_{K} K′, which is an affine open neighborhood in Γ′ of each p′ ∈ T′. In particular, O_{Γ′,p′} is the local ring of A′_{0} at A′_{0} ∩ M_{Γ′,p′}. Since A′_{0} ⊆ A′, this implies that O_{Γ′,p′} = A′_{A′∩M_{Γ′,p′}}.

By [Ro1, p. 181, Thm. 11], A′ is a semilocal ring of F′/K′. Moreover, the prime divisors of F′/K′ with nonzero centers in A′ are extensions of the prime divisors of F/K with nonzero centers at R. Hence, by the second paragraph of §2.2, as p′ ranges over T′, A′ ∩ M_{Γ′,p′} ranges over all maximal ideals of A′. It follows from the preceding paragraph that

A′ = ⋂_{p′∈T′} A′_{A′∩M_{Γ′,p′}} = ⋂_{p′∈T′} O_{Γ′,p′} = R′.

Finally, R = F ∩ R′ follows from [Ro1, Thm. 11].

Lemma 2.6: Let 𝔞 be a Γ-smooth divisor of F/K. Let K′ be an extension of K which is linearly disjoint from F, let F′ = F K′, and let Γ′ = Γ ×_{K} K′. Identify 𝔞 as a Γ′-smooth divisor 𝔞′ of F K′/K′. Denote the semilocal ring of singularities of Γ′ in F′ by O′. Then (a) O′ = OK′ and F ∩ O′ = O;

(b) each K-basis of L_{O}(𝔞) is also a K′-basis of L_{O′}(𝔞′);

(c) ψ_{𝔞′}(Γ′) is obtained from ψ_{𝔞}(Γ) by extension of scalars from K to K′;

(d) 𝔞 is very ample if and only if 𝔞′ is very ample; and

(e) genus(O′/K′) = genus(O/K).

Proof of (a): Let S′ be the set of all points of Γ′ that lie over points of S. By definition O′ = ⋂_{p′∈S′} O_{Γ′,p′}. By Lemma 2.5, O′ = OK′ and O = F ∩ O′.

Proof of (b): Suppose that f_{0}, …, f_{n} form a K-basis of L_{O}(𝔞). By the linear disjointness, f_{0}, …, f_{n} are linearly independent over K′. So, it suffices to prove that L_{O′}(𝔞′) = L_{O}(𝔞)K′. Indeed, it suffices to prove that each g′ ∈ L_{O′}(𝔞′) belongs to L_{O}(𝔞)K′. To that end choose a basis {w_{i} | i ∈ I} for K′/K. By the linear disjointness, the w_{i} are linearly independent over F. As g′ ∈ O′ and O′ = OK′, we can write g′ = ∑_{i∈I} g_{i}w_{i}, with g_{i} ∈ O and almost all of them are 0. It suffices to prove that all g_{i} are in L_{O}(𝔞).

Consider therefore a Γ-smooth prime divisor p of F/K. Our convention identifies p with a smooth point of Γ such that O_{Γ,p} = O_{p}. Hence, by Lemma 2.5, O_{p}K′ is the intersection of all valuation rings O_{p′} of F′, where p′ ranges over all prime divisors of F′/K′ which lie over p. Let π_{p} be an element of F such that v_{p}(π_{p}) = 1. Since p is simple, v_{p′}(π_{p}) = 1 for each p′ as above and n_{p} = v_{p}(𝔞) = v_{p′}(𝔞′). By assumption, v_{p′}(g′) + n_{p} ≥ 0. Hence, v_{p′}(π_{p}^{n_{p}}g′) ≥ 0 for all p′ over p. So, ∑_{i∈I} π_{p}^{n_{p}}g_{i}w_{i} = π_{p}^{n_{p}}g′ ∈ O_{p}K′. It follows that π_{p}^{n_{p}}g_{i} ∈ O_{p} and therefore v_{p}(g_{i}) + v_{p}(𝔞) ≥ 0 for all i ∈ I. Since 𝔞 is Γ-smooth, this implies that the g_{i} are indeed in L_{O}(𝔞).

Proof of (c): Apply (b).

Proof of (d): Let p be a point of Γ and let p′ be a point of Γ′ over p. By (c), ψ_{𝔞} is defined at p if and only if ψ_{𝔞′} is defined at p′. So, assume that this is the case. Let q = ψ_{𝔞}(p) and let q′ = ψ_{𝔞′}(p′). By assumption, O_{Δ,q} ⊆ O_{Γ,p} and O_{Δ′,q′} ⊆ O_{Γ′,p′}. Also, O_{Γ,p} ⊆ O_{Γ′,p′} and O_{Δ,q} ⊆ O_{Δ′,q′}.

Suppose first that ψ′ is very ample at p′. Then O_{Δ′,q′} = O_{Γ′,p′}. By [Lan, p. 92], F ∩O_{Δ′,q′} = O_{Δ,q}. Hence, O_{Δ,q} ⊆ O_{Γ,p} ⊆ F ∩O_{Γ′,p′} = F ∩O_{Δ′,q′} = O_{Δ,q} and therefore O_{Δ,q} = O_{Γ,p}. Conclude that ψ_{𝔞} is very ample at p.

Conversely, suppose that ψ_{𝔞} is very ample at p, i.e., O_{Γ,p} = O_{Δ,q}. Then, O_{Γ,p}K′ = O_{Δ,q}K′. By Lemma 2.5, O_{Γ′,p′} (resp., O_{Δ′,q′}) is the local ring of O_{Γ,p}K′ (resp., O_{Δ,q}K′) at O_{Γ,p}K′ ∩ M_{Γ′,p′} (resp., O_{Δ,q}K′ ∩ M_{Δ′,p′}). Since the intersections are the same, so are the corresponding local rings. Conclude that ψ′ is very ample at p′.

Proof of (e): (See also [Ro1, Thm. 11].) Let 𝔠 (resp., 𝔠′) be the conductor of O (resp., O′). Take a positive Γ-smooth divisor 𝔞 of F/K such that

deg(𝔞) > max (2genus(F/K) − 2 + deg(𝔠), 2genus(F′/K′) − 2 + deg(𝔠′)).

By (4b), dim_{K} L_{O}(𝔞) = deg(𝔞) + 1 − genus(O/K) and dim_{K′} L_{O′}(𝔞′) = deg(𝔞′) + 1 − genus(O′/K′). By (b), dim_{K} L_{O}(𝔞) = dim_{K′} L_{O′}(𝔞′). Also, deg(𝔞) = deg(𝔞′). Hence, genus(O′/K′) = genus(O/K).

Proposition 2.7: Let Γ be a projective curve over a field K. Then there is a positive integer n_{0} such that each Γ-smooth divisor 𝔞 with deg(𝔞) ≥ n_{0} is very ample with respect to Γ.

Proof: Let Γ̃ = Γ ×_{K} K̃, let 𝔠̃ be the conductor of Γ̃, and let g̃ = genus(Γ̃). We claim that n_{0} = max{2g̃ + 1 + deg(𝔠̃), 2g̃ − 1 + 2 deg(𝔠̃)} satisfies the conclusion of the proposition.

Indeed, let 𝔞 be a Γ-smooth divisor with deg(𝔞) ≥ n_{0}. Let 𝔞̃ be the extension of 𝔞 to a Γ̃-smooth divisor. Observe that deg(𝔞̃) = deg(𝔞). Denote the ring of singularities of Γ̃ by Õ. Choose a K-basis f_{0}, …, f_{n} for L_{O}(𝔞) and let ψ_{𝔞} be the associated rational map of Γ. Since F is a regular extension of K, it is linearly disjoint from K̃ over K. So, by Lemma 2.6, f_{0}, …, f_{n} is also a K̃-basis of LÕ(𝔞̃). Hence, the associated rational map ψ𝔞̃ of Γ̃ is the extension of ψ_{𝔞}. By Lemma 2.4, 𝔞̃ is very ample on Γ̃. Hence, by Lemma 2.6, 𝔞 is very ample on Γ.

Example 2.8: We show that the lower bound given in Lemma 2.4(b) is optimal.

Let K be an algebraically closed field and let t be a transcendental element over K. For an integer n ≥ 2 consider the rings S = K[t] and R = K[t^{n}, …, t^{2n−1}]. Thus R consists of all sums a_{0} + ∑_{i=n}^{∞} a_{i}t^{i}, where a_{i} ∈ K and almost all of them are 0. Let Γ_{0} = Spec(R).

Observe that t^{n}S ⊆ R. Hence, S is the integral closure of R in K(t). Let p_{0} = tS and q_{0} = t^{n}S. Then q_{0} = p_{0} ∩ R ∈ Γ_{0} and S_{p_{0}} is the integral closure of R_{q_{0}}. Since t is in S_{p_{0}} but not in R_{q_{0}}, the latter ring is not integrally closed.

If q ∈ Γ_{0} and q ≠ q_{0}, choose p ∈ Spec(S) above q. Since t^{n} ∈ R ∖ q, we have t = \frac{t^{n+1}}{t^{n}} ∈ R_{q}. Hence, R_{q} = S_{p} is integrally closed. In other words, q is simple.

Let O = R_{q_{0}} and let Ō = S_{p_{0}}. Then O is a local ring and Ō is its integral closure. Moreover, t^{n}Ō is the conductor of O in Ō. Also, Ō/t^{n}Ō = S/t^{n}S has 1, t, …, t^{n−1} as a K-basis.

By [Ro1, p. 174, Thm. 5], there exists a projective curve Γ whose function field is K(t) and whose ring of singularities is O. In particular, Γ is birationally equivalent to 𝔸^{1} and has genus 0. (Note that Γ is a projective completion of the affine curve Γ_{0} which has a natural embedding in 𝔸^{n}.) If we denote the zero divisor of t by P_{0}, we find by the preceding paragraph that 𝔠 = nP_{0} is the conductor divisor of Γ and that deg(𝔠) = n. Lemma 2.4 therefore asserts that if 𝔞 is a divisor of K(t)/K which is relatively prime to P_{0} and deg(𝔞) ≥ 2n − 1, then 𝔞 is very ample on Γ. We show below that this is not the case any more if deg(𝔞) = 2n − 2.

To this end denote the pole divisor of t by ∞. Then 𝔞 = (2n − 2)∞ is a Γsmooth divisor of degree 2n − 2. If f ∈ L_{O}(𝔞), then f ∈ O and div(f) + 𝔞 ≥ 0. In particular, f has no poles except ∞. Hence, f ∈ S ∩O = R. Moreover, deg(f) ≤ 2n−2. Conversely, the two latter conditions suffice for an f ∈ K(t) to belong to L_{O}(𝔞). Hence, 1, t^{n}, t^{n+1}, …, t^{2n−2} is a basis of L_{O}(𝔞). Let ψ be the rational map of Γ into ℙ^{n−1} defined by this basis and let Δ = ψ(Γ). We claim that ψ is not an isomorphism. More precisely, ψ is not biregular at q_{0}.

Indeed, let r_{0} = ψ(q_{0}). Since 2n−1 is not a linear combination of n, n+1, …, 2n−2 with non-negative integral coefficients, t^{2n−1} ∈ R ∖ K[t^{n}, …, t^{2n−2}]. Hence, t^{2n−1} is in O_{Γ,q_{0}} = O but not in O_{Δ,r_{0}}. So, the latter local ring is a proper subring of the former one, which proves our claim.

One observes directly that for m ≥ 2n−1, the Γ-smooth divisor m∞ is very ample at P_{0}, as Lemma 2.4 asserts.

3. Iteration of projections from points Consider a set of m + 1 linearly independent linear forms

(1)  l_{i}(X) = ∑_{j=0}^{n} a_{ij}X_{j},  i = 0, …, m,

with coefficients a_{ij} in a field K. Consider the linear variety L in ℙ^{n} of dimension n − m − 1 defined by the following system of equations:

(2)  ∑_{j=0}^{n} a_{ij}X_{j} = 0,  i = 0, …, m.

Let A = (a_{ij}) be the matrix of coefficients. Let π = π_{A}: ℙ^{n} ∖ L → ℙ^{m} be the morphism

(3)  π(x_{0}:x_{1}: ⋯:x_{n}) = (l_{0}(x):l_{1}(x): ⋯:l_{m}(x)).

If we abuse notation and write x also for the (n + 1)-tuple (x_{0}, x_{1}, …, x_{n}), we may rewrite (3) as π(x) = Ax^{t}, where the exponent t is the transpose operation of matrices. The morphism π is uniquely defined by L up to a linear isomorphism of ℙ^{m}. Hence, as far as geometric properties are concerned, π depends only on L. So, we abuse notation, denote π also by π_{L}, and call π_{L} the projection from L. For each closed subvariety V of ℙ^{n} over K which is disjoint from L, the restriction π_{L,V} of π_{L} to V is a finite morphism onto a closed subvariety V′ of ℙ^{m} [Mum, p. 174]. In particular, if V is linear, then so is V′ and π_{L,V}: V → V′ is an isomorphism.

