The fibration method over real function fields

Abstract

Let $\mathbb R(C)$ be the function field of a smooth, irreducible projective curve over $\mathbb R$. Let X be a smooth, projective, geometrically irreducible variety equipped with a dominant morphism f onto a smooth projective rational variety with a smooth generic fibre over $\mathbb R(C)$. Assume that the cohomological obstruction introduced by Colliot-Thélène is the only one to the local-global principle for rational points for the smooth fibres of f over $\mathbb R(C)$-valued points. Then we show that the same holds for X, too, by adopting the fibration method similarly to Harpaz–Wittenberg.

1 Introduction

Let C be a smooth, geometrically irreducible projective curve over ${\mathbb {R}}$. Let ${\mathbb {R}}(C)$ denote the function field of C, and for every $x\in C({\mathbb {R}})$ let ${\mathbb {R}}(C)_x$ be the completion of ${\mathbb {R}}(C)$ with respect to the valuation furnished by x. Now let ${\mathcal {V}}$ be a class of geometrically irreducible projective varieties over ${\mathbb {R}}(C)$. We say that ${\mathcal {V}}$ satisfies the local-global principle for rational points if for every X in ${\mathcal {V}}$ the following holds:

$$\begin{aligned} \prod _{x\in C({\mathbb {R}})}X({\mathbb {R}}(C)_x)\ne \emptyset \text { implies that }X({\mathbb {R}}(C))\ne \emptyset . \end{aligned}$$

There are classes of varieties when this local-global principle holds:

Theorem 1.1

(Witt) The local-global principle for rational points holds for smooth quadric hypersurfaces of dimension at least one over ${\mathbb {R}}(C)$.

Proof

See [15, 16]. $\square$

There is an even more general result due to Scheiderer:

Theorem 1.2

(Scheiderer) The local-global principle for rational points holds for smooth compactifications of homogeneous spaces over connected linear groups over ${\mathbb {R}}(C)$.

Proof

See [14]. $\square$

However similarly to varieties over number fields, there are some reasonably simple counter-examples to this local-global principle. The following counterexample is due to Racinet: let $C=\mathbb P^1_{{\mathbb {R}}}$ and let U be the affine surface over $\mathbb R(C)={\mathbb {R}}(t)$ given by the following equation:

$$\begin{aligned} a^2+b^2=(c^2+t)(tc^2+c-1) \end{aligned}$$

in the variables a, b and c, and let X be a smooth projective model of U. Then $X({\mathbb {R}}(X)_x)$ is non-empty for every $x\in C({\mathbb {R}})$, but $X({\mathbb {R}}(C))$ is empty. The failure of the local-global principle for this X was explained using a simple obstruction, analogous to the Brauer–Manin obstruction in the number field case, by Colliot-Thélène in [3] around 20 years ago, using unramified cohomology groups. He constructed a subset

$$\begin{aligned} \left( \prod _{x\in C({\mathbb {R}})}X({\mathbb {R}}(C)_x) \right) ^{CT}\subseteq \prod _{x\in C({\mathbb {R}})}X({\mathbb {R}}(C)_x) \end{aligned}$$

which contains $X({\mathbb {R}}(C))$. We will review this construction in the next section. Later Ducros has found a simple topological obstruction equivalent to this obstruction which we will describe next. By resolution of singularities there is an integral, smooth, projective variety ${\mathcal {X}}$ equipped with a projective dominant morphism $p:{\mathcal {X}}\rightarrow C$ over ${\mathbb {R}}$ whose generic fibre is $X\rightarrow {\text {Spec}}({\mathbb {R}}(C))$. Then we have:

Theorem 1.3

(Ducros) Let X be as above and assume that $\prod _{x\in C({\mathbb {R}})}X({\mathbb {R}}(C)_x)$ is non-empty. Then $\big (\prod _{x\in C({\mathbb {R}})}X({\mathbb {R}}(C)_x)\big )^{CT}$ is non-empty if and only if there is a continuous semi-algebraic section of p on $C({\mathbb {R}})$ whose image lies in the smooth locus of p.

Proof

This is Théorème 4.3 of [7] on page 86. $\square$

Let again ${\mathcal {V}}$ be a class of geometrically irreducible projective varieties over ${\mathbb {R}}(C)$. We say that for $\mathcal V$ the CT obstruction is the only one to the local-global principle for rational points if for every X in ${\mathcal {V}}$ the following holds:

$$\begin{aligned} \left( \prod _{x\in C({\mathbb {R}})}X({\mathbb {R}}(C)_x) \right) ^{CT}\ne \emptyset \text { implies that }X(\mathbb R(C))\ne \emptyset . \end{aligned}$$

Ducros produced a class of geometrically rational smooth projective surfaces over the rational function field ${\mathbb {R}}(t)$ for which the local-global principle might fail, in fact Racinet’s example belongs to this class, but the CT obstruction is the only one to the local-global principle for rational points:

Theorem 1.4

(Ducros) The CT obstruction is the only one to the local-global principle for rational points for smooth projective varieties over ${\mathbb {R}}(C)$ which are fibrations in Brauer–Severi varieties over ${\mathbb {P}}^1_{{\mathbb {R}}(C)}$.

Proof

The special case of conic fibrations is proved in [7]. The general case is proved in [8]. $\square$

The main result of this article, Theorem 5.3, implies the following generalisation of the theorem above:

Theorem 1.5

Let $f:X\rightarrow Y$ be a dominant morphism between smooth,  proper,  geometrically irreducible varieties over ${\mathbb {R}}(C)$ with a geometrically irreducible and smooth generic fibre,  and assume that Y is rational. If the CT obstruction is the only one to the local-global principle for rational points for the smooth fibres of f over all but finitely many ${\mathbb {R}}(C)$-valued points,  then the same holds for X,  too.

Clearly Theorem 1.4 is a consequence of this result by Theorem 1.1. The result also implies that the CT obstruction is the only one to the local-global principle for rational points for X when the generic fibre of f is the smooth compactification of a homogeneous space by a connected linear algebraic group by Theorem 1.2. We will actually prove a stronger claim incorporating weak approximation (see Theorem 5.3).

Remark 1.6

The reader might wonder why we only consider real points in the formulation of the local-global principle. Note that our main interest is the case of rationally connected varieties (see Conjecture 2.6 below), when by the Graber–Harris–Starr theorem (see [10]) we have a rational point over the completion of ${\mathbb {R}}(X)$ with respect to every complex point of C. So we do not get additional restrictions by considering all closed points of C. However this is not true in general, genus one curves give easy counterexamples.

Remark 1.7

It is natural to ask if this result can be applied inductively. The way we formulate our main result does not lend itself naturally to iteration, since having a rational point does not guarantee X is rational itself. However see [1], which became available after our preprint appeared, where a similar idea is pursued.

Contents 1.8

In the next section we will recall the notion of unramified cohomology and the definition of Colliot-Thélène’s obstruction, then we will show a useful lemma on the birational invariance of this obstruction. Then we review the work of Ducros on the topological reinterpretation of the cohomological obstruction in the third section. In the fourth section we prove a refined version of the Stone–Weierstrass approximation theorem, which also incorporates interpolation. In the fifth section we formulate a stronger form of our main result, Theorem 5.3, and reduce the general case to fibrations over the projective line. We use arguments inspired by the work of Harpaz–Wittenberg in [11] to prove Theorem 5.3 in the sixth section, using the approximation theorem from the fourth section as a crucial ingredient.

2 The local-global principle and Colliot-Thélène’s obstruction

Definition 2.1

Let X be a smooth, irreducible projective variety defined over a field F. Let F(X) and X(d) denote the function field of X and the set of points of X of codimension d, respectively. For every $x\in X(1)$ let ${\mathcal {O}}_{X,x}$ and F(x) denote the discrete valuation ring in F(X) corresponding to x and the residue field of ${\mathcal {O}}_{X,x}$, respectively. Assume that $\text {char}(F)\ne 2$. For the sake of simplicity let ${\mathbb {Z}}/2$ denote the group of order two. We will also denote by the same symbol the constant étale sheaf of order 2 over any scheme. The unramified cohomology group $H_{nr}^i(F(X)/F,{\mathbb {Z}}/2)$ of X over F is by definition (see section 1.1 of [5] on page 6):

where

$$\begin{aligned} \partial _x:H^i_{{\grave{\mathrm{e}}}{\mathrm{t}}}(F(X),{\mathbb {Z}}/2)\longrightarrow H^{i-1}_{{\grave{\mathrm{e}}}{\mathrm{t}}}(F(x),{\mathbb {Z}}/2) \end{aligned}$$

is the residue map associated to the discrete valuation ring ${\mathcal {O}}_{X,x}$.

Next we need some basic facts about the Galois cohomology of function fields of real algebraic curves, and some form of a residue theorem for them.

Definition 2.2

Let C be a smooth, geometrically irreducible projective curve over ${\mathbb {R}}$. Let ${\mathbb {R}}(C)$ denote the function field of C, as above, and for every $x\in C({\mathbb {R}})$ let ${\mathbb {R}}(C)_x$ be the completion of ${\mathbb {R}}(C)$ with respect to the valuation furnished by x. Then we have a residue map

$$\begin{aligned} \partial _x:H^i_{{\grave{\mathrm{e}}}{\mathrm{t}}}({\mathbb {R}}(C)_x,{\mathbb {Z}}/2)\longrightarrow H^{i-1}_{{\grave{\mathrm{e}}}{\mathrm{t}}}({\mathbb {R}},{\mathbb {Z}}/2) \end{aligned}$$

as the residue field ${\mathbb {R}}(x)$ of ${\mathbb {R}}(C)_x$ is ${\mathbb {R}}$. Note that as a graded algebra:

$$\begin{aligned} H^*_{{\grave{\mathrm{e}}}{\mathrm{t}}}({\mathbb {R}},{\mathbb {Z}}/2) \cong {\mathbb {Z}}/2[t], \end{aligned}$$

where t is the generator of the group $H^1_{{\grave{\mathrm{e}}}{\mathrm{t}}}({\mathbb {R}},{\mathbb {Z}}/2)$ of order two. In particular we have a canonical isomorphism $H^i_{{\grave{\mathrm{e}}}{\mathrm{t}}} ({\mathbb {R}},{\mathbb {Z}}/2)\cong {\mathbb {Z}}/2$. So the residue map is a homomorphism:

$$\begin{aligned} \partial _x:H^i_{{\grave{\mathrm{e}}}{\mathrm{t}}}({\mathbb {R}}(C)_x,{\mathbb {Z}}/2)\longrightarrow {\mathbb {Z}}/2. \end{aligned}$$

By slight abuse of notation let $\partial _x$ denote also the composition of the pull-back

$$\begin{aligned} H^i_{{\grave{\mathrm{e}}}{\mathrm{t}}}({\mathbb {R}}(C),{\mathbb {Z}}/2)\longrightarrow H^i_{{\grave{\mathrm{e}}}{\mathrm{t}}}({\mathbb {R}}(C)_x,{\mathbb {Z}}/2) \end{aligned}$$

and this residue map.

The residue theorem, also called the reciprocity law, is the following

Proposition 2.3

Let $V\subseteq C({\mathbb {R}})$ be a connected component, let i be at least 2, and let $h\in H^i_{{\grave{\mathrm{e}}}{\mathrm{t}}}({\mathbb {R}}(C),{\mathbb {Z}}/2)$. Then

$$\begin{aligned} \sum _{x\in V}\partial _x(h)=0.\square \end{aligned}$$

Remark 2.4

It is easy to see that all but finitely many terms of the sum above are zero, so the left hand side is well-defined. For a proof of this fact, and the proposition, see for example Proposition 3.7 of [3] on pages 157–158.

For the sake of simple notation let ${\mathbb {A}}_C$ denote the direct product $\prod _{x\in C({\mathbb {R}})}{\mathbb {R}}(C)_x$. It is an algebra over ${\mathbb {R}}(C)$. Now let X be a smooth, irreducible projective variety defined over ${\mathbb {R}}(C)$. Clearly

$$\begin{aligned} X({\mathbb {A}}_C)=\prod _{x\in C({\mathbb {R}})}X({\mathbb {R}}(C)_x). \end{aligned}$$

Now we are ready to define Colliot-Thélène’s obstruction.

Definition 2.5

Consider the set

$$\begin{aligned} X({\mathbb {A}}_C)^{CT}\subseteq X({\mathbb {A}}_C), \end{aligned}$$

whose elements

$$\begin{aligned} \prod _{x\in C({\mathbb {R}})}M_x\in \left( \prod _{x\in C({\mathbb {R}})}X({\mathbb {R}}(C)_x)\right) ^{CT} \end{aligned}$$

are subject to the following condition:

$$\begin{aligned} \sum _{x\in V}\partial _x(M_x^*(h))=0 \end{aligned}$$

for every $h\in H_{nr}^{\dim (X)+2}( {\mathbb {R}}(C)(X)/\mathbb R(C),{\mathbb {Z}}/2)$ and for every $V\subseteq C({\mathbb {R}})$ connected component, where

$$\begin{aligned} M_x^*:H_{nr}^i({\mathbb {R}}(C)(X)/{\mathbb {R}}(C),{\mathbb {Z}}/2) \rightarrow H_{{\grave{\mathrm{e}}}{\mathrm{t}}}^i({\mathbb {R}}(C)_x,{\mathbb {Z}}/2) \end{aligned}$$

is the pull-back with respect to the map $M_x$. Note that all but finitely many terms of the sum above are zero, so the left hand side is well-defined, and the image of $X({\mathbb {R}}(C))$ in $X({\mathbb {A}}_C)$ under the diagonal embedding is in $X({\mathbb {A}}_C)^{CT}$ by Proposition 2.3 above. One may justify the usage of the more mysterious group $H_{nr}^i({\mathbb {R}}(C)(X)/\mathbb R(C),{\mathbb {Z}}/2)$ instead of $H_{{\grave{\mathrm{e}}}{\mathrm{t}}}^i(X,\mathbb Z/2)$ as follows: for proper varieties having a smooth rational point is a birationally invariant property, so the obstruction should also have birational invariance. This holds for the former group, but not the latter.