Denote the space of r × s matrices by M_{r,s}. It is naturally isomorphic to 𝔸^{rs}. If r ≤ s, let M_{r,s}^{∗} be the set of all matrices in M_{r,s} of rank r, i.e., with independent rows. Thus M_{r,s}^{∗} is a nonempty Zariski-open subset of M_{r,s}.

In particular, A ∈ M_{m+1,n+1}^{∗}(K) acts on K^{n+1} by multiplication from the left and gives a surjective map onto K^{m+1}. Similarly, each B ∈ M_{l+1,m+1}^{∗} gives a surjective map K^{m+1} → K^{l+1}. Hence, C = BA gives a surjective map K^{n+1} → K^{l+1}. So, C ∈ M_{l+1,n+1}^{∗}. By definition, π_{C} = π_{B} ∘ (π_{A}|_{ℙ^{n} ∖ L′}), where L′ is the linear variety of ℙ^{n} defined by Cx^{t} = 0.

Note that the map μ_{l,m,n}: M_{l+1,m+1}^{∗}(K) × M_{m+1,n+1}^{∗}(K) → M_{l+1,n+1}^{∗}(K) defined by μ_{l,m,n}(B, A) = BA is surjective. Indeed, denote the unit (resp., zero) matrix in M_{r,r} (resp., in M_{r,s}) by I_{r} (resp., O_{r,s}) and let C ∈ M_{l+1,n+1}^{∗}. Let also D be an arbitrary matrix in M_{m−l,n+1}. Then

( I_{l+1} O_{l+1,m−l}) ( \begin{matrix}C \\ D \end{matrix} ) = C.

Moreover, rank(I_{l+1} O_{l+1,m−l}) = l+1 and we may choose D such that rank\binom{C}{D} = m+1.

In particular, for each k ≥ 1 let M_{k}^{∗} = M_{k,k+1}^{∗} be the Zariski-open subset of M_{k,k+1} that consists of all matrices of rank k. Let 𝕄 = M_{2}^{∗} × M_{3}^{∗} × ⋯ × M_{n}^{∗} and define a morphism μ: 𝕄 → M_{2,n+1} by multiplication:

μ(A_{2}, A_{3}, …, A_{n}) = A_{2}A_{3} ⋯ A_{n}.

Since multiplication of matrices is associative, μ maps 𝕄(K) onto M_{2,n+1}^{∗}(K).

If A = (a_{ij})_{0≤i≤n−1, 0≤j≤n} ∈ M_{n}^{∗}, then L becomes a point o whose homogeneous coordinates can be obtained by solving (2) according to Cramer’s rule. The map φ_{n}: M_{n}^{∗} → ℙ^{n}, given by φ_{n}(A) = o, is a morphism. Observe that a point (a_{0}:a_{1}: ⋯:a_{n}) ∈ ℙ^{n}(K) with, say, a_{0} ≠ 0 is the unique solution of the system of equations a_{0}X_{i} − a_{i}X_{0} = 0, i = 1, …, n. Hence, φ_{n} maps M_{n}^{∗}(K) onto ℙ^{n}(K). Let

(4)  ℙ = ℙ^{2} × ⋯ × ℙ^{n}, φ = φ_{2} × ⋯ × φ_{n}: 𝕄 → ℙ.

Then φ is a morphism that maps 𝕄(K) onto ℙ(K).

4. Stabilizing elements By a function field of one variable over K we mean a finitely generated regular extension F of K of transcendence degree 1. Thus F has a transcendental element t over K such that F/K(t) is a finite separable extension. Let F̂ be the Galois closure of F/K(t). We say that t symmetrically stabilizes F/K if G(F̂K̃/K̃(t)) ≅ S_{d}, where d = [F : K(t)]. In this case, G(F̂K̃/K̃(t)) ≅ G(F̂/K(t)) [FJ2, Lemma 16.40] and therefore F̂/K is a regular extension.

Given a curve Γ over K, we denote the curve Γ×_{K} K̃ obtained from Γ by extending the field of constants to K̃ by Γ̃. If the function field of Γ is F, then the function field of Γ̃ is F K̃. Let p̃ be a point of Γ̃. Then, for all large n, dimK̃(M_{Γ̃,p̃}^{n}/M_{Γ̃,p̃}^{n+1}) is a fixed positive integer, called the multiplicity of p̃. See [Ful, p. 7, Thm. 2] for the case where Γ̃ is a plane curve. The general case follows from the definition of multiplicity of a prime ideal of an arbitrary Noetherian local ring [Mat2, p. 108]. We say that p̃ is a cusp of Γ̃ if it is singular and if O_{Γ̃,p̃} is contained in a unique valuation ring of F K̃. We say that Γ is a cusp curve if (1a) the only singular points of Γ̃ are cusps; and (1b) Γ̃ has at least one cusp.

We say that Γ is a special cusp curve if there exists an odd prime number q ≥ genus(F/K) + 1 such that (2a) the only singular points of Γ̃ are cusps of multiplicity at most q; and (2b) Γ̃ has at least one cusp of multiplicity q.

Suppose now that Γ is a plane curve. A point p̃ of Γ̃ is a node if Γ̃ has two simple tangents at p̃ [Ful, p. 66]. In particular, p̃ has multiplicity 2 on Γ̃. We say that Γ is a node curve if the singular points of Γ̃ are nodes. We say that Γ is a cusp-node curve if it satisfies (1), except that in addition to cusps we allow also nodes as singularities for Γ̃. We say that Γ is a special cusp-node curve if it satisfies (2), where in addition to cusps, we allow also nodes.

Suppose again that Γ is a curve in ℙ^{n} with a generic point y = (y_{0}:y_{1}: ⋯:y_{n}) over K with y_{0}, y_{1}, …, y_{n} ∈ F. A strange point of Γ̃ is a point of ℙ^{n} through which infinitely many tangents to Γ̃ pass.

Let p be a simple point of Γ̃. Following [GeJ, p. 360], we say that p is an inflection point of Γ̃ if

rank(y(p) y′(p) y^{[2]}(p)) = 2,

where we now consider y, y′, and y^{[2]} as columns of height n + 1, and y′ and y^{[2]} are the first and the second derivatives of y, respectively, à la F. K. Schmidt [GeJ, §2]. In the case n = 2, this definition agrees with the classical one [GeJ, Lemma 3.1].

Finally, we say that Γ is an ordinary curve if Γ̃ has only finitely many inflection points, finitely many double tangents, and no strange points. Lemma 4.1: Let Γ be a projective ordinary node plane curve over a field K. Let F be the function field of Γ and let y_{0}, y_{1}, y_{2} be elements of F such that (y_{0}:y_{1}:y_{2}) is a generic point of Γ over K. Then there is a nonempty Zariski-open subset U of ℙ^{2} such that if o ∈ U(K) and (x_{0}:x_{1}) = π_{o}(y_{0}:y_{1}:y_{2}), then \frac{x_{0}}{x_{1}} symmetrically stabilizes F/K. Here π_{o} is the projection from o defined in §3. Proof: Let L be the set of all double tangents of Γ̃, tangents at inflection points, lines through two distinct singular points, and tangents that pass through a singular point of Γ̃. By assumption, L is a finite set. Let A be the union of Γ̃ with all lines in L. Then U = ℙ^{2} ∖ A is a nonempty Zariski-open subset of ℙ^{2}.

Consider now o ∈ U(K). Let n = deg(Γ). By Bezout’s theorem (see also [FJ1, Lemma 3.3]), finitely many lines in ℙ^{2} through o intersect Γ̃ at exactly n − 1 points and all others intersect Γ̃ by exactly n points. (Although [FJ1, Lemma 3.3] is stated only in characteristic 0, its proof applies almost verbatim to our case.) The proof of [FJ1, Lemma 2.1] implies now that if (x_{0}:x_{1}) = π_{o}(y_{0}:y_{1}:y_{2}), then \frac{x_{0}}{x_{1}} symmetrically stabilizes F/K.

In the following two lemmas we use the notation of §3. Lemma 4.2: Let K be an infinite field and let n ≥ 2 be an integer. Consider a projective curve Δ in ℙ^{n} over K with a function field F. Let y_{0}, y_{1}, …, y_{n} be elements of F such that y = (y_{0}:y_{1}: ⋯:y_{n}) is a generic point of Δ over K and F = K(y_{0}:y_{1}: ⋯:y_{n}). Suppose that (a) Δ is an ordinary curve;

(b) Δ̃ is contained in no hyperplane of ℙ̃^{n};

(c) if n ≥ 3, then Δ is a smooth curve or a special cusp curve; and

(d) if n = 2, then Δ is a node curve or a special cusp-node curve.

Then there exists a nonempty Zariski-open subset U_{i} of ℙ^{i}, i = 2, 3, …, n, such that with U = U_{2} × U_{3} × ⋯ × U_{n} and for each A ∈ φ^{−1}(U(K)) and with μ(A) = \binom{b}{c}, the element t = ∑_{i=0}^{n} b_{i}y_{i}/ ∑_{i=0}^{n} c_{i}y_{i} symmetrically stabilizes F/K.

Proof: Suppose first that n = 2. Lemma 4.1, in case Δ is a node curve, and [Neu, Prop. 2.12], in case Δ is a cusp-node curve, gives a nonempty Zariski-open subset U_{2} of ℙ^{2} such that if o ∈ U_{2}(K) and (x_{0}:x_{1}) = π_{o}(y_{0}:y_{1}:y_{2}), then \frac{x_{0}}{x_{1}} symmetrically stabilizes F/K. Let therefore o ∈ U_{2}(K) and let A = \binom{b_{0} b_{1} b_{2}}{c_{0} c_{1} c_{2}} ∈ M_{2}^{∗}(K) be a matrix such that o is the unique solution of the system of two equations: ∑_{j=0}^{2} b_{j}X_{j} = 0, ∑_{j=0}^{2} c_{j}X_{j} = 0. Then π_{o}(y_{0}:y_{1}:y_{2}) = ( ∑_{j=0}^{2} b_{j}y_{j}: ∑_{j=0}^{2} c_{j}y_{j}). Hence, ∑_{j=0}^{2} b_{j}y_{j}/ ∑_{j=0}^{2} c_{j}y_{j} symmetrically stabilizes F/K.

Assume therefore that n ≥ 3. By (b), Δ̃ is contained in no hyperplane. Use [Neu, Lemma 2.35] to find a nonempty Zariski-open subset U_{n} of ℙ^{n} such that for each o ∈ U_{n}(K) the projection π_{o}: ℙ^{n} ∖{o} → ℙ^{n−1} maps Δ onto a curve Γ with the following properties: (3a) Γ is an ordinary curve. (3b) If n ≥ 4 and Δ is smooth, then so is Γ. (3c) If n ≥ 4 and Δ is a special cusp curve, then so is Γ. (3d) If n = 3 and Δ is smooth, then Γ is a node curve. (3e) If n = 3 and Δ is a special cusp curve, then Γ is a special cusp-node curve.

Consider o ∈ U_{n}(K). Suppose that o = φ_{n}(A_{n}) with A_{n} = (a_{ij}) ∈ M_{n}^{∗}(K). Then Γ is generated over K by z = (z_{0}:z_{1}: ⋯:z_{n−1}), where z_{i} = ∑_{j=0}^{n} a_{ij}y_{j}, i = 0, …, n − 1.

Claim: Γ̃ is contained in no hyperplane of ℙ̃^{n−1}. Indeed, if Γ̃ were contained in a hyperplane, then there would exist c_{0}, …, c_{n−1} ∈ K̃ not all zero such that ∑_{i=0}^{n−1} c_{i}z_{i} = 0. Thus, ∑_{j=0}^{n} ( ∑_{i=0}^{n−1} c_{i}a_{ij})y_{j} = 0. Since Δ̃ is contained in no hyperplane of ℙ̃^{n}, the elements y_{0}, y_{1}, …, y_{n} are linearly independent over K̃. Hence, ∑_{i=0}^{n−1} c_{i}a_{ij} = 0, j = 0, …, n. But this means that rank(A_{n}) < n. This contradiction to the assumption that A_{n} ∈ M_{n}^{∗}(K) proves our claim.

End of proof: An induction hypothesis gives for each i between 0 and n − 1 a nonempty Zariski-open subset U_{i} of ℙ^{i} such that if A_{i} ∈ M_{i}(K) and φ_{i}(A_{i}) ∈ U_{i}(K), then, with \binom{b}{c} = A_{2}A_{3} ⋯ A_{n−1}, the element

(4)  t = \frac{∑_{i=0}^{n−1} b_{i}z_{i}}{∑_{i=0}^{n−1} c_{i}z_{i}}

symmetrically stabilizes F/K.

Let U = U_{2} × U_{3} × ⋯ × U_{n}. Consider a point A = (A_{2}, …, A_{n}) of 𝕄(K) such that φ(A) ∈ U(K). Then A_{n} = (a_{ij}) ∈ M_{n}^{∗}(K) and o = φ_{n}(A_{n}) ∈ U_{n}(K). Hence, π_{o} satisfies (3) and the Claim. Use the notation of the Claim and substitute z_{i} = ∑_{j=0}^{n} a_{ij}y_{j} in (4) to get a new presentation of t: t = ∑_{j=0}^{n} b′_{j}y_{j}/ ∑_{j=0}^{n} c′_{j}y_{j}, where \binom{b′}{c′} = \binom{b}{c}A_{n}. Thus, μ(A) = A_{2} ⋯ A_{n−1}A_{n} = \binom{b′}{c′}, and the induction step is complete.

Remark 4.3: The Veronese curve. Consider the projective line ℙ^{1} with a generic point (x_{0}:x_{1}) over a field K. The Veronese map of degree n is an isomorphism of ℙ^{1} onto a smooth projective curve V_{n} in ℙ^{n} with generic point (x_{0}^{n}:x_{0}^{n−1}x_{1}: ⋯:x_{1}^{n}) [Sha. p. 40] which can also be written as (1:t: ⋯:t^{n}) where t = \frac{x_{1}}{x_{0}} . Since 1, t_{1}, …, t^{n} are linearly independent over K, the Veronese curve V_{n} is contained in no hyperplane of ℙ^{n}.

If char(K) ≠ 2, then V_{2} is an ordinary curve. But if char(K) = 2, then (0:1:0) is a strange point for V_{2}. So, we rather look at V_{3}.

Lemma 4.4: V_{3} is an ordinary curve.

Proof: In the notation of Remark 4.3, t = (1:t:t^{2}:t^{3}) is a generic point of V_{3} over K. One checks that the following homogeneous equations define V_{3} in ℙ^{3}:

(5)  \begin{aligned}Y_{1}Y_{2} − Y_{0}Y_{3} & = 0 \\ Y_{1}^{2} − Y_{0}Y_{2} & = 0 \\ Y_{2}^{2} − Y_{1}Y_{3} & = 0 \end{aligned}

The curve V_{3} has a unique infinite point ∞ = (0:0:0:1). Each finite point has the form a = (1:a:a^{2}:a^{3}). Computing the partial derivatives of the left hand sides of (5) one finds that the following system of linear equations defines the tangent T_{a} to V_{3} at a:

(6)  \begin{aligned}−a^{3}Y_{0} & + a^{2}Y_{1} + aY_{2} − Y_{3} = 0 \\ −a^{2}Y_{0} & + 2aY_{1} − Y_{2}  = 0 \\ & − a^{3}Y_{1} + 2a^{2}Y_{2} − aY_{3} = 0 \end{aligned}

Of course, the third equation of (6) is redundant. If we take another point b = (1:b:b^{2}:b^{3}) on V_{3}(K̃) with b ≠ a, then T_{𝔞} and T_{b} do not intersect. Indeed,

| \begin{matrix}−a^{3} & a^{2} & a & −1 \\ −a^{2} & 2a & −1 & 0 \\ −b^{3} & b^{2} & b & −1 \\ −b^{2} & 2b & −1 & 0 \end{matrix} | = (a − b)^{4} ≠ 0.

Similarly one computes the equation of T_{∞} to be given by the equations Y_{0} = 0 and Y_{1} = 0. One observes that T_{∞} does not intersect T_{a}. It follows that V_{3} has no double tangents nor there exists a strange point for V_{3}.

Finally we prove that the point o = (1:0:0:0) of V_{3} is not an inflection point. Indeed, t is a uniformizing parameter at o. Rule (1f) of [GeJ, §2] says that

(t^{j})^{[i]} = \binom{j}{i}t^{j−i} for each j ≥ i. Hence,

(t t′ t^{[2]}) = [ \begin{matrix}1 & 0 & 0 \\ t & 1 & 0 \\ t^{2} & 2t & 1 \\ t^{3} & 3t^{2} & 3t \end{matrix} ]

In particular,

(t(o) t′(o) t^{[2]}(o)) = [ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ]

is a matrix of rank 3. Hence, o is not an inflection point of V_{3}. Since the set of inflection points is Zariski-closed in V_{3}, this means that V_{3} has only finitely many inflection points. Conclude that V_{3} is an ordinary curve.

Lemma 4.5: Let F be a function field of one variable over an infinite field K. Let Γ be a projective ordinary model for F/K which is either a smooth curve or a special cusp curve. Let O be the ring of singularities of Γ. Then F/K has a positive divisor 𝔞_{0} which is O-smooth such that the following statement holds for each O-smooth divisor 𝔞 ≥ 𝔞_{0}:

There exists a basis y_{0}, y_{1}, …, y_{n} for the linear space L_{O}(𝔞) and there exists a nonempty Zariski-open subset U_{i} of ℙ^{i}, i = 2, 3, …, n such that with U = U_{2} × U_{3} × ⋯ × U_{n}, for each A ∈ φ^{−1}(U(K)), and with μ(A) = \binom{b}{c}, the element t = ∑_{i=0}^{n} b_{i}y_{i}/ ∑_{i=0}^{n} c_{i}y_{i} symmetrically stabilizes F/K. Proof: If Γ is a line, use Remark 4.3 and Lemma 4.4 to replace it by V_{3} (or by V_{2} if char(K) ≠ 2). Thus, in any case, we may embed Γ in ℙ^{m} for some m ≥ 2 such that Γ is contained in no hyperplane of ℙ^{m}. Let x = (x_{0}:x_{1}: ⋯:x_{m}) be a generic point for Γ with x_{0}, x_{1}, …, x_{m} ∈ F such that F = K(x). Then x_{0}, x_{1}, …, x_{m} are linearly independent over K.

Use the notation of §2. By the weak approximation theorem there exists u ∈ F such that v_{P} (u) ≥ k_{P} − min_{0≤i≤m} v_{P} (x_{i}) for each P ∈ PrimDiv(F/K) which lies over a singular point of Γ and where k_{P} satisfies (1) of §2. Then replace x_{i} by ux_{i}, if necessary, to assume that x_{i} ∈ O, i = 0, 1, …, m. In particular, the div_{∞}(x_{i}) is relatively prime to the conductor 𝔠 of O, i = 0, 1, …, m. Let n_{0} be the positive integer that appears in Proposition 2.7. Then 𝔞_{0} = ∑_{i=0}^{m} n_{0}div_{∞}(x_{i}) is a positive O-smooth divisor of F/K with deg(𝔞_{0}) ≥ n_{0}.

Let 𝔞 be a divisor of F/K such that 𝔞 ≥ 𝔞_{0}. Let y_{i} = x_{i}, i = 0, 1, …, m. Then div(y_{i})+𝔞 ≥ −div_{∞}(x_{i})+n_{0}div_{∞}(x_{i}) ≥ 0 and therefore y_{i} ∈ L_{O}(𝔞). Since y_{0}, y_{1}, …, y_{m} are linearly independent over K, we may extend them to a basis y_{0}, y_{1}, …, y_{n} of L_{O}(𝔞). By Proposition 2.7, 𝔞 is very ample on Γ. Thus, the map ψ associated with 𝔞 is an isomorphism of Γ onto the projective curve Δ generated over K in ℙ^{n} by y = (y_{0}:y_{1}: ⋯:y_{n}). In particular, if Γ is smooth, so is Δ; if Γ is a special cusp curve, so is Δ. Since y_{0}, y_{1}, …, y_{n} are linearly independent over K and F/K is regular, Δ̃ is contained in no hyperplane of ℙ̃^{n}.

Consider now the linear variety L in ℙ^{n} defined by the equations Y_{i} = 0, i = 0, 1, …, m. The projection π_{L}: ℙ^{n} ∖ L → ℙ^{m} maps each point q = (q_{0}:q_{1}: ⋯:q_{n}) of (ℙ^{n} ∖ L)(K̃) onto the point (q_{0}: ⋯:q_{m}) of ℙ^{m}(K̃). If q ∈ Δ(K̃), then q = ψ(p), with p = (p_{0}: ⋯:p_{m}) ∈ Γ(K̃). In particular p_{i} ≠ 0 for at least one i and (q_{0}: ⋯:q_{m}) = (p_{0}: ⋯:p_{m}). Thus, π_{L} projects Δ isomorphically onto the ordinary curve Γ. By [Neu, Lemma 2.29], Δ is an ordinary curve. Note that [Neu, Lemma 2.29] is actually stated only in the case where L is a point. In the general case, π_{L} can be factored into successive projections, first onto the first n coordinates, then onto the first n − 1 coordinates, and so on. So, we may apply [Neu, Lemma 2.29] n − m times.

With this, Δ satisfies conditions (a), (b), (c), and (d) of Lemma 4.2. So, Lemma 4.2 supplies the Zariski-open subsets U_{i} of ℙ^{i} with the desired properties.

5. Local existence theorem Let Γ be a projective curve over a field K with a function field F. In [Ro2], Maxwell Rosenlicht constructs an algebraic group J_{Γ} over K, known as the generalized Jacobian variety of Γ, which coincides with the usual Jacobian variety of Γ if Γ is smooth. As in the latter case, there is an intimate connection between divisors of F/K and the group J_{Γ}(K). We survey this connection as well as the properties of J_{Γ} that we need in the proof of the local existence theorem. In this survey we use the terminology of §2.

Let O be a semilocal ring of F/K which is contained in the ring of singularities of Γ in F. A priori, the algebraic group that Rosenlicht constructs depends on O (and not only on Γ). So, we denote it by J_{O} or also by J until we change O. Rosenlicht’s construction depends on the following assumption:

(1) Γ(K) is infinite. Let h = genus(O/K). If h = 0, then J is a point [Ro2, p. 519]. We therefore assume throughout that h > 0.

Choose a point o ∈ Γ_{simp}(K). Then J = J_{O} has the following properties: (2a) J is a commutative group variety over K of dimension h [Ro2, Thm. 7]. (2b) There is a morphism φ: Γ_{simp} → J over K such that φ(o) is the zero point of

J(K) [Ro2, Thm. 7]. (2c) For each extension L of K which is linearly disjoint from F let Div_{O}(F L/L)

(resp., Div_{O,0}(F L/L) ) be the group of all OL-smooth divisors (resp., divisors of degree 0) of F L/L. Then φ: Γ_{simp}(K̃) → J(K̃) extends to an epimorphism φ̃: Div_{O}(F K̃/K̃) → J(K̃) by the rule

φ̃( ∑_{i=1}^{m} n_{i}p_{i}) = ∑_{i=1}^{m} n_{i}φ(p_{i}),

where the sum on the right hand side is addition on the group variety. (2d) Symmetric products: Consider the direct product Γ_{simp}^{h} of h copies of Γ_{simp}. It

is a variety over K of dimension h. The symmetric group S_{h} acts on Γ_{simp}^{h} by permuting the coordinates. The factor variety Γ^{(h)} = Γ_{simp}^{h}/S_{h} is the symmetric product of Γ_{simp} with itself h times. It is a variety over K and there is a Galois cover ρ: Γ_{simp}^{h} → Γ^{(h)} over K with Galois group S_{h} [Ser, p. 48, Prop. 18].

By the generalized Riemann-Roch theorem, φ induces a birational map σ: Γ^{(h)} → J over K. In particular, there are nonempty Zariski-open subsets B_{h}, J_{0} of Γ^{(h)}, J, respectively, such that the restriction of σ to B_{h} is an isomorphism onto J_{0}. Moreover, the dominant rational map ψ = σρ of Γ_{simp}^{h} to J satisfies

(2d1)  ψ(p_{1}, …, p_{h}) = ∑_{i=1}^{h} φ(p_{i})

for all p_{1}, …, p_{h} ∈ Γ_{simp}. In particular, the function field of Γ_{simp}^{h} is a Galois extension of the function field of J with Galois group S_{h}.

(2e) Abel’s theorem. In the notation of (2c) let L be an algebraic extension of K. Let Prin_{O}(F L/L) = {div(f) | f ∈ (OL)^{×}} be the group of all principal divisors of F L/L coming from units of OL. It fits into the following short exact sequence

(2e1)  1 ⟶ L^{×} ⟶ (OL)^{×} ⟶^{div} Prin_{O}(F L/L) → 0.