Now that we made explicit what we mean by the CT obstruction, we can make the following bold conjecture, which is motivated by theorems of Witt, Scheiderer and Ducros mentioned in the introduction, and in analogy with Colliot-Thélène’s celebrated conjecture (see [4]) saying that the Brauer–Manin obstruction is the only one to the local-global principle for rational points for smooth projective rationally connected varieties over number fields:

Conjecture 2.6

The CT obstruction is the only one to the local-global principle for rational points for smooth projective rationally connected varieties over ${\mathbb {R}}(C)$.

All known classes of varieties for which CT obstruction is the only one to the local-global principle for rational points are rationally connected. Our main result Theorem 1.5 is a contribution to this conjecture.

We finish this section with a lemma which will be used in the proof of Theorem 5.3. Let $f:X\rightarrow Y$ be a morphism between smooth, projective, irreducible varieties over ${\mathbb {R}}(C)$. The morphism f induces a map $X({\mathbb {R}}(C)_x)\rightarrow Y({\mathbb {R}}(C)_x)$ for every $x\in C({\mathbb {R}})$, which in turn induces a map $X({\mathbb {A}}_C) \rightarrow Y({\mathbb {A}}_C)$, which we will denote also by f by slight abuse of notation.

Lemma 2.7

Assume that $f:X\rightarrow Y$ is birational and let ${\underline{M}}\in X({\mathbb {A}}_C)$. Then

$$\begin{aligned} {\underline{M}}\in X({\mathbb {A}}_C)^{CT}\text { if and only if } f({\underline{M}})\in Y({\mathbb {A}}_C)^{CT}. \end{aligned}$$

Proof

Let d be the common dimension of X and Y. By assumption the pull-back with respect to f induces an isomorphism ${\mathbb {R}}(C)(Y)\rightarrow {\mathbb {R}}(C)(X)$ and hence an isomorphism $f^*:H^{d+2}_{{\grave{\mathrm{e}}}{\mathrm{t}}}({\mathbb {R}}(C)(Y),\mathbb Z/2)\rightarrow H^{d+2}_{{\grave{\mathrm{e}}}{\mathrm{t}}}({\mathbb {R}}(C)(X),{\mathbb {Z}}/2)$. Under this isomorphism the subgroup $H_{nr}^{d+2}(\mathbb R(C)(Y)/{\mathbb {R}}(C),{\mathbb {Z}}/2)$ maps isomorphically onto $H_{nr}^{d+2}({\mathbb {R}}(C)(X)/{\mathbb {R}}(C),{\mathbb {Z}}/2)$. Write ${\underline{M}}$ as

$$\begin{aligned} {\underline{M}}=\prod _{x\in C({\mathbb {R}})}M_x \end{aligned}$$

and let $h\in H_{nr}^{d+2}({\mathbb {R}}(C)(Y)/{\mathbb {R}}(C),\mathbb Z/2)$ be arbitrary. Then

$$\begin{aligned} M_x^*(f^*(h))=f(M_x)^*(h) \end{aligned}$$

for every $x\in C({\mathbb {R}})$ by naturality, so

$$\begin{aligned} \sum _{x\in V}\partial _x(M_x^*(f^*(h)))=0 \text { if and only if } \sum _{x\in V}\partial _x(f(M_x)^*(h))=0 \end{aligned}$$

for every connected component $V\subseteq C({\mathbb {R}})$. The claim is now clear. $\square$

3 The topological reinterpretation of the obstruction due to Ducros

Notation 3.1

By resolution of singularities there is an integral, smooth, projective variety ${\mathcal {X}}$ equipped with a projective dominant morphism $p:{\mathcal {X}}\rightarrow C$ over ${\mathbb {R}}$ whose generic fibre is $X\rightarrow {\text {Spec}}(\mathbb R(C))$. As usual we will call ${\mathcal {X}}$ a model of X over C. For every closed point x of C let ${\mathcal {O}}_x$ be the valuation ring of ${\mathbb {R}}(C)_x$, let ${\mathcal {X}}_x$ denote the fibre of p over x, let ${\mathcal {X}}_{sm}\subseteq {\mathcal {X}}$ be the smooth locus of p, and let ${\mathcal {X}}_{x,sm}=\mathcal X_{sm}\cap {\mathcal {X}}_x$ be the smooth locus of ${\mathcal {X}}_x$.

Definition 3.2

Equip each connected component of $C({\mathbb {R}})$ with an orientation. For every pair of points $x,y\in C({\mathbb {R}})$ lying on the same connected component $V\subseteq C({\mathbb {R}})$ the interval ]xy[ is the set of points of V which lies to the right of x and lies to the left of y with respect to the chosen orientation. We also set [xy] to be the union of ]xy[ and the points x, y. Now let $P\in {\mathcal {X}}_{x,sm}({\mathbb {R}})$, $Q\in {\mathcal {X}}_{y,sm}({\mathbb {R}})$ and let ${\mathcal {X}}'_{x,y}\subseteq {\mathcal {X}}_{sm}$ denote the open subscheme which is the complement of the union of ${\mathcal {X}}_{x,sm}$ and ${\mathcal {X}}_{y,sm}$. We say that a connected component V of ${\mathcal {X}}'_{x,y}({\mathbb {R}})$ touches P on the right and Q on the left if the image of V under p is ]xy[, and both P and Q lie in the closure of V.

Definition 3.3

For every map $f:A\rightarrow B$ of schemes over ${\mathbb {R}}$ let $f({\mathbb {R}}):A({\mathbb {R}})\rightarrow B({\mathbb {R}})$ denote the map induced by f on ${\mathbb {R}}$-valued points. Let $\sigma$ be a set-theoretical section of the map $p({\mathbb {R}}):{\mathcal {X}}({\mathbb {R}})\rightarrow C({\mathbb {R}})$. We say that the map $\sigma$ is weakly continuous if for every $x\in C({\mathbb {R}})$ the point $\sigma (x)$ lies in ${\mathcal {X}}_{x,sm}({\mathbb {R}})$ and if the following holds: let x and y be two different ${\mathbb {R}}$-valued points of C lying in the same connected component of $C(\mathbb R)$. Then there is a semialgebraic connected component V of ${\mathcal {X}}'_{x,y}({\mathbb {R}})$ touching $\sigma (x)$ on the left and $\sigma (y)$ on the right such that for every $z\in ]xy[$ the point $\sigma (z)$ lies in V.

This terminology is justified by the fact that every continuous section $C({\mathbb {R}})\rightarrow {\mathcal {X}}_{sm}({\mathbb {R}})$ is indeed weakly continuous. The converse is not true, because we have the following

Proposition 3.4

Let $\sigma$ be a weakly continuous section of $p({\mathbb {R}})$. Let $\sigma '$ be another set-theoretical section of $p({\mathbb {R}})$ such that for every $x\in C({\mathbb {R}})$ the points $\sigma (x)$ and $\sigma '(x)$ are in the same connected component of $\mathcal X_{x,sm}({\mathbb {R}})$. Then $\sigma '$ is also weakly continuous.

Proof

This is Proposition 3.3 of [7] on page 82. $\square$

Notation 3.5

By the valuative criterion of properness there is a bijection between the set of sections of the fibre product ${\mathcal {X}}\times _C{\text {Spec}}({\mathcal {O}}_x)\rightarrow {\text {Spec}}({\mathcal {O}}_x)$ and the set of ${\mathbb {R}}(C)_x$-valued points of X. Given an

$$\begin{aligned} {\underline{M}}=\prod _{x\in C({\mathbb {R}})}M_x\in \prod _{x\in C({\mathbb {R}})}X({\mathbb {R}}(C)_x) \end{aligned}$$

let $m_x$ be the special fibre of the section ${\text {Spec}}({\mathcal {O}}_x)\rightarrow \mathcal X\times _C{\text {Spec}}({\mathcal {O}}_x)$ associated to $M_x$ for every $x\in C({\mathbb {R}})$. Since ${\mathcal {X}}$ is regular, the point $m_x$ lies in ${\mathcal {X}}_{x,sm}({\mathbb {R}})$. Whenever convenient, we will denote the map $x\mapsto m_x$ on $C({\mathbb {R}})$ by $\sigma ({\underline{M}})$.

The topological reformulation of the CT obstruction due to Ducros is the following

Theorem 3.6

(Ducros) The following are equivalent : 

• (i) we have ${\underline{M}}\in X({\mathbb {A}}_C)^{CT},$

• (ii) the map $x\mapsto m_x$ on $C({\mathbb {R}})$ is weakly continuous.

Proof

This is Théorème 3.5 of [7] on page 83. $\square$

Proposition 3.7

Let $\sigma$ be a weakly continuous section of $p({\mathbb {R}})$. Then there is a continuous semi-algebraic section $\sigma '$ of $p({\mathbb {R}})$ such that for every $x\in C({\mathbb {R}})$ the points $\sigma (x)$ and $\sigma '(x)$ are in the same connected component of ${\mathcal {X}}_{x,sm}({\mathbb {R}})$.

Remark 3.8

Note that Theorem 1.3 is an immediate corollary of this proposition.

Proof of Proposition 3.7

This is essentially Proposition 4.1 of [7] on page 85, but there it is stated in a weaker form. However the proof actually shows the stronger form above. We will give an even stronger version incorporating interpolation, see Proposition 3.19 below, using essentially the same methods. $\square$

Definition 3.9

Let B be a smooth variety over ${\mathbb {R}}$. For every closed point x of B let ${\mathcal {I}}(x)\subset {\mathcal {O}}_B$ be the corresponding ideal sheaf. For every effective zero cycle $S=\sum n_xx$ on B the zero-dimensional closed subscheme of B defined by S is the zero scheme of the ideal sheaf ${\mathcal {I}}_S=\prod {\mathcal {I}}(x)^{n_x}\subset {\mathcal {O}}_B$. By slight abuse of notation we will let the symbol S denote this closed subscheme, too. When B is a curve this construction furnishes a bijective correspondence between the set of zero-dimensional closed subschemes of B whose closed points are all real and the set of effective zero cycles on B which are supported on $B({\mathbb {R}})$. In this case we will identify these two sets in all that follow.

Definition 3.10

Let $f:A\rightarrow B$ be a map of smooth varieties over ${\mathbb {R}}$. An interpolation condition (for the map f) is a section $\phi :S\rightarrow A\times _BS$ (of the base change of f), defined over ${\mathbb {R}}$, where S is a zero-dimensional closed subscheme of B defined by an effective zero cycle supported on $B({\mathbb {R}})$. We say that $\phi$ is of order $\le k$, where k is a positive integer, if the coefficient of every point in the effective zero cycle defining S is at most $k+1$. We say that an interpolation condition ${\widetilde{\phi }}:{\widetilde{S}}\rightarrow A\times _B{\widetilde{S}}$ subsumes the interpolation condition $\phi :S\rightarrow A\times _BS$ if S is a closed subscheme of ${\widetilde{S}}$ and the pull-back of ${\widetilde{\phi }}$ with respect to the closed immersion $S\hookrightarrow {\widetilde{S}}$ is $\phi$. We say that a set-theoretical section $\sigma$ of the map $f({\mathbb {R}}):A(\mathbb R)\rightarrow B({\mathbb {R}})$ and an interpolation condition $\phi :S\rightarrow A\times _BS$ are weakly compatible with each other if for every $x\in S({\mathbb {R}})$ we have $\sigma (x)=\phi (x)$. We say that a section $\sigma :B\rightarrow A$ of f and an interpolation condition $\phi :S\rightarrow A\times _BS$ are compatible with each other if the composition of the closed immersion $S\hookrightarrow B$ and $\sigma$ is $\phi$.

Definition 3.11

As usual, we say that a function $f:M\rightarrow N$ between two smooth manifolds is a ${\mathcal {C}}^k$-map if it is k-times continuously differentiable. For every M, N as above and for every point p of M let ${\mathcal {C}}_{p}^{k}(M,N)$ denote the set of ${\mathcal {C}}^k$-maps $f:U\rightarrow N$ where U is some open neighbourhood of p in M. For every $p\in {\mathbb {R}}$ and for every $f\in {\mathcal {C}}_{p}^{k}({\mathbb {R}},{\mathbb {R}})$ the k-jet of f at the point $x_0$ is defined to be the polynomial

$$\begin{aligned} (J_p^kf)(z)=f(p)+f'(p)z+\cdots + \frac{f^{(k)}(p)}{k!}z^k. \end{aligned}$$

We say that two maps $f,g\in {\mathcal {C}}_{p}^{k}(M,N)$ are k-equivalent at p if $f(p)=g(p)$ and for every pair of maps $\gamma \in {\mathcal {C}}_{0}^{k}({\mathbb {R}},M)$ and $\phi \in \mathcal C_{f(p)}^{k}(M,{\mathbb {R}})$ such that $\gamma (0)=p$, we have $J_{0}^{k}(\phi \circ f\circ \gamma )=J_{0}^{k}(\phi \circ g\circ \gamma )$.

Definition 3.12

Now let $f:A\rightarrow B$ be as in Definition 3.10. Then $f({\mathbb {R}}):A({\mathbb {R}})\rightarrow B({\mathbb {R}})$ is a morphism of Nash manifolds. Now let $S=\sum n_xx$ be an effective zero cycle on B supported on $B({\mathbb {R}})$, and let $\phi :S\rightarrow A\times _BS$ be an interpolation condition of order $\le k$. Let $U\subset B$ be an open, affine subvariety such that there is a section ${\widetilde{\phi }}:U\rightarrow A\times _BU$ compatible with $\phi$. We say that a ${\mathcal {C}}^k$-section $\sigma :B({\mathbb {R}})\rightarrow A({\mathbb {R}})$ of $f({\mathbb {R}})$ and $\phi$ are compatible with each other if $\sigma$ and ${\widetilde{\phi }}$ are $(n_x-1)$-equivalent at x for every $x\in B({\mathbb {R}})$ in the support of S. Since the notion introduced in Definition 3.11 is indeed an equivalence relation, this notion of compatibility is independent of the choice of ${\widetilde{\phi }}$, and so it is well-defined.

Remarks 3.13

A section $\sigma :B\rightarrow A$ and an interpolation condition $\phi :S\rightarrow A\times _BS$ are compatible in the sense of Definition 3.10 if and only if the underlying Nash section $\sigma ({\mathbb {R}}):B({\mathbb {R}})\rightarrow A({\mathbb {R}})$ and $\phi$ are compatible in the sense of Definition 3.12. Therefore the latter is a generalisation of the former. If a section $\sigma :B\rightarrow A$ and an interpolation condition $\phi :S\rightarrow A\times _BS$ are compatible then $\sigma ({\mathbb {R}}):B({\mathbb {R}})\rightarrow A({\mathbb {R}})$ and $\phi$ are weakly compatible. In particular this terminology is justified. Finally note that when $A=B\times D$ for some smooth variety D over ${\mathbb {R}}$ and $f:A\rightarrow B$ is the projection onto the first factor, sections of f are in a natural bijection correspondence with morphisms $A\rightarrow D$ of schemes over ${\mathbb {R}}$. Similarly an interpolation condition $\phi :S\rightarrow A\times _BS$ is the same data as a morphism $S\rightarrow D$ of schemes over ${\mathbb {R}}$, while ${\mathcal {C}}^k$-sections of $f({\mathbb {R}})$ can be identified with ${\mathcal {C}}^k$-maps $A({\mathbb {R}})\rightarrow D({\mathbb {R}})$. Therefore we will freely apply the concepts of Definitions 3.10 and 3.12 to such functions in all that follows.

Proposition 3.14

Let $\sigma$ be a weakly continuous section of $p({\mathbb {R}})$ and let $\phi : S\rightarrow {\mathcal {X}}\times _CS$ be an interpolation condition of order $\le k$ weakly compatible with $\sigma$. Then there is a $\mathcal C^k$-section $\sigma '$ of $p({\mathbb {R}})$ such that for every $x\in C({\mathbb {R}})$ the points $\sigma (x)$ and $\sigma '(x)$ are in the same connected component of ${\mathcal {X}}_{x,sm}({\mathbb {R}})$ and $\sigma '$ is compatible with $\phi$.

Proof

Note that the fibre of $p({\mathbb {R}})$ over x is a Nash manifold for all but finitely many $x\in C({\mathbb {R}})$. So by the Nash version of the stratification theorem (see Theorem A of [6] on page 349) there is a finite subset ${\mathcal {P}}$ of $C({\mathbb {R}})$, and for every semi-algebraic connected component U of $C({\mathbb {R}})-{\mathcal {P}}$ a Nash manifold $F_U$ such that $p({\mathbb {R}})^{-1}(U)$ is Nash-isomorphic to $F_U\times U$ and the restriction of $p({\mathbb {R}})$ onto $p({\mathbb {R}})^{-1}(U)$ is, modulo the given isomorphism, is the projection onto the second coordinate. By the nature of our construction for every point P in $C({\mathbb {R}})-{\mathcal {P}}$ the fibre ${\mathcal {X}}_P$ is smooth. By adding finitely many points to the set ${\mathcal {P}}$, if it is necessary, we may assume that ${\mathcal {P}}$ has at least two points in each semi-algebraic connected component of $C({\mathbb {R}})$. Similarly we may assume that ${\mathcal {P}}$ contains every closed point of S without loss of generality.

Set ${\widetilde{S}}=(k+1)\sum _{P\in {\mathcal {P}}}P$ and let the same symbol denote the unique closed subscheme defined by this zero cycle. Since there is an interpolation condition ${\widetilde{\phi }}:{\widetilde{S}}\rightarrow {\mathcal {X}}\times _C{\widetilde{S}}$ which subsumes $\phi$, we may assume without loss of generality that the zero cycle defined by S is indeed $(k+1)\sum _{P\in \mathcal P}P$. Write S as a coproduct:

$$\begin{aligned} S=\coprod _{P\in {\mathcal {P}}}S_P, \end{aligned}$$

where $S_P$ is a closed subscheme of S supported on P for each $P\in {\mathcal {P}}$ (possibly empty). For every such P let $\phi _P:S_P\rightarrow {\mathcal {X}}\times _CS_P$ be the interpolation condition which is the pull-back of $\phi$ with respect to the closed imbedding $S_P\hookrightarrow S$. Let P, Q be a pair of consecutive points of ${\mathcal {P}}$. Since $\sigma (P)$ and $\sigma (Q)$ lie in the smooth locus of p, by the implicit function theorem there are two points $P',Q'$ in the open interval ]PQ[ such that $P'$ lies before $Q'$, and $p({\mathbb {R}})$ has a ${\mathcal {C}}^{\infty }$-section $\sigma _P$ (resp. $\sigma _Q$) defined over some open neighbourhood of $[PP']$ (resp. of $[Q'Q]$) such that $\sigma _P(P)=\sigma (P)$ (resp. $\sigma _Q(Q)=\sigma (Q)$), and $\sigma _P$ is compatible with $\phi _P$ (resp. $\sigma _Q$ is compatible with $\phi _Q$). Since $\sigma$ is weakly continuous, there is a semi-algebraic connected component $V_{P,Q}$ of $p({\mathbb {R}})^{-1}(]PQ[)$ which touches P on the right and Q on the left. Clearly $\sigma _P(P')$ and $\sigma _Q(Q')$ lie in $V_{P,Q}$.

On the other hand, because of the way the set ${\mathcal {P}}$ was constructed, the restriction of $p({\mathbb {R}})$ onto $p(\mathbb R)^{-1}(]PQ[)$ is a projection, up to a Nash isomorphism. Therefore there is a ${\mathcal {C}}^{\infty }$-section $\sigma _{P,Q}$ of $p({\mathbb {R}})$ defined over some open neighbourhood of $[P'Q']$ such that $\sigma _{P,Q}$ and $\sigma _P$ are k-equivalent at $P'$, and similarly $\sigma _{P,Q}$ and $\sigma _Q$ are k-equivalent at $Q'$. Therefore the concatenation of $\sigma _P,\sigma _{P,Q}$ and $\sigma _Q$ (restricted to $[PP'],[P'Q']$ and $[Q'Q]$, respectively) is a ${\mathcal {C}}^k$-section $\widehat{\sigma }_{P,Q}$ of $p(\mathbb R)$ defined over [PQ] such that $\widehat{\sigma }_{P,Q}(P)=\sigma (P)$ and $\widehat{\sigma }_{P,Q}(Q)=\sigma (Q)$.

Now let P, Q, R be three consecutive points of ${\mathcal {P}}$ (where $P=R$ is allowed). Since both $\widehat{\sigma }_{P,Q}$ and $\widehat{\sigma }_{Q,R}$ have extensions to an open neighbourhood of their definitions which are compatible with $\phi _Q$, we get that their concatenation is ${\mathcal {C}}^k$ at Q. We get that the concatenation of the different sections $\widehat{\sigma }_{P,Q}$ for all couples P, Q of consecutive points of ${\mathcal {P}}$ is a ${\mathcal {C}}^k$-section $\widehat{\sigma }$ of $p({\mathbb {R}})$ defined over all of $C({\mathbb {R}})$ such that for each point $x\in C(\mathbb R)$ the point $\widehat{\sigma }(x)$ lies in the same connected component of ${\mathcal {X}}_{x,sm}({\mathbb {R}})$ as $\sigma (x)$, and $\widehat{\sigma }$ is compatible with $\phi$. $\square$

We will need a variant of the claim above with Nash sections, since for technical reasons it will be more convenient to work with the latter in the next section. In order to do so we will show two interpolation lemmas first.

Lemma 3.15

Let $V\subseteq {\mathbb {A}}^n_{{\mathbb {R}}}$ be a non-singular closed affine subvariety such that $V({\mathbb {R}})$ is compact. Let $\phi :Y\rightarrow \mathbb A^m_{{\mathbb {R}}}$ be an interpolation condition for some subscheme $Y\subseteq V$. Then every ${\mathcal {C}}^{\infty }$-function $g:V({\mathbb {R}})\rightarrow {\mathbb {R}}^m$ compatible with $\phi$ can be approximated in the ${\mathcal {C}}^{\infty }$-topology by regular functions $V\rightarrow {\mathbb {A}}_{{\mathbb {R}}}^m$ compatible with $\phi$.

Proof

Let $\pi _j:{\mathbb {A}}^m_{{\mathbb {R}}}\rightarrow {\mathbb {A}}^1_{{\mathbb {R}}}$ be the projection onto the j-th coordinate, where $j=1,2,\ldots ,m$. It will be sufficient to show the claim for $\phi _j=\pi _j\circ \phi$ and $g_j=\pi _j\circ g$ for each j. In other words we may assume that $m=1$ without loss of generality. Since V is affine there is a regular map $\psi :V\rightarrow {\mathbb {A}}^1_{{\mathbb {R}}}$ compatible with $\phi$. By replacing g with $g-\psi$ we may assume without loss of generality that $\phi$ is the zero map. In this form the claim is a mild variant of Lemma 12.5.5 of [2] on page 321. We include the proof for the reader’s convenience. Let $h_1,h_2,\ldots ,h_n$ be the generators of the defining ideal of Y. Since V is non-singular, we can represent the germ of g at a point $x\in V$ in the form $g_x=\lambda _{1,x}h_1+\cdots +\lambda _{n,x}h_n$, where the $\lambda _{i,x}$ are the germs of ${\mathcal {C}}^{\infty }$-functions at x. Using a partition of unity and the compactness of $V(\mathbb R)$, this allows us to represent g globally as $g=\lambda _1h_1+\cdots +\lambda _nh_n$, where $\lambda _i\in \mathcal C^{\infty }(V({\mathbb {R}}))$. Then it suffices to apply Nachbin’s version of the Stone–Weierstrass theorem to the functions $\lambda _i$ (see [13]). $\square$

Definition 3.16

Let V be a nonsingular variety over ${\mathbb {R}}$, and let $W\subset {\mathbb {A}}^m({\mathbb {R}})={\mathbb {R}}^m$ be a Nash manifold. An interpolation condition $\phi :S\rightarrow W$ for some subscheme $S\subset V$ of the type considered above is an interpolation condition $\phi :S\rightarrow {\mathbb {A}}^m$ which is compatible with some smooth function $U\rightarrow W$, where U is a neighborhood of the support of S in $V({\mathbb {R}})$.

Lemma 3.17

Let $V\subseteq {\mathbb {A}}^n_{{\mathbb {R}}}$ be a non-singular closed affine subvariety such that $V({\mathbb {R}})$ is compact,  and let $W\subset {\mathbb {R}}^m$ be a compact Nash submanifold. Let $\phi :Y\rightarrow W$ be an interpolation condition of order $\le k$ for some subscheme $Y\subseteq V$. Then every ${\mathcal {C}}^{\infty }$-function $g:V({\mathbb {R}})\rightarrow W$ compatible with $\phi$ can be approximated in the $\mathcal C^{\infty }$-topology by Nash functions $V({\mathbb {R}})\rightarrow W$ compatible with $\phi$.

Proof

(Compare with Corollary 8.9.7 of [2] on page 200.) By the Nash version of the tubular neighbourhood theorem (see Corollary 8.9.5 of [2] on page 199) there is an open semi-algebraic neighbourhood U of W in ${\mathbb {R}}^m$ and a Nash retraction $\rho :U\rightarrow W$. Since for ${\mathcal {C}}^{\infty }$-functions $h:V({\mathbb {R}})\rightarrow {\mathbb {R}}^m$ the condition of having image in U is open for the $\mathcal C^1$-topology, by Lemma 3.15 above any given function $g:V({\mathbb {R}})\rightarrow W$ compatible with $\phi$ can be approximated in the ${\mathcal {C}}^{\infty }$-topology by a sequence $g_n$ of Nash functions $V({\mathbb {R}})\rightarrow U$ compatible with $\iota \circ \phi$, where $\iota :W\rightarrow U$ is the inclusion map. The compositions $\rho \circ g_n$ are Nash functions $V({\mathbb {R}})\rightarrow W$ compatible with $\phi$. Since W is compact, each derivative of $\rho$ is bounded in some fixed $\epsilon$-neighbourhood of W, and hence sequence $\rho \circ g_n$ approximates g in the $\mathcal C^{\infty }$-topology. $\square$

Remark 3.18

Note that for every pair of conjugate point $P,{\overline{P}}\in C({\mathbb {C}})-C({\mathbb {R}})$ the complement $C'=C-P-{\overline{P}}$ is an affine curve, and $C'({\mathbb {R}})=C({\mathbb {R}})$, so this set is compact. Therefore we may apply Lemmas 3.15 and 3.17 to $C'$. In particular the conclusion of Lemma 3.17 holds for C, too.