Let φ_{L} be the restriction of φ̃ to Div_{O,0}(F L/L). By [Ro2, p. 517 and p. 519], Ker(φ_{L}) = Prin_{O}(F L/L) and Im(φ_{L}) ⊆ J(L). By (2d), Im(φ_{L}) contains a nonempty Zariski-open subset of J(L). But as Im(φ_{L}) is a subgroup of J(L), it is all of J(L). In other words, the following short sequence is exact:

(2e2)  0 ⟶ Prin_{0}(F L/L) ⟶ Div_{O,0}(F L/L) ⟶^{φ_{L}} J(L) ⟶ 0.

(2f) Changing O: Intersecting O with finitely many valuation rings of prime divisors of F/K does not change J_{0}. We may therefore use J_{Γ} instead of J_{O}. Indeed, it suffices to consider a prime divisor q of F/K which does not lie over O, to write R = O ∩ O_{q}, and to prove that J_{O} = J_{R}. This follows from (2e2) (with L = K̃), (2f1), and (2f2): (2f1) Prin_{O}(F K̃/K̃) ∩ Div_{R,0}(F K̃/K̃) = Prin_{R}(F K̃/K̃) and (2f2) Prin_{O}(F K̃/K̃) + Div_{R,0}(F K̃/K̃) = Div_{O,0}(F K̃/K̃). In order to prove (2f1), we apply our convention from §2.1 to q and identify it

with a simple point of Γ such that O_{Γ,q} = O_{q}. Then we prove that OK̃ ∩ O_{q}K̃ = RK̃. To this end let {w_{i} | i ∈ I} be a basis for K̃/K. Each g ∈ OK̃ ∩ O_{q}K̃ can be written as g = ∑_{i∈I} g_{i}w_{i} with g_{i} ∈ O and g = ∑_{i∈I} g′_{i}w_{i} with g′_{i} ∈ O_{q}. So, g_{i} = g′_{i} belongs to R and therefore g ∈ RK̃.

Consider now f ∈ (OK̃)^{×} such that div(f) ∈ Div_{R,0}(F K̃/K̃). Then, vq̃(f) = 0 for each prime divisor q̃ of F K̃/K̃ that lies over q. By Lemma 2.5, O_{q}K̃ is the intersection of all valuation rings of F K̃ that lie over O_{q}. So, by the preceding paragraph, f, f^{−1} ∈ OK̃ ∩ O_{q}K̃ = RK̃. In other words, f ∈ (RK̃)^{×}. It follows that div(f) ∈ Prin_{R}(F K̃/K̃).

In order to prove (2f2) consider 𝔟 ∈ Div_{O,0}(F K̃/K̃). Use the weak approximation theorem to find f ∈ F K̃ such that vQ̃(f) = vQ̃(𝔟) for each prime divisor Q̃ of F K̃/K̃ over q and vP̃(f − 1) is sufficiently large for all P̃ which lie over OK̃. By (1) of §2, applied to the semilocal ring OK̃, f ∈ (OK̃)^{×}. Also, 𝔟_{1} = −div(f) + 𝔟 ∈ Div_{R,0}(F K̃/K̃), as needed.

(2g) Comparing two curves: Let π: Γ′ → Γ be a birational morphism of projective curves over K. By (1), Γ′(K) is infinite. For each object associated with Γ add a tag on the letter denoting it to denote the corresponding object associated with Γ′. If π(p′) = p, then O_{Γ,p} ⊆ O_{Γ′,p′}. Moreover, if p′ is singular, then so is p. Let O (resp., O′) be the ring of singularities of Γ (resp., Γ′) and let R = O′ ∩ ⋂_{P} O_{P} where P ranges over all prime divisors of F/K that lie over singular points of Γ. Then O ⊆ R ⊆ O′ and a divisor of F/K is O-smooth if and only if it is R-smooth. In particular, by Lemma 2.5, Div_{OK̃,0}(F K̃/K̃) = Div_{RK̃,0}(F K̃/K̃). By (2f), J′ = J_{R}. By [Ro2, Thm. 8], there exists a surjective homomorphism θ: J → J′ over K of the corresponding group varieties. Let H = Ker(θ) and let θ̃: J(K̃) → J′(K̃) be the homomorphism induced by θ. The short exact sequences (2e1) and (2e2) for L = K̃ yield a commutative diagram

[figure: 0; 0; 0; Prin_{O}(F K̃/K̃); Prin_{RK̃}(F K̃/K̃); (RK̃)^{×}/(OK̃)^{×}; 1; Div_{O,0K̃}(F K̃/K̃); φ_{K}; Div_{RK̃,0}(F K̃/K̃); φK̃; 1; H(K̃); J(K̃); θ̃; J′(K̃); 1; 0; 0]

An application of the snake lemma [Eis, p. 640] to the vertical short exact sequences proves that H(K̃) ≅ (RK̃)^{×}/(OK̃)^{×}. This gives the following short exact sequence

(2g1)  1 ⟶ (RK̃)^{×}/(OK̃)^{×} ⟶ J(K̃) ⟶^{θ̃} J′(K̃) ⟶ 0.

(2h) The genera satisfy the inequality genus(F/K) ≤ h′ ≤ h. Indeed, choose a Γsmooth divisor 𝔞 of large degree. Let g = genus(F/K). Then, the Riemann-Roch theorem and the generalized Riemann-Roch theorem (4) of §2 give the following relations:

dim (L_{F/K}(𝔞)) = deg(𝔞) + 1 − g,
dim (L_{O′/K}(𝔞)) = deg(𝔞) + 1 − h′,
dim (L_{O/K}(𝔞)) = deg(𝔞) + 1 − h.

Since O ⊆ O′ ⊆ F, we have L_{O/K}(𝔞) ⊆ L_{O′/K}(𝔞) ⊆ L_{F/K}(𝔞). Hence, g ≤ h′ ≤ h. (2i) The map θ: J → J′ of (2g) is separable. Indeed, by (2h), h′ ≤ h. This gives rise

to the following commutative diagram:

\xymatrix{Γ_{simp}^{h} \ar[r]^{pr} \ar[d]_{ψ} & Γ_{simp}^{h′} & Γ_{simp}^{′h′} \ar[d]^{ψ′} \ar[l]_{π^{h′}} \\ J \ar[rr]^{θ} &  & J′}

In this diagram pr is a regular map (i.e., the extension of the corresponding function fields is regular), π^{h′} is a birational map, and ψ′ is a Galois map (by (2d)). Hence, ψ′ ∘ (π^{h′})^{−1} ∘ pr is a separable map. It follows that θ is also a separable map.

Remark 5.1: At the beginning of [Ro2, §3], Rosenlicht considers a smooth projective curve C and a semilocal ring O of the function field F of C. Then he constructs the generalized Jacobian variety of C with respect to the equivalence relation defined by O. However, the construction works even if one allows singularities and only demands that O is contained in the ring of singularities of C. This comes into evidence in [Ro2, Thm. 12], where Rosenlicht explicitly allows singularities on C. The only result which possibly depends on the smoothness of C is [Ro2, Thm. 10] (see the top of p. 523 of [Ro2]).

We are using the letter Γ instead of C. Our only application of Theorem 10 of [Ro2] appears in (2g1) which we have been careful to reprove directly.

Assumption 5.2: (3a) K has a local prime divisor 𝔭 with respect to which it is either Henselian or real

closed. Let K̂ = K̂_{𝔭}. (3b) The curve Γ has a simple K-rational point. (3c) Γ is K-normal or char(K) > 0 and Γ is a cusp curve.

Conditions (3a) and (3b) imply, by the density theorem 8.2(b), that Γ(K) is Zariski-dense in Γ, which means that Γ(K) is infinite (condition (1)). This adds an additional information to (2a)-(2i):

(4) By (2d), Γ_{simp}^{h} has a Zariski-open subset D such that ψ|_{D}: D → J is an étale map [Hrt, p. 271, Lemma 10.5]. (Note that Hartshorne states the lemma only in characteristic 0. However, the proof makes use only of the separability of the map.) Hence, by Proposition 8.2(c), ψ|_{D(K)}: D(K) → J(K) is a local 𝔭-homeomorphism.

Lemma 5.3: Make Assumption 5.2 and let b ∈ J(K). Then, for each 𝔭-open neighborhood A of 0 in J(K) there exist infinitely many positive integers n such that nb ∈ A.

Proof: Suppose first that the theorem holds for K̂. Then there exist infinitely many positive integers n such that nb ∈ A(K̂). For each of these n we also have nb ∈ J(K). Hence, nb ∈ A. So, we have to prove the theorem only in the case where K is locally compact.

Under this assumption consider first the case that Γ is K-normal. In this case the lemma coincides with [GPR, Lemma 2.2].

Suppose therefore that char(K) = p > 0 and that Γ is a cusp curve. Let π: Γ′ → Γ be the normalization of Γ over K. Let θ: J → J′ be the homomorphism of the corresponding generalized Jacobians which is introduced in (2g).

Part A: There is a power q of p such that q · Ker(θ̃) = 0, where Ker(θ̃) = {a ∈ J(K̃) | θ̃(a) = 0}. Indeed, let O be the ring of singularities of Γ. Then, in the terminology of (2g1), Ker(θ̃) ≅ (RK̃)^{×}/(OK̃)^{×}. By Lemma 2.5, OK̃ is the intersection of the local rings O_{Γ̃,p}, where p ranges on Γ_{sing}. Each p ∈ Γ̃_{sing} is a cusp and therefore O_{Γ̃,p} is contained in a valuation ring O_{P} of some unique prime divisor P of F K̃/K̃. The ring RK̃ is contained in the intersection of all those O_{P}. It follows that (RK̃)^{×}/(OK̃)^{×} ⊆ ∏_{p} O_{P}^{×}/O_{Γ̃,p}^{×}, where, again, p ranges on Γ̃_{sing}. Hence, it suffices to prove that for each p ∈ Γ̃_{sing} there is a power q of p such that (O_{P}^{×}/O_{Γ̃,p}^{×})^{q} = 1.

Since O_{P} is the unique valuation ring of F K̃/K̃ over O_{Γ̃,p}, it is the integral closure of O_{Γ̃,p} in F. Hence there exists a positive integer k such that M_{P}^{k} ⊆ M_{Γ̃,p} (as follows for example from (1) of §2). Let q ≥ k be a power of p. Then M_{P}^{q} ⊆ M_{Γ̃,p}. Also, O_{P} /M_{P} = K̃ = O_{Γ̃,p}/M_{Γ̃,p}, i.e., O_{P} = O_{Γ̃,p} + M_{P}. In particular, each x ∈ (O_{P})^{×} has the form x = a + m where a ∈ O_{Γ̃,p} ∖ M_{Γ̃,p} and m ∈ M_{P}. So, x^{q} = a^{q} + m^{q} ∈ O_{Γ̃,p}^{×}, as desired.

Part B: A 𝔭-open map. Since θ: J → J′ is defined over K, it defines a map θ_{K}: J(K) → J′(K). We claim that this map is 𝔭-open. Indeed, since, by (2i), θ: J → J′ is a separable map, J and J′ have nonzero Zariski-open subsets J_{0} and J′_{0} such that the restriction of θ to J_{0} is a smooth map onto J′_{0} [Hrt, p. 271, Lemma 10.5]. By the open map theorem (Proposition 8.2(c)), θ_{K}: J(K) → J′(K) is 𝔭-open in a neighborhood of each point a ∈ J_{0}(K). Since translations by points of J(K) and of J′(K) are 𝔭-isomorphisms, θ_{K}: J(K) → J′(K) is 𝔭-open in a neighborhood of each point of J(K) (see also [GPR, p. 51, J5]). So, θ_{K}: J(K) → J′(K) is a 𝔭-open map.

Consider now a 𝔭-open neighborhood A of 0 in J(K). Let A_{0} be a 𝔭-open neighborhood of 0 in A such that qA_{0} ⊆ A. Then θ(A_{0}) is a 𝔭-open neighborhood of 0 in J′(K). By [GPR, Lemma 2.2] there exist infinitely many positive integers l such that lθ(b) ∈ θ(A_{0}). Thus lb ∈ A_{0} + Ker(θ). Conclude from Part A that qlb ∈ A.

Lemma 5.4 ([GPR, Lemma 2.3]): Let G be a topological additive commutative group, let k ≥ 3, and let x_{1}, …, x_{k} be elements of G. For each j let W_{j} be a neighborhood of x_{j} and for each i ≠ j let Δ_{ij} be a subset of W_{i} × W_{j} whose complement is open and dense in W_{i} × W_{j}. Then there are y_{j} ∈ W_{j} such that ∑_{j=1}^{k} x_{j} = ∑_{j=1}^{k} y_{j} and (y_{i}, y_{j}) ∉ Δ_{ij} if i ≠ j.

The following result gives an element of O^{×} with control on both his pole divisor and his zero divisor.