Proposition 3.19

Let $\sigma$ be a weakly continuous section of $p({\mathbb {R}})$ and let $\phi : S\rightarrow {\mathcal {X}}\times _CS$ be an interpolation condition weakly compatible with $\sigma$. Then there is a Nash section $\sigma '$ of $p({\mathbb {R}})$ such that for every $x\in C({\mathbb {R}})$ the points $\sigma (x)$ and $\sigma '(x)$ are in the same connected component of ${\mathcal {X}}_{x,sm}({\mathbb {R}})$ and $\sigma '$ is compatible with $\phi$.

Proof

We may assume that $\phi$ is an interpolation condition of order $\le k$, where k is a positive integer. By Lemma 3.14 there is a ${\mathcal {C}}^k$-section ${\widetilde{\sigma }}$ of $p({\mathbb {R}})$ such that for every $x\in C({\mathbb {R}})$ the points $\sigma (x)$ and ${\widetilde{\sigma }}(x)$ are in the same connected component of ${\mathcal {X}}_{x,sm}({\mathbb {R}})$ and ${\widetilde{\sigma }}$ is compatible with $\phi$. By the usual approximation theorems in theory of smooth manifolds the section ${\widetilde{\sigma }}$ can be arbitrarily well approximated by ${\mathcal {C}}^{\infty }$-maps $s:C({\mathbb {R}})\rightarrow {\mathcal {X}}_{sm}({\mathbb {R}})$ which are compatible with $\phi$.

Note that if such an s is sufficiently close to ${\widetilde{\sigma }}$ in the ${\mathcal {C}}^1$-topology then $p(\mathbb R)\circ s$ is a diffeomorphism and its inverse is very close to the identity map of $C({\mathbb {R}})$. Therefore $s\circ (p({\mathbb {R}})\circ s)^{-1}$ is a ${\mathcal {C}}^{\infty }$-section of $p({\mathbb {R}})$ which can be arbitrarily close to ${\widetilde{\sigma }}$. Also note that $(p({\mathbb {R}})\circ s)^{-1}$, if it exists, is Nash if s is. For any smooth section $\sigma ':C({\mathbb {R}})\rightarrow {\mathcal {X}}_{sm}(\mathbb R)$ sufficiently close to ${\widetilde{\sigma }}$ in the $\mathcal C^0$-topology and for every $x\in C({\mathbb {R}})$ the points ${\widetilde{\sigma }}(x)$ and $\sigma '(x)$ are in the same connected component of ${\mathcal {X}}_{x,sm}({\mathbb {R}})$, and hence the same holds for $\sigma (x)$ and $\sigma '(x)$. The claim now follows at once from Lemma 3.17, as we explained in Remark 3.18. $\square$

We finish this section with a convenient condition for weak continuity.

Definition 3.20

We say that a set-theoretical section $\sigma$ of the map $p({\mathbb {R}})$ is mildly continuous at $x\in C({\mathbb {R}})$ if $\sigma (x)$ lies in ${\mathcal {X}}_{x,sm}({\mathbb {R}})$ and the intersection of the closure of the image of $\sigma$ with ${\mathcal {X}}_x({\mathbb {R}})$ lies in the connected component of ${\mathcal {X}}_{x,sm}({\mathbb {R}})$ containing $\sigma (x)$. Note that $\sigma$ is mildly continuous at $x\in C({\mathbb {R}})$ if it is continuous there, since in the latter case the image of $\sigma$ is closed, and its intersection with $\mathcal X_x({\mathbb {R}})$ is $\sigma (x)$. Therefore the terminology is justified.

Proposition 3.21

Let $\sigma$ be a semi-algebraic section of $p({\mathbb {R}})$ which is mildly continuous at every $x\in C({\mathbb {R}})$. Then $\sigma$ is weakly continuous.

Proof

Let x and y be two arbitrary different ${\mathbb {R}}$-valued points of C lying in the same connected component of $C({\mathbb {R}})$. We will need the following $\square$

Lemma 3.22

Let V be a semi-algebraic connected component of ${\mathcal {X}}'_{x,y}({\mathbb {R}})$ and let ${\overline{V}}$ be its closure in ${\mathcal {X}}({\mathbb {R}})$. Then the intersection ${\overline{V}}\cap {\mathcal {X}}_{x,sm}({\mathbb {R}})$ (respectively ${\overline{V}}\cap {\mathcal {X}}_{y,sm}({\mathbb {R}}))$ is the union of connected components of ${\mathcal {X}}_{x,sm}({\mathbb {R}})$ (respectively of ${\mathcal {X}}_{y,sm}({\mathbb {R}})).$

Proof

We may assume without loss of generality that V lies in $p({\mathbb {R}})^{-1}(]xy[)$; otherwise we only need to reverse the roles of x and y. It is also enough to prove the claim for the fibre above x; the proof for the fibre above y is similar. Since ${\overline{V}}$ is closed, the intersection ${\overline{V}}\cap {\mathcal {X}}_{x,sm}({\mathbb {R}})$ is closed in ${\mathcal {X}}_{x,sm}({\mathbb {R}})$. Therefore it will be enough to show that it is also open in ${\mathcal {X}}_{x,sm}({\mathbb {R}})$. Let n be the relative dimension of ${\mathcal {X}}$ over C and let $z\in {\overline{V}}\cap {\mathcal {X}}_{x,sm}({\mathbb {R}})$ be arbitrary. By the implicit function theorem there is a small connected open neighbourhood $I\subset C({\mathbb {R}})$ of x, a connected open set $O\subset {\mathbb {R}}^n$, an open neighbourhood $U\subset \mathcal X_{sm}({\mathbb {R}})$ of z, and a diffeomorphism $\phi :I\times O\rightarrow U$ such that the composition $p({\mathbb {R}})\circ \phi$ is the projection $I\times O\rightarrow I$ onto the first factor.

Let $J\subset I$ be the set of points in I lying to the right of x. By shrinking I, if it is necessary, we may assume that $J=I\cap ]xy[$. By assumption

$$\begin{aligned} \phi (J\times O)\cap V=\phi (I\times O)\cap p({\mathbb {R}})^{-1}(]xy[)\cap V= \phi (I\times O)\cap V=U\cap V \end{aligned}$$

is non-empty. Since both J and O are connected, the product $J\times O$ is also connected, and hence $\phi (J\times O)$ is connected, too. Moreover $\phi (J\times O)$ lies in $p({\mathbb {R}})^{-1}(]xy[)$, so the connected component V must contain it. Therefore ${\overline{V}}$ contains the closure of $\phi (J\times O)$, which contains $\phi (\{x\}\times O)$. The latter is an open neighbourhood of z in ${\mathcal {X}}_{x,sm}({\mathbb {R}})$. $\square$

Since $\sigma$ is semi-algebraic, it is continuous at all but finitely many points of $C({\mathbb {R}})$. Therefore there is a finite sequence of points $z_1,z_2,\ldots ,z_n\in [xy]$ such that $\sigma$ is continuous at every $z\in ]xy[$ not on this list. We may even assume that $z_1=x$, $z_n=y$, and $z_i$ lies to the left of $z_j$ for every pair of indices $i<j$. For every $i=1,2,\ldots ,n$ let $E_i$ denote the connected component of ${\mathcal {X}}_{z_i,sm}({\mathbb {R}})$ containing $\sigma (z_i)$ and for every $i=1,2,\ldots ,n-1$ let $J_i$ denote the closure of the image of $]z_iz_{i+1}[$ with respect to $\sigma$.

Since the restriction of $\sigma$ onto $]z_iz_{i+1}[$ is continuous for every index $i<n$, the image of $]z_iz_{i+1}[$ with respect to $\sigma$ is connected, and hence its closure $J_i$ is connected, too. Since $J_i$ is also compact, its image under $p({\mathbb {R}})$ is the closure $[z_iz_{i+1}]$ of $]z_iz_{i+1}[$. Therefore $J_i\cap X_{z_i}({\mathbb {R}})$ is non-empty. By our assumptions the latter intersection lies in $E_i$, hence $J_i\cap E_i$ is non-empty, too. A similar argument shows that $J_i\cap E_{i+1}$ is also non-empty. Therefore the set

$$\begin{aligned} L=p({\mathbb {R}})^{-1}(]xy[)\cap \left( \bigcup _{i=1}^{n-1}J_i\cup \bigcup _{i=2}^{n-1}E_i\right) \end{aligned}$$

is connected. Let V be the unique semi-algebraic connected component of ${\mathcal {X}}'_{x,y}({\mathbb {R}})$ containing L and let ${\overline{V}}$ be its closure in $\mathcal X({\mathbb {R}})$. Since

$$\begin{aligned} {\overline{V}}\cap E_{z_1}\supseteq J_1\cap E_{z_1}\ne \emptyset , \end{aligned}$$

by Lemma 3.22 the intersection ${\overline{V}}\cap \mathcal X_{x,sm}({\mathbb {R}})$ contains $E_{z_1}$, and hence $\sigma (x)$. We may argue similarly to deduce that ${\overline{V}}\cap \mathcal X_{y,sm}({\mathbb {R}})$ contains $\sigma (y)$. In other words V touches $\sigma (x)$ on the right and $\sigma (y)$ on the left. Since x and y are arbitrary, we get that $\sigma$ is weakly continuous. $\square$

4 The Stone–Weierstrass approximation theorem with interpolation

Definition 4.1

Let $\pi :C\times {\mathbb {P}}^1({\mathbb {R}})= C({\mathbb {R}})\times {\mathbb {P}}^1({\mathbb {R}})\rightarrow C({\mathbb {R}})$ denote the projection onto the first factor. Let $R\subseteq C(\mathbb R)\times {\mathbb {P}}^1({\mathbb {R}})$ be a semi-algebraic subset. We say that R is admissible if it is the union of an open semi-algebraic set and finitely many points. The kissing points of an admissible semi-algebraic set R as above are all points of R which are not in the interior of R. We say that R does not have topological obstruction if there is a Nash section $s:C({\mathbb {R}})\rightarrow C\times {\mathbb {P}}^1({\mathbb {R}})$ whose image lies in R.

The key result we need is an analogue of Conjecture 9.1 in [11] which we will formulate next. It is essentially a refined version of the classical the Stone–Weierstrass approximation theorem with interpolation conditions.

Theorem 4.2

Let $R\subseteq C({\mathbb {R}}) \times {\mathbb {P}}^1({\mathbb {R}})$ be an admissible semi-algebraic subset,  and for some closed subscheme $S\subset C$ let $\phi :S\rightarrow ({\mathbb {P}}^1\times C)\times _CS={\mathbb {P}}^1\times S$ be an interpolation condition such that there is a Nash section $s:C({\mathbb {R}})\rightarrow C\times {\mathbb {P}}^1({\mathbb {R}})$ compatible with $\phi$ and whose image lies in R. Then there is a regular section $f:C\rightarrow C\times {\mathbb {P}}^1_{{\mathbb {R}}}$ of the first projection compatible with $\phi$ such that $f(C({\mathbb {R}}))$ lies in R.

In particular we get that if $R\subseteq C({\mathbb {R}}) \times \mathbb P^1({\mathbb {R}})$ is an admissible semi-algebraic subset which does not have topological obstruction then there is a morphism $f:C\rightarrow {\mathbb {P}}^1_{{\mathbb {R}}}$ of schemes over ${\mathbb {R}}$ such that $f(C({\mathbb {R}}))$ lies in R. We are going to prove the theorem above via a sequence of lemmas.

Lemma 4.3

For some closed subscheme $Z\subset C$ let $h:Z\rightarrow {\mathbb {P}}^1_{{\mathbb {R}}}$ be an interpolation condition. Then there is a morphism $f:C\rightarrow {\mathbb {P}}^1_{{\mathbb {R}}}$ of schemes over ${\mathbb {R}}$ such that f is compatible with h and f has no poles on $C({\mathbb {R}})$ outside of Z.

Proof

First assume that h is actually a map $h:Z\rightarrow {\mathbb {A}}^1_{{\mathbb {R}}}$. By our usual abuse of notation let Z also denote the effective divisor defining this closed subscheme and let d denote its degree. Choose an effective real divisor D on C which is supported outside of $C({\mathbb {R}})$ and whose degree is bigger than $2g+d$ where g is the genus of C. Then by the Riemann–Roch theorem we have $\dim H^1(C,{\mathcal {O}}_C(D))=0$, so the pull-back induces a surjection:

$$\begin{aligned} H^0(C,{\mathcal {O}}_C(D+Z))\longrightarrow H^0(Z,{\mathcal {O}}_C(D+Z)|_Z). \end{aligned}$$

Therefore there is a real rational function f on C compatible with h whose polar divisor is a sub-divisor of D.