Proposition 5.5 (Local existence theorem for normal curves and cusp curves): Under Assumption 5.2 let U be a nonempty 𝔭-open subset of Γ_{simp}(K). Let 𝔞 be a Γ-smooth positive divisor of F/K. Let O be the ring of singularities of Γ. Then there exists a positive integer r_{0} such that for each positive integer k there exists f ∈ O^{×} such that div(f) = ∑_{i=1}^{s} p_{i} − kr_{0}𝔞, where p_{1}, …, p_{s} are distinct points in U ∖ Supp(𝔞). In particular, each of them belong to Γ(K).

Proof: We follow the proof of [GPR, Thm. 2.1] which handles the case where Γ is K-normal. Our unified proof handles both the case where Γ is K-normal and the case where Γ is a cusp curve.

By shrinking U, if necessary, we may assume that U contains no point that belongs to the support of 𝔞. By assumption, U^{h} is a nonempty 𝔭-open subset of Γ_{simp}^{h}(K). By (4), Γ_{simp}^{h} has a nonempty Zariski-open subset D such that ψ|_{D(K)} is a local 𝔭homeomorphism into J(K). By Proposition 8.2(b), U^{h} is Zariski-dense in Γ_{simp}^{h}. Hence, U^{h} ∩ D(K) has a nonempty 𝔭-open subset V which ψ maps 𝔭-homeomorphically onto a 𝔭-open subset W of J(K). Choose (q_{1}, …, q_{h}) ∈ V. Then q_{i} ∈ Γ_{simp}(K) and W_{0} = W −ψ(q_{1}, …, q_{h}) is a 𝔭-open neighborhood of 0 in J(K). Also, 𝔟 = h𝔞−deg(𝔞) ∑_{j=1}^{h} q_{j} is a divisor of F/K of degree 0. Hence b = φ̃(𝔟) ∈ J(K) (by (2e) ).

By Lemma 5.3 there exists an integer n such that nb ∈ W_{0} and n deg(𝔞) ≥ 3. In other words, there exists (q′_{1}, …, q′_{h}) ∈ V such that nb = ψ(q′_{1}, …, q′_{h})−ψ(q_{1}, …, q_{h}). Hence, nhφ̃(𝔞) − n deg(𝔞) ∑_{j=1}^{h} φ(q_{j}) = nφ̃(𝔟) = nb = ψ(q′_{1}, …, q′_{h}) − ψ(q_{1}, …, q_{h}) = ∑_{j=1}^{h} φ(q′_{j}) − ∑_{j=1}^{h} φ(q_{j}) (use (2d1) ). Hence

(5)  nhφ̃(𝔞) = ∑_{j=1}^{h} φ(q′_{j}) + ∑_{j=1}^{h}(n deg(𝔞) − 1)φ(q_{j}).

Let k be a positive integer. Then m = kn deg(𝔞) ≥ 3. Multiply (5) by k:

(6)  knhφ̃(𝔞) = ∑_{j=1}^{h} kφ(q′_{j}) + ∑_{j=1}^{h} k(n deg(𝔞) − 1)φ(q_{j}).

Thus, with a = knhφ̃(𝔞), there exist q_{ij} ∈ Γ_{simp}(K), i = 1, …, m, j = 1, …, h, such that

(7)  a = ∑_{i=1}^{m} ∑_{j=1}^{h} φ(q_{ij})

and

(8)  (q_{i1}, …, q_{ih}) ∈ V,  i = 1, …, m.

Indeed, we may take q_{ij} = q′_{j} for i = 1, …, k and q_{ij} = q_{j} for i = k + 1, …, m.

For each pair (i, r) of distinct integers between 1 and m let

Δ′_{ir} = {(p_{i1}, …, p_{ih}, p_{r1}, …, p_{rh}) ∈ Γ^{h} × Γ^{h} | p_{ij} ≠ p_{rs} for j, s = 1, …, h and

p_{ij} ≠ p_{ij′} and p_{rj} ≠ p_{rj′} if j ≠ j′}.

As Δ′_{ir} is a nonempty Zariski-open subset of Γ^{h} × Γ^{h}, the density theorem (Proposition 8.2(b)) implies that Δ′_{ir}(K)∩(V ×V) is 𝔭-open and 𝔭-dense in V ×V. Hence, ψ(Δ′_{ir}(K)∩ (V×V)) is 𝔭-open and 𝔭-dense in W×W. By (7) and by (2d1), ∑_{i=1}^{m} ψ(q_{i1}, …, q_{ih}) = a. Lemma 5.4 gives (q′_{i1}, …, q′_{ih}) ∈ V, i = 1, …, m, such that (q′_{i1}, …, q′_{ih}, q′_{r1}, …, q′_{rh}) ∈ Δ′_{ir}(K) and such that

∑_{i=1}^{m} ∑_{j=1}^{h} φ(q′_{ij}) = ∑_{i=1}^{m} ψ(q′_{i1}, …, q′_{ih}) = a.

Replace q_{ij} by q′_{ij}, if necessary, to assume that the q_{ij} are distinct. Rewrite (7) in the form

φ̃(−knh𝔞 + ∑_{i=1}^{m} ∑_{j=1}^{h} q_{ij}) = 0.

By Abel’s theorem (2e), there exists f ∈ O^{×} such that

div(f) = −knh𝔞 + ∑_{i=1}^{m} ∑_{j=1}^{h} q_{ij}.

By assumption, 𝔞 > 0. Hence, div_{∞}(f) = knh𝔞, the geometric zeros of f are q_{ij}, they are distinct and belong to U. Thus, r_{0} = nh satisfies the conclusion of the proposition.

6. Rumely elements Let S be a finite set of local primes of a field K. Consider a projective curve Γ over K. Let F be the function field of Γ. Denote the ring of singularities of Γ in F by O. Let 𝔞 be a positive divisor of F/K. Denote the support of 𝔞 by Supp(𝔞). For each 𝔭 ∈ S let U_{𝔭} be a nonempty 𝔭-open subset of Γ_{simp}(K_{𝔭}) and let U = ⋂_{𝔭∈S} ⋂_{σ∈G(K)} U_{𝔭}^{σ}. Lemma 6.1: U is an S-open subset of Γ_{simp}(K_{tot,S}). Proof: By the definition of K_{tot,S} (beginning of §1), U is a subset of Γ_{simp}(K_{tot,S}). In order to prove that U is S-open in Γ_{simp}(K_{tot,S}) we may assume that Γ has an affine part Γ_{0}, say embedded in 𝔸^{n}, such that U_{𝔭} ⊆ Γ_{0}(K_{𝔭}) for each 𝔭 ∈ S. Similarly, we may as well assume that U_{𝔭} is a basic 𝔭-open set:

U_{𝔭} = {p ∈ Γ_{0}(K_{𝔭}) | |p − a_{𝔭}|_{𝔭} < ε_{𝔭}},

where a_{𝔭} ∈ Γ_{0}(K_{𝔭}) and ε_{𝔭} > 0. Let L be a finite normal extension of K such that a_{𝔭} is L-rational for each 𝔭 ∈ S. For each 𝔭 ∈ S choose an extension of | |_{𝔭} to an absolute value of K̃. Then choose a′_{𝔭} ∈ K^{n} such that |a′_{𝔭} − a_{𝔭}|_{𝔭} < ε_{𝔭} and let V_{𝔭} = {p ∈ K̃^{n} | |p − a′_{𝔭}|_{𝔭} < ε_{𝔭}}. Then V = ⋂_{𝔭∈S} ⋂_{σ∈G(K)} V_{𝔭}^{σ} is an S-open subset of K̃^{n} (Remark 1.1) and U = V ∩ Γ_{0}(K_{tot,S}). Conclude that U is S-open in Γ_{simp}(K_{tot,S}).

A separating transcendental element t of F/K is said to be a Rumely element with respect to S, 𝔞, U if

(1) div(t) = ∑_{k=1}^{m} b_{k} − 𝔞, where b_{1}, …, b_{m} are distinct points in U ∖ Supp(𝔞). The existence of Rumely elements implies that U is nonempty. This is assured by the following criterion. Proposition 6.2: Let K, S, Γ, F, O, and U be as above. Suppose that Γ is K-normal or char(K) > 0 and Γ is a cusp curve. Let 𝔞_{0} be a positive Γ-smooth divisor of F/K. Then there exists a positive integer k_{0} such that for each multiple k of k_{0}, for 𝔞 = k𝔞_{0}, and for each basis u_{0}, …, u_{n} of L_{O}(𝔞) there exists a nonempty S-open set A ⊆ K^{n+1} such that for each a ∈ A, ∑_{i=0}^{n} a_{i}u_{i} is a Rumely element with respect to S, 𝔞, U. Proof: The proof naturally breaks up into two parts.

Part A: The local case. Consider 𝔭 ∈ S. Let Γ_{𝔭} = Γ ×_{K} K_{𝔭} be the projective curve obtained from Γ by extending K to K_{𝔭}. Since F, as a regular extension of K, is linearly disjoint from K_{𝔭} over K, F K_{𝔭} is the function field of Γ_{𝔭} and we may view 𝔞_{0} also as a divisor of F K_{𝔭}/K_{𝔭}. By [Mum, p. 158], Γ_{𝔭}(K_{𝔭}) = Γ(K_{𝔭}). By Lemma 2.6, OK_{𝔭} is the ring of singularities of Γ_{𝔭} in F K_{𝔭}.

By Proposition 5.5 there exists a positive integer k_{𝔭} such that for each multiple k of k_{𝔭} there exists a function f_{𝔭} ∈ OK_{𝔭} with the following property:

(2) div(f_{𝔭}) = ∑_{i=1}^{m} b_{𝔭,i}−k𝔞_{0}, where b_{𝔭,1}, …, b_{𝔭,m} are distinct points in U_{𝔭} ∖ Supp(𝔞_{0}). Suppose now that u_{0}, …, u_{n} is a basis of L_{OK_{𝔭}}(k𝔞_{0}). Write f_{𝔭} = ∑_{j=0}^{n} a_{𝔭,j}u_{j}

with a_{𝔭,j} ∈ K_{𝔭}. Let a_{𝔭} = (a_{𝔭,0}, …, a_{𝔭,n}). By Proposition 8.3

(3) there exists a 𝔭-open neighborhood A_{𝔭} of a_{𝔭} in K_{𝔭}^{n+1} such that if a′ ∈ A_{𝔭} and f′_{𝔭} = ∑_{j=0}^{n} a′_{j}u_{j}, then (2) remains valid if we replace f_{𝔭} by f′_{𝔭}.

Part B: The global case. Let k_{0} be the least common multiple of all k_{𝔭} from Part A with 𝔭 ∈ S. Consider a multiple k of k_{0} and let 𝔞 = k𝔞_{0}. Let u_{0}, …, u_{n} be a basis of L_{O}(𝔞). By Lemma 2.6, u_{0}, …, u_{n} is also a basis of L_{OK_{𝔭}}(𝔞). For each 𝔭 ∈ S let f_{𝔭} ∈ OK_{𝔭} be a function as in (2). Condition (3) gives a 𝔭-open neighborhood A_{𝔭} of a_{𝔭} in K_{𝔭}^{n+1} such that if a = (a_{0}, …, a_{n}) ∈ A_{𝔭} and f = ∑_{j=0}^{n} a_{j}u_{j}, then

(4) div(f) = ∑_{k=1}^{m} b_{k} − 𝔞 with distinct b_{1}, …, b_{m} ∈ U_{𝔭} ∖ Supp(𝔞).

To conclude the proof use the 𝔭-density of K in K_{𝔭} and the weak approximation theorem [CaF, p. 48] to choose a nonempty S-open subset A of K^{n+1} which is contained in A_{𝔭} for each 𝔭 ∈ S. Consider a ∈ A and let f = ∑_{j=0}^{n} a_{j}u_{j}. Then (4) holds for each 𝔭 ∈ S. In particular, b_{1}, …, b_{m} ∈ U_{𝔭} for each 𝔭 ∈ S.

Since F/K is regular, we may extend each σ ∈ G(K) to an automorphism of F K̃/F. Since f ∈ F, σ permutes b_{1}, …, b_{m}. Hence, b_{1}, …, b_{m} ∈ U_{𝔭}^{σ}. Conclude that b_{1}, …, b_{m} ∈ U.

The combination of Proposition 6.2 and Lemma 4.5 yields a stabilizing element for F/K with control on the location of its zeros and poles.

Theorem 6.3: Let K, S, Γ, F, and O be as above. Suppose that Γ is smooth or char(K) > 0 and Γ is an ordinary special cusp curve. For i = 1, 2 and for each 𝔭 ∈ S let U_{i,𝔭} be a nonempty 𝔭-open subset of Γ_{simp}(K_{𝔭}). Write U_{i} = ⋂_{𝔭∈S} ⋂_{σ∈G(K)} U_{i,𝔭}^{σ}, i = 1, 2. Let d_{0} be a positive integer. Then there exists d ≥ d_{0} and F/K has a stabilizing element t such that

(a) div_{0}(t) = ∑_{i=1}^{d} z_{i}, where z_{1}, …, z_{d} are distinct points in U_{1}; and

(b) div_{∞}(t) = ∑_{i=1}^{d} p_{i}, where p_{1}, …, p_{d} are distinct points in U_{2}.

(c) Moreover, let F̂ be the Galois closure of F/K(t). Then G(F̂/K(t)) ≅ S_{d}. Proof: Let 𝔞_{0} be the positive divisor of F/K that Lemma 4.5 gives. For i = 1, 2, Proposition 6.2 gives a positive integer k_{i} such that for each multiple k of k_{i}, for 𝔞 = k𝔞_{0}, and for each basis y_{0}, y_{1}, …, y_{n} of L_{O}(𝔞)

(5) there exists a nonempty S-open set A_{i} ⊆ K^{n+1} such that for each b ∈ A_{i}, ∑_{j=0}^{n} b_{j}y_{j} is a Rumely element with respect to S, 𝔞, U_{i}. Let k be a common multiple of k_{1} and k_{2} and let 𝔞 = k𝔞_{0}. Section 3 defines two

morphisms of varieties over K:

ℙ ⟵^{φ} 𝕄 ⟶^{μ} M_{2,n+1}^{∗}.

In particular, M_{2,n+1}^{∗} consists of all 2 × (n + 1) matrices of rank 2. By Lemma 4.5, L_{O}(𝔞) has a basis y_{0}, y_{1}, …, y_{n} and there exists a nonempty Zariski-open subset U of ℙ such that

(6) for each A ∈ φ^{−1}(U(K)) and with μ(A) = \binom{b_{1}}{b_{2}} the element

t = \frac{∑_{j=0}^{n} b_{1j}y_{j}}{∑_{j=0}^{n} b_{2j}y_{j}}

symmetrically stabilizes F/K.

Make U_{i,𝔭} smaller, if necessary, to assume that U_{1} and U_{2} are disjoint. For i = 1, 2 let A_{i} be the set that (5) gives. Then A_{1} × A_{2} is a nonempty S-open subset of M_{2,n+1}^{∗}(K). Since μ(𝕄(K)) = M_{2,n+1}^{∗}(K) (by §3), the S-open subset μ^{−1}(A_{1} × A_{2}) of 𝕄(K) is not empty.

Since 𝕄 is a Zariski-open subset of an affine space, μ^{−1}(A_{1} × A_{2}) is Zariski-dense in 𝕄 (by the weak approximation theorem). In particular, there exists

A ∈ μ^{−1}(A_{1} × A_{2}) ∩ φ^{−1}(U(K)).

Let μ(A) = \binom{b_{1}}{b_{2}}. Then b_{i} ∈ A_{i}, i = 1, 2. By (5), t_{i} = ∑_{j=0}^{n} b_{ij}y_{j} is a Rumely element with respect to S, 𝔞, U_{i}. Thus, div_{∞}(t_{i}) = 𝔞 and div_{0}(t_{i}) = ∑_{j=1}^{d} p_{ij}, where p_{i1}, …, p_{id} are distinct points of U_{i}, i = 1, 2. Let t = t_{1}/t_{2}. Then div(t) = div_{0}(t_{1}) − div_{∞}(t_{1}) − div_{0}(t_{2})+div_{∞}(t_{2}) = div_{0}(t_{1})−div_{0}(t_{2}). Since U_{1}∩U_{2} = ∅, the positive divisors div_{0}(t_{1}) and div_{0}(t_{2}) are relatively prime. Hence div_{0}(t) = div_{0}(t_{1}) = ∑_{j=1}^{d} p_{1j}. Similarly, div_{∞}(t) = div_{0}(t_{2}) = ∑_{j=1}^{d} p_{2j}. So both (a) and (b) hold.

Finally, since φ(A) ∈ U(K), (6) implies that the element t symmetrically stabilizes F/K. For the Galois closure F̂ of F/K(t), this means that G(F̂K̃/K̃(t)) ≅ S_{d}, where d = [F : K(t)] = deg(div_{0}(t)) = deg(div_{0}(t_{1})) = deg(div_{∞}(t_{1})) = deg(𝔞) = k deg(𝔞_{0}). Hence, G(F̂/K(t)) ≅ S_{d}. Taking k large, d becomes large. This proves (c) and concludes the proof of the theorem.

7. The main result Let A be a subset of a field M. We say that M is PAC over A if for each dominant separable rational map φ: V → 𝔸^{r} of varieties of dimension r over M there exists x ∈ V (M) such that φ(x) ∈ A^{r}.

Proposition 7.1 (Razon [Ra1, Thm. 2.3]): Let S be a finite set of local primes of a field K such that none of them is complex. Let M be a field which is PAC over K. Then M is PAC over each nonempty S-open subset of K.

Lemma 7.2 ([Ra1, Lemma 3.6]): Let S be a finite set of local primes of an infinite field K. Let τ: Γ → ℙ^{1} be a rational map of degree d of projective curves over K. Suppose that there exists p ∈ ℙ^{1}(K) such that τ^{−1}(p)(K̃) consists of d distinct points which are in Γ_{simp}(K_{tot,S}). Then ℙ^{1}(K) has a nonempty S-open set A such that for each p′ ∈ A, the fibre τ^{−1}(p′)(K̃) consists of d distinct points which are in Γ_{simp}(K_{tot,S}).

Remark 7.3: Sections 1–2 and 3.1–3.6 of [Ra1] are independent of our work. The use of Theorem 2.3 and Lemma 3.6 of [Ra1] here is therefore valid.

Likewise, Sections 1–6 and 8 of our work are independent of [Ra1]. Hence, the use of Proposition 8.2(b) and Proposition 6.2 of our work in [Ra1, Lemma 3.7 and Prop. 3.8] is justified.

Theorem 7.4: Let S be a finite set of local primes of a field K. Let M be a field which is PAC over K. Denote the maximal Galois extension of K which is contained in M ∩ K_{tot,S} by N. Then N is PSC.

Proof: Let C be a curve over K. If char(K) = 0, take Γ to be the projective K-normal curve which is birationally equivalent over K to C. Then Γ is smooth. If char(K) > 0, then C is birationally equivalent over K to an ordinary special cusp curve Γ [Neu, Prop. 2.22]. Suppose that for each 𝔭 ∈ S there exists a_{𝔭} ∈ Γ_{simp}(K_{𝔭}). By Lemma 1.6, it suffices to prove that Γ(N) is Zariski-dense in Γ. So, consider a nonempty Zariski-open subset Γ_{0} of Γ. We have to prove that Γ_{0}(N) ≠ ∅.

By Proposition 8.2(a), a_{𝔭} has a 𝔭-open neighborhood U_{𝔭} in Γ_{simp}(K_{𝔭}). Then U = ⋂_{𝔭∈S} ⋂_{σ∈G(K)} U_{𝔭}^{σ} is an S-open subset of Γ_{simp}(K_{tot,S}) (Lemma 6.1).

Let F be the function field of Γ over K. Theorem 6.3 with U_{1,𝔭} = U_{2,𝔭} = U_{𝔭} gives a stabilizing element t for F/K such that all geometric zeros of t belong to U and each of them has multiplicity 1. In other words, let d = [F : K(t)] = [F K̃ : K̃(t)]. Let F̂ be the Galois closure of F/K(t). Then the following holds: (1a) F̂ is a regular extension of K. (1b) If we consider t as an element of F K̃, then div_{0}(t) = ∑_{i=1}^{d} p_{0i}, where p_{01}, …, p_{0d}

are distinct and each of them is in U, hence in Γ_{simp}(K_{tot,S}).

As an element of F, t may be identified with a separable rational map τ: Γ → ℙ^{1} over K, where

deg(τ) = [F : K(t)] = [F K̃ : K̃(t)] = deg(div_{0}(t)) = d.

Moreover, denote the zero of t, when considered as a rational function of ℙ^{1}, by p_{0}. Then τ^{−1}(p_{0})(K̃) = {p_{01}, …, p_{0d}} ⊆ Γ_{simp}(K_{tot,S}).

Let Γ̂ be a model of F̂/K. Since F̂/K is regular, Γ̂ is a curve (i.e., absolutely irreducible). Also, there exist separable rational maps π: Γ̂ → Γ and τ̂: Γ̂ → ℙ^{1} over K such that τ ∘ π = τ̂. In addition, there exists a nonempty Zariski-open subset A of ℙ^{1} such that if q̂ ∈ Γ̂(K̃) and τ̂(q̂) ∈ A(K), then K(q̂)/K is a Galois extension, π is defined at q̂, and π(q̂) ∈ Γ_{0}(K̃).

Let A be as in Lemma 7.2. By Proposition 7.1 there exists q̂ ∈ Γ̂_{simp}(M) such that p = τ̂(q̂) ∈ A ∩ A(K). By Lemma 7.2, τ^{−1}(p)(K̃) ⊆ Γ(K_{tot,S}). Let q = π(q̂). Then q is in τ^{−1}(p)(K̃) and therefore also in Γ(K_{tot,S}). Moreover, K(q) ⊆ K(q̂) ∩ K_{tot,S} ⊆ M ∩ K_{tot,S}. Since K(q̂) is Galois over K, so is K(q̂) ∩ K_{tot,S}. Hence, K(q) ⊆ N. Conclude that q ∈ Γ_{0}(N).

Let S be a set of local primes of a field K. Let N be a field between K and K_{tot,S}. We say that N has the S-density property if it satisfies the following condition: Let V be a variety over K. For each 𝔭 ∈ S let U_{𝔭} be a nonempty 𝔭-open subset of V_{simp}(K_{𝔭}). Then

(2)  V (N) ∩ ⋂_{𝔭∈S} ⋂_{σ∈G(K)} U_{𝔭}^{σ} ≠ ∅.

Corollary 7.5 (S-density property): Let K, S, M, and N be as in Theorem 7.4. Then N has the S-density property.

Proof: Since N is PSC, the corollary follows from Theorem 7.4 and from Razon’s density theorem [Ra1, Thm. 3.9]. Alternatively, one can use the arguments of the proof of Theorem 7.4 to prove the corollary directly.

Corollary 7.6: Let K be a field and let S be a finite set of local primes of K. Then K_{tot,S} is PSC. Moreover, K_{tot,S} has the S-density property.

Proof: By the Hilbert Nullstellensatz, the field M = K_{s} is PAC over K. Moreover, in the notation of Theorem 7.4, N = K_{tot,S}. So, Corollary 7.6 is a special case of Theorem 7.4 and Corollary 7.5.

Note that the first statement of Corollary 7.6 is the main result of [Po1] (if char(K) = 0). So, Theorem 7.4 is a generalization of Pop’s result.

Corollary 7.7: Let K be a countable separably Hilbertian field and let S be a finite set of local primes of K. Let e be a positive integer. Then, for almost all σ σ σ ∈ G(K)^{e}, the field N = K_{s}[σ σ σ] ∩ K_{tot,S} is PSC. Moreover, N has the S-density property.

Proof: Recall that for each σ σ σ ∈ G(K)^{e} the field K_{s}[σ σ σ] is the maximal Galois extension of K which is contained in K_{s}(σ σ σ). Hence, N is the maximal Galois extension of K which is contained in K_{s}(σ σ σ) ∩ K_{tot,S}. By [JR1, Prop. 3.1], K_{s}(σ σ σ) is PAC over K for almost all σ σ σ ∈ G(K)^{e}. The corollary is therefore a special case of Theorem 7.4 and Corollary 7.5.

Recall that a field N is ample if every curve C over N with a simple N-rational point has infinitely many N-rational points [Po2, p. 2 or HJ2, Def. 6.3]. If N is PSC and C is a curve over N with a simple N-rational point, then this point is also N_{𝔭}- rational for each 𝔭 ∈ S. By Proposition 1.5, C(N) is infinite. Thus N is ample. As an ample field, N has several nice properties, which we may apply to the field appearing in Corollary 7.7.

Corollary 7.8: Let K be a countable separably Hilbertian field and let S be a finite set of local primes of K. Let e be a positive integer. Then, for almost all σ σ σ ∈ G(K)^{e}, the field N = K_{s}[σ σ σ] ∩ K_{tot,S} has the following properties:

(a) N is ample; in particular, N is existentially closed in the field of formal power series N((x)).

(b) Let E be a finitely generated regular extension of N of transcendence degree 1. Then every finite split embedding problem over E is solvable (even in a regular way).

Proof of (a): Note that the statement about the field of formal power series is actually equivalent to N being ample [Po2, Prop. 1].