Now consider the general case. We may assume without loss of generality that Z is non-empty. Since Z is a finite scheme over ${\mathbb {R}}$ there is an interpolation condition $\widetilde{h}:Z\rightarrow {\mathbb {A}}^2_{{\mathbb {R}}}-\{0\}$ such that the composition of ${\widetilde{h}}$ and the projection ${\mathbb {A}}^2_{\mathbb R}-\{0\}\rightarrow {\mathbb {P}}^1_{{\mathbb {R}}}$ given by the rule $(a,b)\mapsto (a:b)$ is h. Let ${\widetilde{h}}_1:Z\rightarrow \mathbb A^1_{{\mathbb {R}}}$ and ${\widetilde{h}}_2:Z\rightarrow {\mathbb {A}}^1_{{\mathbb {R}}}$ denote the first, respectively the second, coordinate of $\widetilde{h}$. By the above there is a rational function $f_2$ on C compatible with ${\widetilde{h}}_2$. Let O be the zero divisor of $f_2$. We can write it as $O=O_1+O_2+O_3$, where $O_1$ is supported on the support of Z, the divisor $O_2$ is supported on the complement of the support of Z in $C({\mathbb {R}})$, and $O_3$ is supported outside of $C({\mathbb {R}})$. Now let

$$\begin{aligned} h_1:Z\coprod 2O_2\rightarrow {\mathbb {A}}^1_{{\mathbb {R}}} \end{aligned}$$

be the unique map such that $h_1|_Z$ is ${\widetilde{h}}_1$, and $h_1|_{2O_2}$ is $f_2|_{2O_2}$. By the above there is a rational function $f_1$ on C compatible with $h_1$ whose polar divisor is supported outside of $C({\mathbb {R}})$. Since Z is non-empty, the functions $f_1,f_2$ are not both identically zero, and hence there is a non-empty Zariski-open $U\subseteq C$ such that the rule $z\mapsto (f_1,f_2)$ furnishes a map ${\widetilde{f}}:U\rightarrow \mathbb A^2_{{\mathbb {R}}}-\{0\}$. The composition of ${\widetilde{f}}$ and the projection ${\mathbb {A}}^2_{{\mathbb {R}}}-\{0\}\rightarrow {\mathbb {P}}^1_{{\mathbb {R}}}$ extends uniquely to a morphism $f:C\rightarrow {\mathbb {P}}^1_{{\mathbb {R}}}$ by the valuative criterion of properness. By our choice of $f_1$ and $f_2$ it is clear that f is compatible with h. It is also clear that on $C({\mathbb {R}})-Z$ the rational function f can only have poles on the support of $O_2$. However the zero divisors of $f_1$ and $f_2$ are equal on the support of $O_2$, in fact it is $O_2$ in both cases, so f does not have a pole on the support of $O_2$ either. $\square$

Lemma 4.4

Let A and B be smooth irreducible algebraic varieties over ${\mathbb {R}},$ and let $f:A({\mathbb {R}})\rightarrow B({\mathbb {R}})$ be a Nash map,  and let $Z\subset B$ be a closed subscheme of positive codimension. Then the intersection of $f^{-1}(Z)({\mathbb {R}})$ with each connected component of $A({\mathbb {R}})$ is either a semi-algebraic subset of positive codimension,  or the connected component itself.

Proof

Let $D\subset A({\mathbb {R}})$ be a semi-algebraic connected component, and let $D_1\subset D$ be the collection of all points $x\in D$ such that x has a semi-algebraic open neighbourhood $U\subset D$ such that the semi-algebraic set $U\cap f^{-1}(Z)$ has dimension strictly less than D. Clearly $D_1$ is open (in the usual semi-algebraic topology). Set $D_2=f^{-1}(Z)-D_1$; since $f^{-1}(Z)$ is closed, the set $D_2$ is closed, too. It will be sufficient to show that $D_2$ is also open. Clearly we only need to verify the latter Zariski-locally, that is, we may assume without loss of generality that A is affine.

Now let $x\in D_2$ be arbitrary; then there are a Zariski-open affine neighbourhood $V\subset B$ of f(x) and a finite set of regular maps $g_1,g_2,\ldots ,g_n$ from V to ${\mathbb {A}}^1_{\mathbb R}$ such that $Z\cap V$ is the common zero locus of $g_1,g_2,\ldots ,g_n$. Let $U\subset f^{-1}(V({\mathbb {R}}))$ be a connected semi-algebraic open neighbourhood of x in D. Then $g_i({\mathbb {R}})\circ f: U\rightarrow {\mathbb {R}}$ is a Nash function for each index i, so by Proposition 8.1.10 of [2] on page 166 its zero set $Z_i\subset U$ has either dimension strictly less than U, or this set $Z_i$ is equal to U. Since $Z_i$ contains $f^{-1}(Z)({\mathbb {R}})\cap U$ the former is not possible, so $Z_i=U$ for every index i. This implies that $f^{-1}(Z)({\mathbb {R}})\cap U$ also equal to U. $\square$

Lemma 4.5

In the proof of Theorem 4.2 we may assume that $R\subseteq C({\mathbb {R}}) \times {\mathbb {A}}^1({\mathbb {R}})$ without loss of generality.

Proof

By Lemma 4.4 for every connected component $D\subset C({\mathbb {R}})$ either s is constant on D, or s takes every value only finitely many times on D. Therefore for all but finitely many $x\in {\mathbb {P}}^1({\mathbb {R}})$ the function s takes x only finitely many times on $C({\mathbb {R}})$, and hence after applying an automorphism of ${\mathbb {P}}^1$ over ${\mathbb {R}}$, if this is necessary, we may assume that the set T of points $t\in C({\mathbb {R}})$ where $s(t)=\infty$ is finite without loss of generality. We may even assume that $\phi$ does not take $\infty$ as a value. Since Nash maps are analytic, there are an effective zero divisor Z with support on $C({\mathbb {R}})$ and an interpolation condition $h:Z\rightarrow {\mathbb {P}}^1_{{\mathbb {R}}}$ compatible with s such that for every Nash map $r:C({\mathbb {R}})\rightarrow {\mathbb {P}}^1({\mathbb {R}})$ compatible with h the limit

$$\begin{aligned} \lim _{x\rightarrow t}(s(x)-r(x)) \end{aligned}$$

(4.5.1)

exists and finite for every $t\in T$. We may even assume that h subsumes $\phi$ without loss of generality by choosing a Z such that $Z-S$ is effective. By Lemma 4.3 there is a morphism $r:C\rightarrow {\mathbb {P}}^1_{{\mathbb {R}}}$ of schemes over ${\mathbb {R}}$ such that r is compatible with h and r has no poles on $C({\mathbb {R}})$ outside of Z.

By our assumptions both s and r take values in ${\mathbb {R}}$ on $C({\mathbb {R}})-T$, so their difference $\widetilde{s}=s-r$ is a Nash function $C({\mathbb {R}})-T\rightarrow {\mathbb {R}}$. Since the limit (4.5.1) exists and finite for every $t\in T$, this map $\widetilde{s}$ extends uniquely to a continuous map $C({\mathbb {R}})\rightarrow {\mathbb {R}}$ which we will also denote by $\widetilde{s}$ by slight abuse of notation. The latter is also Nash and it is compatible with a unique interpolation condition ${\widetilde{\phi }}:Z\rightarrow {\mathbb {A}}^1_{\mathbb R}$. Let ${\widetilde{R}}\subseteq C({\mathbb {R}})\times \mathbb A^1({\mathbb {R}})$ be the union of the image of ${\widetilde{s}}$ and the set

$$\begin{aligned} \{(x,y)\in C({\mathbb {R}})\times {\mathbb {A}}^1({\mathbb {R}})\mid x\not \in T\ \text {and}\ (x,y+r(x))\in R\}. \end{aligned}$$

The set $\widetilde{R}$ is admissible, and it contains the image of ${\widetilde{s}}$. So if we assume that the claim of the theorem holds for ${\widetilde{R}}$ then there is a map ${\widetilde{f}}:C\rightarrow {\mathbb {P}}^1_{{\mathbb {R}}}$ of schemes over ${\mathbb {R}}$ compatible with ${\widetilde{\phi }}$ such that ${\widetilde{f}}(C({\mathbb {R}}))$ lies in ${\widetilde{R}}$. The rational function ${\widetilde{f}}+r:C\dashrightarrow \mathbb P^1_{{\mathbb {R}}}$ extends to a map $f:C\rightarrow {\mathbb {P}}^1_{{\mathbb {R}}}$ which is compatible with $\phi$ and $f(C({\mathbb {R}}))$ lies in R. $\square$

Proposition 4.6

Let $R\subseteq C\times {\mathbb {A}}^1({\mathbb {R}})$ be an admissible semi-algebraic subset, and let $s : C({\mathbb {R}})\rightarrow C\times {\mathbb {A}}^1({\mathbb {R}})$ be a Nash section whose image lies in R. Then there are an open neighbourhood U of s in the ${\mathcal {C}}^\infty$-topology, a finite subscheme $Y\subset C$ and an interpolation condition $\psi : Y\rightarrow C\times {\mathbb {A}}^1_{\mathbb {R}}$ compatible with s with the following property: for every ${\mathcal {C}}^\infty$-section $r : C({\mathbb {R}}) \rightarrow C \times {\mathbb {A}}^1({\mathbb {R}})$ which lies in U and is compatible with $\psi$, the image of r lies in R.

Proof

Let $s_2,r_2:C({\mathbb {R}})\rightarrow {\mathbb {R}}$ denote the second coordinate functions of s and r. The set of kissing points is $Z=R\setminus R^\circ$. Let $T=\pi (Z)\subset C({\mathbb {R}})$, and let $V\subset C({\mathbb {R}})$ be an open neighborhood of T which is the union of disjoint open intervals. We fix a real analytic imbedding $V\hookrightarrow (0,1)\subset {\mathbb {R}}$, this induces a real analytic imbedding of $G=V\times {\mathbb {A}}^1({\mathbb {R}})$ into the Euclidean plane. In particular, U and G inherit the Euclidean metric. Let $A=G\setminus R^\circ$, it is a closed semi-algebraic set in G. Let $K\subset V$ be a compact neighborhood of T in V, and let S denote the maximum of $\big |r_2(t)\big |$ for $t\in K$. Let $E\subset A$ denote the compact subset $E=\big (K\times [-S-1,S+1]\big )\setminus R^\circ$. Consider the real analytic function

$$\begin{aligned} f:G=V\times {\mathbb {A}}^1({\mathbb {R}})\rightarrow {\mathbb {R}}, \quad f(t,a)=|a-s_2(t)|^2. \end{aligned}$$

The zero set of f is s(V), hence the zero set of the restriction $f|_A$ is just Z. By the Łojasiewicz inequality (in the form of Corollaire to Théorème 1 of section 18 in [12]) applied to our f, G, A, E, and Z, there are $d,N>0$ such that

$$\begin{aligned} \big |f(x)\big |\ge d\cdot \,\mathrm{dist}\big (x,Z\big )^N \quad \text {for all }x\in E. \end{aligned}$$

(4.2)

Choose an integer $M\ge N/2$, and set ${\tilde{d}}=\min \big (\sqrt{d},1\big )$. Then

$$\begin{aligned} \big |a-s_2(t)\big |\ge {\tilde{d}}\cdot \,\mathrm{dist}\big (t,T)^M \quad \text {whenever }t\in K,\ (t,a)\in A. \end{aligned}$$

(4.3)

Indeed, the projection $\pi$ is a contraction, and $\mathrm{dist}(t,T)\le 1$ since U was imbedded into (0, 1). So (4.2) implies (4.3) for $(t,a)\in E$. On the other hand if $(t,a)\notin E$ then $|a|>S+1$, hence $\big |a-s_2(t)\big |>1$, and (4.3) follows again.

With the divisor $Y=MT$ let $\psi :Y\rightarrow C\times {\mathbb {A}}^1_{\mathbb {R}}$ be the (unique) interpolation condition (of order M) compatible with s. Suppose now that a section r is compatible with $\psi$, and its ${\mathcal {C}}^{M+1}$-distance from s is less than, say, 1. If D denotes the maximum of $\big |s_2^{(M+1)}(t)\big |$ for $t\in K$, then $\big |r_2^{(M+1)}(t)\big |<D+1$ for all $t\in K$. By Taylor’s theorem (with the Lagrange form of remainder)

$$\begin{aligned} \big |r_2(t)-s_2(t)\big |<2\frac{D+1}{(M+1)!}\,\mathrm{dist}\big (t,T)^{M+1} \quad \text {for all }t\in K. \end{aligned}$$

(4.4)

Combining (4.3) and (4.4) we obtain that, after possibly shrinking K, $r(t)=\big (t,r_2(t)\big )\notin A$ for $t\in K\setminus T$, hence $r(K)\subset R$.

On the other hand $Q=s\big (C({\mathbb {R}})\setminus K^\circ \big )$ is a compact set in $R^\circ$. If the ${\mathcal {C}}^0$-distance of r from s is smaller than $\mathrm{dist}(Q,A)$, then $r\big (C({\mathbb {R}})\setminus K\big )\subset R$ as well. $\square$

Proof of Theorem 4.2

By Lemma 4.5 we may assume that $R\subseteq C({\mathbb {R}})\times {\mathbb {A}}^1({\mathbb {R}})$ without loss of generality. By Proposition 4.6 there is an open neighbourhood U of s in the $C^{\infty }$-topology and an interpolation condition $\psi :Y\rightarrow {\mathbb {A}}^1_{{\mathbb {R}}}$ compatible with s with the following property: for every ${\mathcal {C}}^{\infty }$-section $r:C({\mathbb {R}})\rightarrow C\times {\mathbb {A}}^1({\mathbb {R}})$ which lies in U and compatible with $\psi$, the image of r lies in R. Let T be a zero-dimensional closed subscheme of C whose closed points are all real and which contains both S and Y as a closed subscheme. Let ${\widetilde{\phi }}:T\rightarrow {\mathbb {A}}^1_{{\mathbb {R}}}$ be the unique interpolation condition compatible with s. Since s is compatible both with $\phi$ and $\psi$, the interpolation condition ${\widetilde{\phi }}$ subsumes both $\phi$ and $\psi$. By Lemma 3.15 there is a regular section $r:C\rightarrow C\times {\mathbb {A}}^1$ compatible with ${\widetilde{\phi }}$ such that $r({\mathbb {R}})$ lies in U. Since ${\widetilde{\phi }}$ subsumes $\phi$, the section r is compatible with $\phi$. Since ${\widetilde{\phi }}$ subsumes $\psi$, the section r is compatible with $\psi$, too. Therefore the image of $r({\mathbb {R}})$ lies in R. $\square$

5 The main theorem and some easy reductions

Definition 5.1

Note that for every $x\in C({\mathbb {R}})$ the discrete valuation of ${\mathbb {R}}(C)_x$ induces a topology on the projective space ${\mathbb {P}}^n({\mathbb {R}}(C)_x)$, and hence on the ${\mathbb {R}}(C)_x$-valued points of any quasi-projective variety defined over ${\mathbb {R}}(C)_x$. Moreover this topology is canonical in the sense that it does not depend on the choice of the embedding into a projective space. We will call this the x-adic topology. Now let X be again a smooth, irreducible projective variety defined over ${\mathbb {R}}(C)$. We will equip the direct product

$$\begin{aligned} X({\mathbb {A}}_C)=\prod _{x\in C({\mathbb {R}})}X({\mathbb {R}}(C)_x) \end{aligned}$$

with the direct product of the x-adic topologies.

Remark 5.2

Let X be as above, and let ${\mathcal {X}}$ be a model of X over C. It is possible to give a simple description of a basis for the topology on $X({\mathbb {A}}_C)$ defined above in terms of ${\mathcal {X}}$ as follows. By slight abuse of notation let ${\mathcal {X}}({\mathcal {O}}_x)$ denote the set of sections ${\text {Spec}}({\mathcal {O}}_x)\rightarrow {\mathcal {X}}\times _C{\text {Spec}}({\mathcal {O}}_x)$ for every $x\in C({\mathbb {R}})$. As we already noted we have a bijection $X({\mathbb {R}}(C)_x)\cong {\mathcal {X}}({\mathcal {O}}_x)$ for every $x\in C({\mathbb {R}})$ by the valuative criterion of properness, so we have a bijection

$$\begin{aligned} X({\mathbb {A}}_C)\cong \prod _{x\in C({\mathbb {R}})} {\mathcal {X}}(\mathcal O_x), \end{aligned}$$

(5.2.1)

too. For every interpolation condition $\phi : S\rightarrow \mathcal X\times _CS$ let

$$\begin{aligned} U(S,\phi )\subseteq \prod _{x\in C({\mathbb {R}})}{\mathcal {X}}({\mathcal {O}}_x) \end{aligned}$$

be the subset of all those sections whose pull-back under the closed immersion

$$\begin{aligned} S\rightarrow \coprod _{x\in C({\mathbb {R}})} {\text {Spec}}({\mathcal {O}}_x) \end{aligned}$$

is $\phi$. These sets form a basis for the topology of $X({\mathbb {A}}_C)$ under the map in (5.2.1).

For every morphism $f:X\rightarrow Y$ of varieties over ${\mathbb {R}}(C)$ and for every $c\in Y({\mathbb {R}}(C))$ let $X_c$ denote the fibre of f above c. Our main result Theorem 1.5 follows from the following

Theorem 5.3

Assume that $C({\mathbb {R}})$ is non-empty. Let Y be a smooth,  projective,  rational variety over ${\mathbb {R}}(C),$ and let X be a smooth,  projective,  irreducible variety over ${\mathbb {R}}(C)$ endowed with a dominant morphism $f:X\rightarrow Y$ with a geometrically irreducible and smooth generic fibre. Let

$$\begin{aligned} {\underline{M}}=\prod _{x\in C({\mathbb {R}})}M_x\in X({\mathbb {A}}_C)^{CT}, \end{aligned}$$

and let $U\subset X({\mathbb {A}}_C)$ be an open neighbourhood of ${\underline{M}}$. Then there exist

• (i) a point $c\in Y({\mathbb {R}}(C))$ such that $X_c$ is smooth,  and

• (ii) an ${\underline{N}}\in X_c({\mathbb {A}}_C)^{CT}$ such that ${\underline{N}}\in U$.

Remark 5.4

Note that Theorem 1.5 is trivially true when $C({\mathbb {R}})$ is empty. Indeed in this case there is no CT obstruction both for the smooth fibres of f over rational points and for X itself. By assumption all such fibres will have rational points, so X has rational points, too.

Proof

We start the proof with two easy reduction steps. For the sake of simple notation set

$$\begin{aligned} Y_n=\underbrace{{\mathbb {P}}^1_{{\mathbb {R}}(C)}\times {\mathbb {P}}^1_{{\mathbb {R}}(C)}\times \cdots \times {\mathbb {P}}^1_{\mathbb R(C)}}_{n\text {-times}}. \end{aligned}$$

$\square$

Proposition 5.5

We may assume that $Y=Y_n$ for some n without loss of generality.

Proof

Using resolutions of singularities it follows from the assumption that there is a diagram

of birational morphisms between smooth projective varieties over ${\mathbb {R}}(C)$ (for some n). Let ${\widetilde{X}}\rightarrow X\times _Y{\widetilde{Y}}$ be a desingularisation of the pull-back $X\times _Y{\widetilde{Y}}$ of X via $\phi$ which is isomorphic over the nonsingular part, and let ${\widetilde{f}}:\widetilde{X}\rightarrow {\widetilde{Y}}$ be the composition of this desingularisation and the base change of f with respect to $\phi$. Let $V\subseteq {\widetilde{Y}}$ be a non-empty Zariski-open subset such that the restrictions of both $\phi$ and $\psi$ onto V are isomorphisms onto their images, and f is smooth over $\phi (V)$. Then we have a commutative diagram:

such that $\rho$ maps $\widetilde{f}^{-1}(V)$ isomorphically onto $f^{-1}(\phi (V))$. In particular $\rho$ is birational. Let $Z\subset X$ be the complement of $f^{-1}(\phi (V))$, and let ${\widetilde{Z}}\subset {\widetilde{X}}$ be the complement of ${\widetilde{f}}^{-1}(V)$. Now let ${\underline{M}}$ be an element of $X({\mathbb {A}}_C)^{CT}$, as in the claim above. We may assume without loss of generality that its given open neighbourhood U is of the form

$$\begin{aligned} U=\prod _{x\in C({\mathbb {R}})}U_x \end{aligned}$$

where $U_x$ is an x-adic open neighbourhood of $M_x$ in $X(\mathbb R(C)_x)$ for every $x\in C({\mathbb {R}})$. Let $I\subseteq C(\mathbb R)$ be the set of those $x\in C({\mathbb {R}})$ where $U_x\ne X(\mathbb R(C)_x)$. The set I is finite.

Now let ${\mathcal {X}}$ be a model of X over C and for every $x\in C({\mathbb {R}})$ let $T_x\subseteq X({\mathbb {R}}(C)_x)$ be the x-adic open neighbourhood of $M_x$ which under the bijection $X(\mathbb R(C)_x)\cong {\mathcal {X}}({\mathcal {O}}_x)$ corresponds to those sections ${\text {Spec}}({\mathcal {O}}_x)\rightarrow \mathcal X\times _C{\text {Spec}}({\mathcal {O}}_x)$ whose special fibre is the same as the special fibre of the section corresponding to $M_x$. Since Z is a proper Zariski-closed subscheme of X, the set $Z({\mathbb {R}}(C)_x)$ is nowhere dense in $X({\mathbb {R}}(C)_x)$ with respect to the x-adic topology, so there is a non-empty $W_x\subseteq U_x\cap T_x$, open with respect to the x-adic topology, such that $W_x$ and $Z({\mathbb {R}}(C)_x)$ have empty intersection, for every $x\in C({\mathbb {R}})$. By the above for every

$$\begin{aligned} {\underline{M}}'=\prod _{x\in C({\mathbb {R}})}M_x'\in \prod _{x\in C({\mathbb {R}})}W_x \end{aligned}$$

we have $\sigma (\underline{M}')=\sigma ({\underline{M}})$ (see Notation 3.5). Pick such an ${\underline{M}}'$. Then by Theorem 3.6 we have $\underline{M}'\in X({\mathbb {A}}_C)^{CT}$. Clearly it will be enough to show the claim for ${\underline{M}}'$ and for any of its open neighbourhoods contained in

$$\begin{aligned} W=\prod _{x\in I}W_x\times \prod _{x\in C({\mathbb {R}})-I} X({\mathbb {R}}(C)_x). \end{aligned}$$

Therefore we may assume, without loss of generality, that $M_x\not \in Z({\mathbb {R}}(C)_x)$ for every $x\in C({\mathbb {R}})$, the set I is non-empty, and $U_x\cap Z(\mathbb R(C)_x)=\emptyset$ for every $x\in I$. Since $\rho$ maps $\widetilde{f}^{-1}(V)$ isomorphically onto $f^{-1}(\phi (V))$ we get that there is a ${\widetilde{M}}_x\in {\widetilde{X}}({\mathbb {R}}(C)_x)$ for every $x\in C({\mathbb {R}})$ such that ${\widetilde{M}}_x\not \in {\widetilde{Z}}(\mathbb R(C)_x)$ and $\rho ({\widetilde{M}}_x)=M_x$. By Lemma 2.7 we know that

$$\begin{aligned} \underline{{\widetilde{M}}}=\prod _{x\in C({\mathbb {R}})} {\widetilde{M}}_x \in {\widetilde{X}}({\mathbb {A}}_C)^{CT}. \end{aligned}$$

Moreover for every $x\in I$ there is an x-adic open neighbourhood $\widetilde{U}_x\subseteq {\widetilde{X}}({\mathbb {R}}(C)_x)$ of ${\widetilde{M}}_x$ such that ${\widetilde{U}}_x\cap {\widetilde{Z}}({\mathbb {R}}(C)_x)=\emptyset$ and ${\widetilde{U}}_x$ maps to $U_x$ homeomorphically with respect to $\rho$. Set

$$\begin{aligned} {\widetilde{U}}=\prod _{x\in I}{\widetilde{U}}_x\times \prod _{x\in C({\mathbb {R}})-I}{\widetilde{X}}({\mathbb {R}}(C)_x); \end{aligned}$$

it is an open neighbourhood of $\underline{{\widetilde{M}}}$. Since the map $\psi \circ {\widetilde{f}}:{\widetilde{X}}\rightarrow Y_n$ satisfies the conditions of theorem, we get that there is a point $c\in Y_n({\mathbb {R}}(C))$ such that ${\widetilde{X}}_c$ is smooth, and an $\widetilde{{\underline{N}}}\in {\widetilde{X}}_c({\mathbb {A}}_C)^{CT}$ such that $\widetilde{{\underline{N}}}\in {\widetilde{U}}$. Let x be now an element of I. Then c lies in the image of ${\widetilde{U}}_x$ with respect to $\psi \circ {\widetilde{f}}$, so it must lie in $\psi (V({\mathbb {R}}(C)_x))$. Therefore there is a unique $\overline{c}\in V({\mathbb {R}}(C))$ such that ${\widetilde{c}}=\psi ({\overline{c}})$. Set $c=\phi ({\overline{c}})$ and ${\underline{N}}= \rho (\widetilde{{\underline{N}}})$. Clearly $c\in \phi (V({\mathbb {R}}(C)))$ and hence $X_c$ is smooth. Moreover $\rho$ maps ${\widetilde{X}}_c$ isomorphically onto $X_c$, so ${\underline{N}}\in X_c(\mathbb A_C)^{CT}$. Finally ${\underline{N}}\in U$ since $\rho$ maps ${\widetilde{U}}$ into U. $\square$

Lemma 5.6

We may assume that $Y={\mathbb {P}}^1_{{\mathbb {R}}(C)}$ without loss of generality.

Proof

We may immediately reduce the case when $Y=Y_n$ to the case when $Y={\mathbb {P}}^1_{{\mathbb {R}}(C)}$ via an easy induction on n, so the claim follows from the proposition above. $\square$

6 The fibration method

Let us begin the main part of the proof of Theorem 5.3. By the above we may assume without loss of generality that $Y={\mathbb {P}}^1_{{\mathbb {R}}(C)}$. By resolution of singularities there is an integral, smooth, projective variety ${\mathcal {X}}$ equipped with a projective dominant morphism ${\mathbf {f}}:{\mathcal {X}}\rightarrow C\times {\mathbb {P}}^1_{{\mathbb {R}}}$ over ${\mathbb {R}}$ whose generic fibre is $f:X\rightarrow {\mathbb {P}}^1_{{\mathbb {R}}(C)}$. In particular ${\mathcal {X}}$ is a model of X over C with respect to the composition p of ${\mathbf {f}}$ with the projection $\pi :C\times {\mathbb {P}}^1_{\mathbb R}\rightarrow C$ onto the first factor.