Proof of (b): We refer the reader to [HJ3, Thm. B] for an exact formulation of a split embedding problem and for an algebraic proof of the statement. Alternatively, see [Po3] for a proof which uses methods of rigid analytical geometry.

Another application of the main result is motivated by a peculiar result of Razon. He proves in [Ra2, Thm. 4.8] that for each finite set S of prime numbers and for almost all σ σ σ ∈ G(ℚ)^{e} the absolute Galois group of each field between (ℚ̃(σ σ σ) ∩ ℚ_{tot,S})_{cycl} and ℚ̃(σ σ σ)_{cycl} is isomorphic to F̂_{ω}. (Here L_{cycl} denotes the field obtained from L by adjoining all roots of unity.) The proof of this result relies, among others, on the property of ℚ̃(σ σ σ) being PAC over ℚ. We do not know whether ℚ̃[σ σ σ] is PAC over ℚ. Nevertheless, we can substitute the later property of ℚ̃(σ σ σ) by ℚ̃[σ σ σ]/ℚ being Galois and by a recent result of Haran:

Corollary 7.9: Let K be an infinite field which is finitely generated over its prime field. Let S be a finite set of local primes of K. Let e be a positive integer. Then, for almost all σ σ σ ∈ G(K)^{e} and for each field E between (K_{s}[σ σ σ] ∩ K_{tot,S})_{cycl} and K_{s}[σ σ σ]_{cycl} we have: G(E) ≅ F̂_{ω}.

Proof: By [FJ2, Cor. 12.8 and Thm. 12.10], K is Hilbertian. By Corollaries 7.7 and 7.8, for almost all σ σ σ ∈ G(K)^{e} the field N = K_{s}[σ σ σ]∩K_{tot,S} is PSCand therefore ample. Since K_{cycl} is an infinite extension of K, we may assume, in addition, that K_{cycl} ⊈ K_{s}[σ σ σ].

Let E be a field between N_{cycl} and K_{s}[σ σ σ]_{cycl}. As a separable algebraic extension of an ample field, E is ample [Po2, Prop. 1.2]. In addition, let E = {EK_{𝔭}^{σ} | 𝔭 ∈ S, σ ∈ G(K)}. By [Ja2, Lemma 7.2], E is PEC.

We prove below that since E contains K_{cycl}, the absolute Galois group G(EK_{𝔭}^{σ}) is projective for each 𝔭 ∈ S and all σ ∈ G(K). It will follow from Proposition 1.7(b) that G(E) is projective. As K_{𝔭} is, any way, determined only up to K-conjugation, we may as well assume that σ = 1.

Suppose first that K̂_{𝔭} is a finite extension of ℚ_{p} for some prime number p. Since E contains all roots of unity, l^{∞} divides [EK̂_{𝔭} : E] for each prime number l. By [Rib, p. 291, Cor. 7.4], G(EK̂_{𝔭}) is projective. Since restriction maps G(K̂_{𝔭}) bijectively onto G(K_{𝔭}) (by Krasner’s Lemma), it maps G(EK̂_{𝔭}) bijectively onto G(EK_{𝔭}). Hence, G(EK_{𝔭}) is projective.

If K̂_{𝔭} is ℝ or ℂ, then EK_{𝔭} is algebraically closed and therefore G(EK_{𝔭}) is projective.

The third possibility is that K̂_{𝔭} is a finite extension of 𝔽_{p}((t)). Then EK̂_{𝔭} is a separable algebraic extension of 𝔽̃_{p}((t)). The latter field is complete with respect to a discrete valuation and with residue field 𝔽̃_{p} of cohomological dimension 0. Hence, by [Rib, p. 277, Thm. 6.1], G(𝔽̃_{p}((t))) is projective. Let L = K_{s} ∩ 𝔽̃_{p}((t)). Then G(L) is isomorphic to G(𝔽̃_{p}((t))) and is therefore also projective. Finally, as EK_{𝔭} contains L, its absolute Galois group is also projective.

Finally, a recent result of Haran [Har, Thm. 4.1] says that if L_{1} and L_{2} are Galois extensions of a Hilbertian field L and F is a field between L and L_{1}L_{2} which is not contained in L_{1} or in L_{2}, then F is Hilbertian. In our case, K is Hilbertian and both K_{cycl} and K_{s}[σ σ σ] are Galois extensions of K. By the choice of σ σ σ, E is not contained in K_{s}[σ σ σ]. Hence, if E is also not contained in K_{cycl}, then, by Haran’s result, E is Hilbertian.

By avoiding another set of measure 0, one may choose σ σ σ such that K_{s}[σ σ σ]∩K_{tot,S} ⊈ K_{cycl} and hence E ⊈ K_{cycl}. We do not prove this fact here and instead note that if E = K_{cycl}, then E is Hilbertian (by a theorem of Kuyk [FJ2, Thm. 15.6]).

In any case, it follows now from [HJ2, Thm. 6.5] that G(E) ≅ F̂_{ω}.

Remark 7.10: (a) It is interesting to note that, in the setup of Corollary 7.8, G(E) ≅ F̂_{ω} for each field E between K_{s}[σ σ σ] and K_{s}[σ σ σ]_{cycl}.

Indeed, for almost all σ σ σ ∈ G(K)^{e}, K_{s}[σ σ σ] is PAC and G(K_{s}[σ σ σ]) ≅ F̂_{ω} [Ja4, Thm. 2.7], K_{cycl} ⊈ K_{s}[σ σ σ], and K_{s}[σ σ σ] ⊈ K_{cycl} (because K_{s}[σ σ σ]/K has Galois subextensions L/K with G(L/K) ≅ S_{n}, as follows e.g. from [JR1, Prop. 6.2]). If K_{s}[σ σ σ] ⊂ E ⊆ K_{s}[σ σ σ]_{cycl} = K_{cycl} · K_{s}[σ σ σ], then, by Haran’s result, E is Hilbertian. As E is an algebraic extension of a PAC field, E is itself PAC [FJ2, Cor. 10.7]. By [HJ2, Thm. A], G(E) ≅ F̂_{ω}.

(b) If we take K = ℚ in Corollary 7.9 and let S be the set whose only element is the ordering of ℚ, then ℚ_{tot,S} = ℚ_{tr} is the maximal totally real extension of ℚ. For almost all σ σ σ ∈ G(ℚ)^{e} Corollary 7.7 asserts that E = ℚ̃[σ σ σ]∩ℚ_{tr} is PRC (Remark 1.4(f)). Hence, F = (ℚ̃[σ σ σ] ∩ ℚ_{tr})(\sqrt{−1}) is a PRC field [Pre, Thm. 3.1] and therefore PAC. By a result of Weissauer [FJ2, Cor. 12.15] (which preceded that of Haran), F = E(\sqrt{−1}) is Hilbertian. So, again by [HJ2, Thm. 6.5], G(F) ≅ F̂_{ω}. This is an analog of a result of Pop which says that G(ℚ_{tr}(\sqrt{−1})) ≅ F̂_{ω}.

8. Appendix — Generalized local fields A generalized local field is a pair (K, 𝔭) which satisfies one of the following three conditions: (1a) K is a field and 𝔭 is an equivalence class of Henselian valuations of K. In this case

we choose a valuation v_{𝔭} representing 𝔭. (1b) K is a real closed field and 𝔭 is the unique ordering < of K. The latter defines a

generalized metric absolute value | | of K by |x| = x if x ≥ 0 and |x| = −x if x ≤ 0.

(1c) K is the algebraic closure of a real closed field R and 𝔭 is the corresponding generalized metric absolute value | |. Thus, if z = x + y\sqrt{−1} with x, y ∈ R, then |z| = \sqrt{x^{2} + y^{2}}.

In particular, each local field is a generalized local field.

In each case 𝔭 defines a topology on K. A basic 𝔭-open neighborhood of an element a ∈ K is {x ∈ K | v_{𝔭}(x − a) > α} for some α in the value group of 𝔭 in the Henselian case, and {x ∈ K | |x − a| < ε} for some positive ε ∈ R in the generalized metric case.

Two of the main tools that we use in the proof of the main result of this work are the density theorem and the open map theorem for varieties over local fields. In addition, we use the continuity of zeros of algebraic functions. Proofs of these theorems for Henselian valuations appear in [GPR]. However, they are also true for generalized metric absolute values. Here we reduce the proof of the latter case to the Henselian case. The reduction depends on the following construction.

Remark 8.1: Ultrapowers. Let (K, | |) be a generalized metric absolute valued field. Choose a nonprincipal ultrafilter D of ℕ and let (K^{∗}, | |^{∗}) = (K, | |)^{ℕ}/D be the corresponding ultrapower. (See [FJ2, §6.7] for definition and basic properties of ultrapowers.) Then (K^{∗}, | |^{∗}) is an elementary extension of (K, | |) [FJ2, Prop. 6.15]. In particular K^{∗} is real (resp., algebraically) closed if K is real (resp., algebraically) closed and | |^{∗} is a generalized metric absolute value. Let

O = \{x ∈ K^{∗} | ∃n ∈ ℕ: |x| ≤ n\} and M = \{x ∈ K^{∗} | ∀n ∈ ℕ: |x| ≤ \frac{1}{n}\}.

Then O is a valuation ring of K^{∗} and M is the maximal ideal of O. In particular, the equivalence class in K^{∗} of (1, 2, 3, 4, …) is not in O. Denote the corresponding valuation by v.

If K^{∗} is real closed, then, by [Ja1, Lemma 16.2], v is Henselian. Alternatively note that if x ∈ O ∖ M, then x^{2} + 1 ≢ 0 mod M. Thus, O/M is not algebraically closed. Now use the inequality efg ≤ [K̃^{∗} : K^{∗}] for extensions of valuations [Rbn, p. 228] to conclude that v has a unique extension to K̃^{∗}. In other words, (K^{∗}, v) is Henselian.

Claim: The v-topology of K^{∗} coincides with the | |^{∗}-topology. It suffices to prove that each v-open neighborhood of 0 in K^{∗} is contained in an | |^{∗}-open neighborhood of K^{∗} and vica versa.

Indeed, if a ∈ K^{∗}, a ≠ 0, then v(x) > v(a) implies that v(\frac{x}{a}) > 0, hence \frac{x}{a} ∈ M, hence |\frac{x}{a}| < 1 and therefore |x| < |a|.

Conversely, choose b ∈ M, b ≠ 0. If |x| < |a|, then |\frac{x}{a}| < 1, hence |\frac{xb}{a}| < |b| < \frac{1}{n} for all n ∈ ℕ. It follows that \frac{xb}{a} ∈ M and therefore v(x) > v(\frac{a}{b}).

As usual, each of the above field topologies induce compatible topologies on the sets of rational points of varieties [Mum, p. 81]. Recall that rational functions f_{1}, …, f_{r} of a variety V of dimension r are local parameters at a simple point a of V if they generate the maximal ideal of the local ring O_{V,a}.

Proposition 8.2: Let (K, 𝔭) be a generalized local field. Let V be a variety over K and let a ∈ V_{simp}(K).

(a) Local parameters: Let t_{1}, …, t_{r} be a system of local parameters of V at a. View t = (t_{1}, …, t_{r}) as a rational map of V into 𝔸^{r}. Then a has a 𝔭-open neighborhood V in V (K) which t maps 𝔭-homeomorphically onto a 𝔭-open neighborhood of 0 in K^{r}.

(b) Density theorem: Each 𝔭-open neighborhood of a in V (K) is Zariski-dense in V. In particular, V (K) is Zariski-dense in V.

(c) Open map theorem: Let φ: V → W be a dominant morphism of varieties over K. Suppose that φ is smooth at a simple point a of V (K). Then a has a 𝔭-open neighborhood V in V (K) such that φ|_{V} is a 𝔭-open map onto a 𝔭-open subset W of W (K). In particular, if dim(V) = dim(W), then V and W can be chosen such that φ maps V 𝔭-homeomorphically onto W.

Proof: As all parts of the theorem are of Zariski-local nature, we may assume that both V and W are affine.

Proof of (a): The case where (K, 𝔭) is Henselian is [GPR, Thm. 9.2]. So, suppose that (K, 𝔭) is a generalized metric absolute valued field.

Take a non-principal ultrapower (K^{∗}, | |^{∗}) of (K, | |). By Remark 8.1, K^{∗} has a valuation v such that the v-topology of K^{∗} coincides with the | |^{∗}-topology of K^{∗}. Moreover, K^{∗} is Henselian with respect to v.