Since the generic fibre of f is smooth, the same holds for ${\mathbf {f}}$, too. Therefore there is a closed subscheme $\mathcal Z\subset C\times {\mathbb {P}}^1_{{\mathbb {R}}}$ of positive codimension such that ${\mathbf {f}}|_{{\mathbf {f}}^{-1}({\mathcal {U}})}:\mathbf{f}^{-1}({\mathcal {U}})\rightarrow {\mathcal {U}}$ is smooth, where $\mathcal U=C\times {\mathbb {P}}^1_{{\mathbb {R}}}-{\mathcal {Z}}$ is the complement of ${\mathcal {Z}}$. Now let ${\underline{M}}$ be an element of $X(\mathbb A_C)^{CT}$, and let $U\subset X({\mathbb {A}}_C)$ be an open neighbourhood of ${\underline{M}}$, as in the claim above. We may assume without loss of generality that there is an interpolation condition $\phi :S\rightarrow {\mathcal {X}}\times _CS$ such that the open neighbourhood U given is of the form $U(S,\phi )$ by Remark 5.2.

Proposition 6.1

There is an ${\underline{M}}' \in X({\mathbb {A}}_C)^{CT}\cap U$ such that

• (i) the section $\sigma ({\underline{M}}'):C({\mathbb {R}})\rightarrow {\mathcal {X}}_{sm}({\mathbb {R}})$ is Nash,

• (ii) the section $\sigma ({\underline{M}}')$ is compatible with $\phi$,

• (iii) the section ${\mathbf {f}}({\mathbb {R}})\circ \sigma ({\underline{M}}')$ only intersects ${\mathcal {Z}}({\mathbb {R}})$ in finitely many points.

Proof

By our assumption U is of the form

$$\begin{aligned} U=\prod _{x\in C({\mathbb {R}})}U_x \end{aligned}$$

where $U_x$ is an x-adic open neighbourhood of $M_x$ in $X(\mathbb R(C)_x)$ for every $x\in C({\mathbb {R}})$. Let $I\subseteq C(\mathbb R)$ be the set of those $x\in C({\mathbb {R}})$ where $U_x\ne X(\mathbb R(C)_x)$. The set I is finite. For every connected component $D\subset C({\mathbb {R}})$ for all but finitely many $x\in D$ the intersection ${\mathcal {Z}}\cap \pi ^{-1}(x)$ is a zero dimensional scheme, since ${\mathcal {Z}}$ has positive codimension in $C\times {\mathbb {P}}^1_{{\mathbb {R}}}$. Since p is generically smooth and has geometrically irreducible fibres, for all but finitely many $x\in D$ the fibre ${\mathcal {X}}_x$ is smooth and geometrically irreducible. Therefore for all D as above we may choose a point $x(D)\in D$ which does not lie in I, the intersection $\mathcal Z({\mathbb {R}})\cap \pi ^{-1}(x_D)({\mathbb {R}})$ is finite, and $\mathcal X_{x(D)}$ is smooth and geometrically irreducible.

For every D as above let $E_D\subset {\mathcal {X}}_{x(D)}({\mathbb {R}})$ denote the connected component containing $m_{x(D)}=\sigma ({\underline{M}})(x(D))\in {\mathcal {X}}_{x(D)}(\mathbb R)$. We claim that there is an $n_D\in E_D$ such that $n_D\not \in {\mathbf {f}}^{-1}({\mathcal {Z}})({\mathbb {R}})$. Assume that this is not the case for some $D\in \pi _0(C({\mathbb {R}}))$. Then the image of $E_D$ under ${\mathbf {f}}({\mathbb {R}})$ lies in the finite set ${\mathcal {Z}}({\mathbb {R}})\cap \pi ^{-1}(x(D))({\mathbb {R}})$. But $E_D$ is connected, so is its image under the continuous map $\mathbf{f}({\mathbb {R}})$, which therefore must be a point $p_D\in \mathcal Z({\mathbb {R}})\cap \pi ^{-1}(x(D))({\mathbb {R}})$. Note that $E_D$ is Zariski-dense in ${\mathcal {X}}_{x(D)}$. Indeed suppose that this is not the case; as the latter is geometrically irreducible, the Zariski-closure of $E_D$ has dimension strictly less than the dimension d of ${\mathcal {X}}_{x(D)}$. Therefore the dimension of the semi-algebraic set $E_D$ is also less than d. But $\mathcal X_{x(D)}$ is smooth, so the dimension of $E_D$ is d by the inverse function theorem, which is a contradiction. We get that $\mathcal X_{x(D)}$ lies in ${\mathbf {f}}^{-1}(p_D)$, and hence the fibre of ${\mathbf {f}}$ over any other point in $\pi ^{-1}(x(D))$ is empty. But this is a contradiction, so our original assumption on the image of $E_D$ with respect to ${\mathbf {f}}({\mathbb {R}})$ is false.

For every D as above choose an $N_{x(D)}\in X(\mathbb R(C)_{x(D)})$ such that the special fibre of the section ${\text {Spec}}({\mathcal {O}}_ {x(D)})\rightarrow \mathcal X\times _C{\text {Spec}}({\mathcal {O}}_{x(D)})$ associated to $N_{x(D)}$ is the point $n_D\in E_D-{\mathbf {f}}^{-1}({\mathcal {Z}})({\mathbb {R}})$ found above. (This is possible by Hensel’s lemma.) Let $\underline{N}\in X({\mathbb {A}}_C)^{CT}$ be the point we get from ${\underline{M}}$ by replacing $M_{x(D)}$ with $N_{x(D)}$ for every D as above. Clearly ${\underline{N}}\in U$, and by Proposition 3.4 we have ${\underline{N}}\in X({\mathbb {A}}_C)^{CT}$. Let T be the effective zero cycle $S+\sum _{D\in \pi _0(C({\mathbb {R}}))}x_D$ and let $\psi :T\rightarrow {\mathcal {X}}\times _CT$ be the unique interpolation condition which subsumes $\phi$ and $\psi (y)$ is the special fibre of the section ${\text {Spec}}({\mathcal {O}}_y)\rightarrow \mathcal X\times _C{\text {Spec}}({\mathcal {O}}_y)$ associated to $N_y$ for every $y\in T$. This interpolation condition $\psi$ is weakly compatible with $\sigma ({\underline{N}})$, so there is a Nash section $\rho :C({\mathbb {R}})\rightarrow {\mathcal {X}}_{sm}({\mathbb {R}})$ compatible with $\psi$ by Proposition 3.19. Since $\rho (x_D)\not \in {\mathcal {Z}}({\mathbb {R}})$ for every $D\in \pi _0(C({\mathbb {R}}))$ by construction, by Lemma 4.4 the section $\rho$ only intersects ${\mathcal {Z}}({\mathbb {R}})$ in finitely many points. Again by Hensel’s lemma there is an ${\underline{M}}'\in X({\mathbb {A}}_C)^{CT}$ such that $\sigma (\underline{M}')=\rho$ and $M'_x=N_x=M_x$ for every $x\in I$. Clearly $\underline{M}'\in U$ and by Theorem 3.6 we have ${\underline{M}}'\in X({\mathbb {A}}_C)^{CT}$. $\square$

Replacing ${\underline{M}}$ by ${\underline{M}}'$ we may assume without loss of generality that $\sigma ({\underline{M}})$ is Nash, compatible with $\phi$, and ${\mathbf {f}}({\mathbb {R}})\circ \sigma ({\underline{M}})$ only intersects ${\mathcal {Z}}({\mathbb {R}})$ in finitely many points. Recall that a Nash map $a:A\rightarrow B$ of Nash manifolds over ${\mathbb {R}}$ is Nash trivial if there is a Nash manifold L and a Nash diffeomorphism $b:L\times B\rightarrow A$ (over ${\mathbb {R}}$) such that $a\circ b:L\times B\rightarrow B$ is the projection onto the second coordinate.

Proposition 6.2

There is an admissible semi-algebraic subset $R\subseteq C({\mathbb {R}}) \times \mathbb P^1({\mathbb {R}})$ such that

• (i) the image ${\mathbf {I}}$ of ${\mathbf {f}}({\mathbb {R}})\circ \sigma ({\underline{M}})$ lies in R,

• (ii) the set $R^o$ lies in ${\mathcal {U}}({\mathbb {R}})$,

• (iii) there is a Nash retraction $\rho :R^o\rightarrow R^o\cap {\mathbf {I}}$,

• (iv) for every semi-algebraic connected component $O\subset R^o$ the map

  $$\begin{aligned} {\mathbf {f}}({\mathbb {R}})|_{{\mathbf {f}}({\mathbb {R}})^{-1}(O)}: {\mathbf {f}}({\mathbb {R}})^{-1}(O)\rightarrow O \end{aligned}$$

  is Nash-trivial.

Proof

Since ${\mathcal {U}}({\mathbb {R}})$ is a Nash manifold over ${\mathbb {R}}$, there is a finite covering $\{V_j\}_{j\in J}$ of ${\mathcal {U}}({\mathbb {R}})$ by open semi-algebraic subsets which are Nash diffeomorphic to an affine space over ${\mathbb {R}}$, necessarily of dimension 2, by Lemma 3.2 of [9] on page 1217. For every $j\in J$ the restriction

$$\begin{aligned} {\mathbf {f}}({\mathbb {R}})|_{{\mathbf {f}}({\mathbb {R}})^{-1}(V_j)}: {\mathbf {f}}({\mathbb {R}})^{-1}(V_j)\rightarrow V_j \end{aligned}$$

is a proper surjective submersion of Nash manifolds, so it is Nash trivial by Proposition 5.2 of [6] on page 368 and the main theorem of [9] on page 1209. For every $j\in J$ the intersection $V_j\cap {\mathbf {I}}$ is semi-algebraic and open in ${\mathbf {I}}$, so its image $Q_j$ under the projection $\pi ({\mathbb {R}})$ is semi-algebraic and open in $C(\mathbb R)$. Therefore $Q_j$ is the union of finitely many pairwise disjoint open intervals or circles, i.e. open and connected sets. By possibly refining the covering $\{V_j\}_{j\in J}$ we may achieve that we have only intervals, no circles.

Let B be the union of the end-points of these open intervals for all $j\in J$, where by end-points we mean accumulation points not in the interval. Since the set of these intervals is finite, the set B is also finite. The complement of B in $C({\mathbb {R}})$ is the union of finitely many pair-wise disjoint open intervals; let K denote the set of these open intervals. Since the sets $\{V_j\}_{j\in J}$ cover all but finitely many points of $\mathbf{I}$, for every $H\in K$ the pre-image $\pi (\mathbb R)^{-1}(H)\cap {\mathbf {I}}$ lies in $V_j$ for some $j\in J$. For every such H fix such a $V_j$ and let $W_H\subset V_j$ be a semi-algebraic tubular neighbourhood of $\pi (\mathbb R)^{-1}(H)\cap {\mathbf {I}}$ in $\pi ({\mathbb {R}})^{-1}(H)\cap V_j$. (The latter exists by Corollary 8.9.5 of [2] on page 199.)

Let R be the union of the $\{W_H\}_{H\in K}$ and ${\mathbf {I}}$. Since B is finite, the set R is admissible, and (i) holds for R by construction. Since the elements of K are pair-wise disjoint, the elements of $\{W_H\}_{H\in K}$ are also pair-wise disjoint. For every $H\in K$ the semi-algebraic set $\pi (\mathbb R)^{-1}(H)\cap {\mathbf {I}}$ is connected, so the same holds for its tubular neighbourhood $W_H$. As $R^o=\bigcup _{H\in K}W_H$, we get that the sets $\{W_H\}_{H\in K}$ are the connected components of $R^o$. Since $W_H\subset V_j$ for some j, condition (iv) holds. As the sets $\{V_j\}_{j\in J}$ lie in ${\mathcal {U}}({\mathbb {R}})$, condition (ii) is true. Moreover $W_H$ is a semi-algebraic tubular neighbourhood of $W_H\cap {\mathbf {I}}=\pi ({\mathbb {R}})^{-1}(H)\cap \mathbf{I}$, so there is a Nash retraction $\rho _H:W_H\rightarrow W_H\cap {\mathbf {I}}$ for every $H\in K$. Therefore (iii) holds, too. $\square$

Definition 6.3

For every $x\in C\times {\mathbb {P}}^1({\mathbb {R}})$ and every 1-dimensional subspace L of the tangent space of $C\times {\mathbb {P}}^1$ at x let ${\mathcal {X}}_{L,sm}\subset {\mathcal {X}}_x$ be the largest open sub-scheme such that for every closed point P of ${\mathcal {X}}_{L,sm}$ the image of the differential of ${\mathbf {f}}$ at P contains L. Let $\sigma :C({\mathbb {R}})\rightarrow {\mathcal {X}}({\mathbb {R}})$ be a Nash section of $p({\mathbb {R}})$ and let $R_+\subseteq C({\mathbb {R}})\times {\mathbb {P}}^1({\mathbb {R}})$ be an admissible semi-algebraic subset which contains the image $\mathrm {Im}({\mathbf {f}}({\mathbb {R}})\circ \sigma )$ of ${\mathbf {f}}({\mathbb {R}})\circ \sigma$. For every point x on $\mathrm {Im}({\mathbf {f}}({\mathbb {R}})\circ \sigma )$ let L(x) denote the tangent line of ${\mathbf {f}}({\mathbb {R}})\circ \sigma$ at x; it is a 1-dimensional subspace of the tangent space of $C\times {\mathbb {P}}^1$ at x. A butterfly extension of $\sigma$ on $R_+$ is a semi-algebraic section $\beta :R_+\rightarrow {\mathcal {X}}({\mathbb {R}})$ of ${\mathbf {f}}({\mathbb {R}})$ such that $\sigma$ restricted to $R_+^o$ is continuous, the composition $\beta \circ {\mathbf {f}}({\mathbb {R}})\circ \sigma$ is $\sigma$, and for every kissing point x of $R_+$ on $\mathrm {Im}({\mathbf {f}}({\mathbb {R}})\circ \sigma )$ the intersection of the closure of the image of $\beta$ with ${\mathcal {X}}_x({\mathbb {R}})$ lies in the connected component of $\beta (x)$ in ${\mathcal {X}}_{L(x),sm}({\mathbb {R}})$. (Note that $\beta (x)=\sigma (\pi ({\mathbb {R}})(x))$, so $\beta (x)$ does lie in ${\mathcal {X}}_{L(x),sm}({\mathbb {R}})$, since $\sigma$ is a Nash section.)