The point a belongs also to V_{simp}(K^{∗}) and we can view t_{1}, …, t_{r} as local parameters of V over K^{∗} (i.e., of V ×_{K} K^{∗}) at a. By the Henselian case, a has a v-open neighborhood V in V (K^{∗}) which t maps v-homeomorphically onto a v-open neighborhood W of 0 in (K^{∗})^{r}. Since the | |^{∗}-topology of K^{∗} coincides with its v-topology, V and W are also | |^{∗}-open sets and t is an | |^{∗}-homeomorphism. In other words, the following statement holds:

(2) There exists ε > 0 such that t maps the set E = {x ∈ V (K^{∗}) | |x − a|^{∗} <^{∗} ε} injectively into (K^{∗})^{r}, 0 ∈ t(E), and for each a′ ∈ E and for each ε′ > 0 there exists δ > 0 such that for all y ∈ (K^{∗})^{r} we have

{y ∈ (K^{∗})^{r} | |y − t(a′)|^{∗} < δ} ⊆ t({x ∈ E | |x − a′|^{∗} < ε′}).

Since (K^{∗}, | |^{∗}) is an elementary extension of (K, | |) and (2) is an elementary statement on K^{∗} (in the language of ordered fields) with parameters in K, it is also true if we replace (K^{∗}, | |^{∗}) by (K, | |). Conclude that a has an | |-open neighborhood in V (K) which t maps | |-homeomorphically onto an | |-open neighborhood of 0 in K^{r}.

Proof of (b): Let V be a 𝔭-open neighborhood of a in V (K). By (a) we can make V smaller, if necessary, to assume that t induces a 𝔭-homeomorphism of V onto a 𝔭-open neighborhood W of 0 in K^{r}. Suppose that V is a Zariski-closed subset of 𝔸^{n} and let x = (x_{1}, …, x_{n}) be a generic point of V over K. Then t can be also considered as a separating transcendence base of K(x)/K.

Let now g(x) be a nonzero element of K[x]. Then h(t) = Norm_{K(x)/K(t)}g(x) is a nonzero element of K(t). Since K is infinite and W contains a 𝔭-ball, h(t) does not vanish on W. Hence, g(x) does not vanish on V. Conclude that V is Zariski-dense in V.

Proof of (c): The case where (K, 𝔭) is Henselian is [GPR, Thm. 9.4]. Assume again that (K, 𝔭) is a generalized metric absolute valued field and, as in the proof of (a), consider a non-principal ultrapower (K^{∗}, | |^{∗}) of (K, | |).

We may consider φ: V → W as a morphism of the corresponding varieties over K^{∗}. By the Henselian case, a has a v-open neighborhood V in V (K^{∗}) such that φ|_{V} is a v-open map into W (K^{∗}). Since φ is defined by polynomials, with coefficients in K, we may proceed as in (a) to conclude that φ induces an | |-open map of a | |-open neighborhood of a in V (K) into W (K).

Proposition 8.3 (Continuity of zeros of algebraic functions): Let 𝔞 be a positive, Γsmooth divisor of F/K. Let (K, 𝔭) be a generalized local field. Let Γ be a projective curve over K. Denote its function field (resp., semilocal ring of singularities) by F (resp., O). Let 𝔞 be a positive Γ-smooth divisor of F/K. Let f be an element of O such that div(f) = ∑_{i=1}^{m} b_{i} − 𝔞, where b_{1}, …, b_{m} are distinct points in Γ_{simp}(K) ∖ Supp(𝔞). For each i let U_{i} be a 𝔭-open neighborhood of b_{i} in Γ_{simp}(K). Let u_{0}, …, u_{n} be a K-basis of L_{O}(𝔞). Write f = ∑_{j=0}^{n} a_{j}u_{j} with a_{j} ∈ K. Then (a_{0}, …, a_{n}) has a 𝔭open neighborhood A in K^{n+1} such that for each (a′_{0}, …, a′_{n}) ∈ A the function f′ = ∑_{j=0}^{n} a′_{j}u_{j} satisfies div(f′) = ∑_{i=1}^{m} b′_{i} − 𝔞, where b′_{i} ∈ U_{i} ∖ Supp(𝔞), i = 1, …, m. Moreover, b′_{1}, …, b′_{m} are distinct.

Proof: The case where (K, 𝔭) is Henselian follows from [GPR, Cor. 7.2]. So, suppose that (K, 𝔭) = (K, | |) is a generalized metric absolute valued field. Let (K^{∗}, 𝔭^{∗}) and v be as in the second paragraph of the proof of Proposition 8.2(a). Shrink U_{1}, …, U_{m}, if necessary, to assume that they are mutually disjoint and also disjoint from Supp(𝔞). Moreover, suppose that Γ ⊆ ℙ^{r}, choose a generic point x = (x_{0}: …:x_{r}) for Γ over K with x_{0}, …, x_{r} ∈ F and choose homogeneous polynomials of the same degree g_{j}, h_{j} ∈ K[X_{0}, …, X_{r}] such that h_{j}(x) ≠ 0 and u_{j} = \frac{g_{j}(x)}{h_{j}(x)} for j = 0, …, n. Since none of the points b_{1}, …, b_{m} is a pole of u_{j}, we may choose h_{j} such that h_{j}(b_{i}) ≠ 0 for all i and j. Then shrink U_{1}, …, U_{m} further until h_{j}(X) does not vanish on U_{i} for all i and j.

Note that OK^{∗} is the semilocal ring of singularities of Γ ×_{K} K^{∗} and u_{0}, …, u_{n} is also a K^{∗}-basis for L_{OK^{∗}}(𝔞) (Lemma 2.6). Each of the sets U_{i}(K^{∗}) is 𝔭^{∗}-open and therefore also v-open. Hence, by the Henselian case, (a_{0}, …, a_{n}) has a v-open neighborhood A^{∗} in (K^{∗})^{n+1} such that for each (a_{0}^{∗}, …, a_{n}^{∗}) ∈ A^{∗} the element f^{∗} = ∑_{j=0}^{n} a_{j}^{∗}u_{j} of F K^{∗} has for each i exactly one zero b_{i}^{∗} in each U_{i}(K^{∗}). Thus ∑_{j=0}^{n} a_{j}^{∗} \frac{g_{j}(b_{i}^{∗})}{h_{j}(b_{i}^{∗})} = f^{∗}(b_{i}^{∗}) = 0.

Since the v-topology coincides with the 𝔭^{∗}-topology, we may assume that A^{∗} is a basic 𝔭^{∗}-open neighborhood of (a_{0}, …, a_{n}) in (K^{∗})^{n+1}. As (K^{∗}, 𝔭^{∗}) is an elementary extension of (K, 𝔭), the point (a_{0}, …, a_{n}) has a 𝔭^{∗}-open neighborhood A in K^{n+1} such that if (a′_{0}, …, a′_{n}) ∈ A and f′ = ∑_{j=0}^{n} a′_{j}u_{j}, then for each i there exists a unique b′_{i} ∈ U_{i} such that f′(b′_{i}) = ∑_{j=0}^{n} a′_{j} \frac{g_{j}(b′_{i})}{h_{j}(b′_{i})} = 0. Thus ∑_{i=1}^{m} b′_{i} ≤ div_{0}(f′). Since u_{j} ∈ L_{O}(𝔞), div_{∞}(f′) ≤ 𝔞. By assumption, deg(𝔞) = deg(div_{∞}(f)) = deg(div_{0}(f)) = m. It follows that m ≤ deg(div_{0}(f′)) = deg(div_{∞}(f′)) ≤ deg(𝔞) = m. Hence, div(f′) = ∑_{i=1}^{m} b′_{i} − 𝔞, as desired.

References

[Art] E. Artin, Algebraic Numbers and Algebraic Functions, Gordon and Breach, New York, 1967.

[Bou] N. Bourbaki, Commutative Algebra, Chapters 1–7, Springer, Berlin, 1989.

[CaF] J.W.S. Cassels and A. Fröhlich, Algebraic Number Theory, Academic Press, London, 1967.

[Che] C. Chevalley, Introduction to the theory of algebraic functions of one variable, Mathematical Surveys VI, AMS, Providence, 1951.

[Deu] M. Deuring, Algebren, Ergebnisse der Mathematik und ihrer Grenzgebiete 4, Springer, Berlin, 1935.

[Eis] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995.

[FJ1] M. Fried and M. Jarden, Diophantine properties of subfields of ℚ̃, American Journal of Mathematics 100 (1978), 653–666.

[FJ2] M. D. Fried and M. Jarden, Field Arithmetic, Ergebnisse der Mathematik (3) 11, Springer, Heidelberg, 1986.

[Ful] W. Fulton, Algebraic Curves, Addison Weseley, Redwood City, 1989.

[GeJ] W.-D. Geyer and M. Jarden, On stable fields in positive characteristic, Geometria Dedicata 29 (1989), 335–375.

[GPR] B. Green, F. Pop, and P. Roquette, On Rumely’s local-global principle, Jahresbericht der Deutschen Mathematiker-Vereinigung 97 (1995), 43–74.

[Har] D. Haran, Hilbertian fields under separable algebraic extensions, to appear in Inventiones Mathematicae.

[HJ1] D. Haran and M. Jarden, The absolute Galois group of a pseudo p-adically closed field, Journal für die reine und angewandte Mathematik 383 (1988), 147–206.

[HJ2] D. Haran and M. Jarden, Regular split embedding problems over complete valued fields, Forum Mathematicum 10 (1998), 329–351.

[HJ3] D. Haran and M. Jarden, Regular split embedding problems over function fields of one variable over ample fields, Journal of Algebra 208 (1998), 147-164.

[Hrt] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer, New York, 1977.

[Ja1] M. Jarden, Intersection of local algebraic extensions of a Hilbertian field (A. Barlotti et al., eds), NATO ASI Series C 333 343–405, Kluwer, Dordrecht, 1991.

[Ja2] M. Jarden, Algebraic realization of p-adically projective groups, Compositio Mathematica 79 (1991), 21–62.

[Ja3] M. Jarden, The inverse Galois problem over formal power series fields, Israel Journal of Mathematics 85 (1994), 263–275.

[Ja4] M. Jarden, Large normal extensions of Hilbertian fields, Mathematische Zeitschrift 224 (1997), 555-565.

[Jac] N. Jacobson, Finite Dimensional Division Algebras over Fields, Springer-Verlag, Berlin, 1996

[JR1] M. Jarden and A. Razon, Pseudo algebraically closed fields over rings, Israel Journal of Mathematics 86 (1994), 25–59.

[JR2] M. Jarden and A. Razon, Rumely’s local global principle for algebraic PSC fields over rings, Transactions of AMS 350 (1998), 55–85.

[Kow] H. J. Kowalsky, Zur topologischen Kennzeichnung von Körpern, Mathematische Nachrichten 9 (1953), 261–320.

[Lan] S. Lang, Introduction to algebraic geometry, Interscience Publishers, New York, 1958.

[Mat1] H. Matsumura, Commutative Algebra, Benjamin, New York, 1970.

[Mat2] H. Matsumura, Commutative ring theory, Cambridge studies in advanced mathematics 8, Cambridge University Press, Cambridge 1994.

[Mum] D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Mathematics 1358, Springer, Berlin, 1988.

[Neu] K. Neumann, Every field is stable, Israel Journal of Mathematics 104 (1998), 221–260.

[Po1] F. Pop, Fields of totally Σ-adic numbers, manuscript, Heidelberg, 1992.

[Po2] F. Pop, Embedding problems over large fields, Annals of Mathematics 144 (1996), 1–34.

[Po3] F. Pop, The geometric case of a conjecture of Shafarevich, — G_{κ̃(t)} is profinite free —, preprint, Heidelberg, 1993.

[Pre] A. Prestel, Pseudo real closed fields, Set Theory and Model Theory, Lecture Notes 872, (1981), 127–156, Springer.

[Ra1] A. Razon, On the density property of PSC fields, manuscript, Essen, 1998.

[Ra2] A. Razon, Abundance of Hilbertian domains, manuscripta mathematica 94 (1997), 531–542.

[Rbn] P. Ribenboim, Théorie des valuations, Les Presses de l’Université de Montréal, Montréal, 1964.

[Rib] L. Ribes, Introduction to Profinite Groups and Galois Cohomology, Queen’s papers in Pure and Applied Mathematics 24, Queen’s University, Kingston, 1970.

[Ro1] M. Rosenlicht, Equivalence relations on algebraic curves, Annals of Mathematics 56 (1952), 169–191.

[Ro2] M. Rosenlicht, Generalized Jacobian varieties, Annals of Mathematics 59 (1954), 505– 530.

[Ser] J.-P. Serre, Algebraic Groups and Class Fields, Graduate Texts in Mathematics 117, Springer, New-York, 1988.

[Sha] I. R. Shafarevich, Basic Algebraic Geometry, Grundlehren der mathematischen Wissenschaften 213, Springer Berlin, 1977.

[Wei] A. Weil, Foundation of Algebraic Geometry, American Mathematical Society, Providence, 1962.

[ZaS] O. Zariski and P. Samuel, Commutative Algebra I, Springer, New York, 1975.

27 March, 1999