Proposition 6.4

There is an admissible semi-algebraic subset $R_+\subseteq R$ containing $\mathrm {Im}({\mathbf {f}}({\mathbb {R}})\circ \sigma ({\underline{M}}))={\mathbf {I}}$ such that there is a butterfly extension $s:R_+\rightarrow {\mathcal {X}}({\mathbb {R}})$ of $\sigma ({\underline{M}})$.

Proof

We will need the following easy semi-algebraic separation lemma. $\square$

Lemma 6.5

Let F and G be two disjoint closed semi-algebraic subsets of ${\mathcal {X}}({\mathbb {R}})$. Then there are two disjoint open semi-algebraic subsets $A,B\subset {\mathcal {X}}({\mathbb {R}})$ such that $F\subset A$ and $G\subset B$.

Proof

Since ${\mathcal {X}}$ is projective, by Theorem 3.4.4 of [2] on page 72 there is a continuous semi-algebraic embedding $\iota :{\mathcal {X}}({\mathbb {R}})\rightarrow {\mathbb {R}}^m$ for some positive integer m. Because ${\mathcal {X}}$ is projective, the semi-algebraic set ${\mathcal {X}}({\mathbb {R}})$ is compact, so the same holds for its closed subsets F and G. Therefore $\iota (F)$ and $\iota (G)$ are closed in ${\mathbb {R}}^m$, and by elimination of quantifiers these sets are also semi-algebraic. Then $d=\mathrm {dist}(F,G)>0$, where $\mathrm {dist}$ stands for the Euclidean distance. The sets

$$\begin{aligned} A=\left\{ {\underline{x}}\in {\mathbb {R}}^m\mid \mathrm {dist}({\underline{x}},F)<\frac{d}{3}\right\} , \quad B=\left\{ {\underline{x}}\in {\mathbb {R}}^m\mid \mathrm {dist}(\underline{x},G)<\frac{d}{3}\right\} \end{aligned}$$

are open, semi-algebraic, contain F and G, respectively, and $A\cap B=\emptyset$. $\square$

By removing every kissing point of R which does not lie on ${\mathbf {I}}$ we may assume that every kissing point of R lies on ${\mathbf {I}}$ without loss of generality. Let $\rho :R^o\rightarrow R^o\cap {\mathbf {I}}$ be a Nash retraction. For every semi-algebraic connected component $O\subset R^o$ let $M_O$ be a compact Nash manifold such that there is a Nash diffeomorphism $b_O:M_O\times O\rightarrow {\mathbf {f}}({\mathbb {R}})^{-1}(O)$ such that ${\mathbf {f}}(\mathbb R)\circ b_O:M_O\times O\rightarrow O$ is the projection onto the first coordinate. For every such O let $a_O:M_O\times O\rightarrow M_O$ be the projection onto the second coordinate, let $\rho _O:O\rightarrow O\cap \mathbf{I}$ be the restriction of $\rho$ onto O, let $r_O:O\rightarrow M_O$ denote the composition:

$$\begin{aligned} a_O\circ b_O^{-1}\circ \sigma ({\underline{M}})\circ \pi ({\mathbb {R}})\circ \rho _O, \end{aligned}$$

and let $s_O:O\rightarrow {\mathbf {f}}({\mathbb {R}})^{-1}(O)$ be the composition of the map:

$$\begin{aligned} O\rightarrow O\times M_O\text { given by }t\mapsto (t,r_O) \end{aligned}$$

and $b_O$. Let $s^o$ be the coproduct of the maps $s_O$ for all such O. The map $s^o$ is a Nash section of

$$\begin{aligned} {\mathbf {f}}({\mathbb {R}})|_{{\mathbf {f}}({\mathbb {R}})^{-1}(R^o)}: {\mathbf {f}}({\mathbb {R}})^{-1}(R^o)\rightarrow R^o \end{aligned}$$

which extends $\sigma ({\underline{M}})\circ \pi ({\mathbb {R}})|_{R^o\cap {\mathbf {I}}}$. It has a unique extension $s:R\rightarrow {\mathcal {X}}({\mathbb {R}})$ such that $s\circ f({\mathbb {R}})\circ \sigma ({\underline{M}})$ is $\sigma (\underline{M})$.

For every point x on ${\mathbf {I}}$ let L(x) denote the tangent line of ${\mathbf {f}}({\mathbb {R}})\circ \sigma ({\underline{M}})$ at x. Note that for every $x\in {\mathbf {I}}$ we have $s(x)=\sigma (\underline{M})(\pi ({\mathbb {R}})(x))$, and $\sigma ({\underline{M}})$ is a Nash section, so s(x) lies in ${\mathcal {X}}_{L(x),sm}({\mathbb {R}})$. Therefore it will be enough to show that there is an admissible semi-algebraic subset $R_+\subseteq R$ containing ${\mathbf {I}}$ such that for every kissing point x of $R_+$ on ${\mathbf {I}}$ the intersection of the closure of the image of s with $\mathcal X_x({\mathbb {R}})$ lies in the connected component of s(x) in ${\mathcal {X}}_{L(x),sm}$. Let K denote the set of kissing points of R.

For every x on ${\mathbf {I}}$ let ${\mathcal {X}}_{L(x),bad}$ be the complement of ${\mathcal {X}}_{L(x),sm}$ in ${\mathcal {X}}_x$. For every $x\in K$ the subscheme ${\mathcal {X}}_{L(x),bad}$ of ${\mathcal {X}}$ is Zariski-closed, so the semi-algebraic set $\mathcal X_{L(x),bad}({\mathbb {R}})$ is closed in ${\mathcal {X}}({\mathbb {R}})$, and does not contain s(x), hence by Lemma 6.5 we may pick two disjoint open semi-algebraic subsets $A_x,B_x\subset \mathcal X({\mathbb {R}})$ such that ${\mathcal {X}}_{L(x),bad}({\mathbb {R}})\subset A_x$ and $s(x)\in B_x$. Now let E be any semi-algebraic connected component of ${\mathcal {X}}_{L(x),sm}({\mathbb {R}})$ which does not contain s(x), and let ${\overline{E}}$ denote its closure in ${\mathcal {X}}_x({\mathbb {R}})$. Since E is closed in $\mathcal X_{L(x),sm}({\mathbb {R}})$, and s(x) lies in $\mathcal X_{L(x),sm}({\mathbb {R}})$, the set ${\overline{E}}$ does not contain s(x), so by Lemma 6.5 we may pick two disjoint open semi-algebraic subsets $A_E,B_E\subset {\mathcal {X}}({\mathbb {R}})$ such that $E\subset A_E$ and $s(x)\in B_E$.

For every $x\in K$ let $W_x$ be the intersection of $B_x$ and the $B_E$ for all E as above. The set of connected components of ${\mathcal {X}}_{L(x),sm}({\mathbb {R}})$ is finite, therefore the set $W_x$ is an open semi-algebraic neighbourhood of s(x) such the intersection of its closure with ${\mathcal {X}}_x({\mathbb {R}})$ lies in the connected component of s(x) in ${\mathcal {X}}_{L(x),sm}$. Since $\sigma ({\underline{M}})$ is continuous, for every x as above $\pi ({\mathbb {R}})(x)$ has an open connected neighbourhood $V_x$ in $C({\mathbb {R}})$ such that $\sigma ({\underline{M}})$ maps $V_x$ into $W_x$. We may assume that the sets $V_x$ are pair-wise disjoint by shrinking them, if this is necessary. For every $x\in K$ let $R_x$ be the intersection of $\pi ({\mathbb {R}})^{-1}(V_x)$ with the image of $W_x\cap {\mathbf {f}}({\mathbb {R}})^{-1}(R^o)$ with respect to ${\mathbf {f}}$. Since $R^o\subset {\mathcal {U}}({\mathbb {R}})$, the restriction of $\mathbf{f}$ onto $W_x\cap {\mathbf {f}}({\mathbb {R}})^{-1}(R^o)$ is a Nash submersion, and hence open by the implicit function theorem. Hence $R_x$ is open and semi-algebraic. Since $\sigma ({\underline{M}})$ is a section, the set $R_x$ also contains the image of $V_x-\pi (\mathbb R)(x)$ with respect to ${\mathbf {f}}({\mathbb {R}})\circ \sigma (\underline{M})$.

Let T be the interior of the complement of the union of $V_x$ for all $x\in K$ in $C({\mathbb {R}})$. Then the set

$$\begin{aligned} R_{++}=\Big (\pi ({\mathbb {R}})^{-1}(T)\cap R^o\Big )\cup \bigcup _{x\in K} R_x\subset R^o \end{aligned}$$

is open, semi-algebraic and contains all but finitely many points of ${\mathbf {I}}$. Therefore $R_+=R_{++}\cup {\mathbf {I}}$ is an admissible subset of R. For every $x\in K$ the intersection of the closure of the image of $s|_{R_+}$ with ${\mathcal {X}}_x({\mathbb {R}})$ lies in the connected component of s(x) in ${\mathcal {X}}_{L(x),sm}$ by construction. If x is a kissing point of $R_+$ not in K, then $x\in R^o$, so s is continuous at x, and hence the intersection of the closure of the image of $s|_{R_+}$ with ${\mathcal {X}}_x({\mathbb {R}})$ is just s(x). $\square$

Let $X_{sm}\subseteq X$ be the smooth locus of f.

Lemma 6.6

Let x be a point in $C({\mathbb {R}}),$ let $y\in X_{sm}({\mathbb {R}}(C)_x)$ be a point,  and let ${\mathbf {U}}\subset X({\mathbb {R}}(C)_x)$ be an open neighbourhood of y in the x-adic topology. Then there is an open neighbourhood ${\mathbf {V}}$ of f(y) in the x-adic topology such that for every $z\in {\mathbf {V}}$ the set $f^{-1}(z)({\mathbb {R}}(C)_x)\cap {\mathbf {U}}$ is non-empty.

Proof

By the implicit function theorem there is an open neighbourhood ${\mathbf {W}}$ of f(y) in the x-adic topology and a section ${\mathbf {s}}:{\mathbf {W}}\rightarrow X_{sm}({\mathbb {R}}(C)_x)$ of the x-adic analytic map $X_{sm}({\mathbb {R}}(C)_x)\rightarrow {\mathbb {P}}^1({\mathbb {R}}(C)_x)$ induced by f such that ${\mathbf {s}}(f(y))=y$. This is continuous with respect to the x-adic topology. Therefore there is an open neighbourhood ${\mathbf {V}}\subset {\mathbf {W}}$ of f(y) in the x-adic topology such that ${\mathbf {s}}({\mathbf {V}})\subset {\mathbf {U}}$. For every $z\in {\mathbf {V}}$ we have ${\mathbf {s}}(z)\in f^{-1}(z)({\mathbb {R}}(C)_x)\cap {\mathbf {U}}$. $\square$

For every $x\in I$ let $P_x$ denote the formal completion of the Nash section $\sigma ({\underline{M}})$ around $\sigma (\underline{M})(x)$; it is a section ${\text {Spec}}({\mathcal {O}}_x)\rightarrow \mathcal X\times _C{\text {Spec}}({\mathcal {O}}_x)$. By slight abuse of notation let the same symbol $P_x$ denote its generic fibre, too. Since $\sigma ({\underline{M}})$ only intersects ${\mathcal {Z}}({\mathbb {R}})$ in finitely many points, we have $P_x\in X_{sm}({\mathbb {R}}(C)_x)$. By property (ii) of Proposition 6.1 we have $P_x\in U_x$ for every $x\in I$, so by Lemma 6.6 for every $x\in I$ there is an open neighbourhood $V_x$ of $f(P_x)$ in the x-adic topology such that for every $z\in V_x$ the set $f^{-1}(z)({\mathbb {R}}(C)_x)\cap U_x$ is non-empty. We may assume that $V_x={\mathbb {P}}^1({\mathbb {R}}(C)_x)$ whenever $U_x=X({\mathbb {R}}(C)_x)$. There is an interpolation condition ${\widetilde{\kappa }}:\widetilde{T}\rightarrow {\mathbb {P}}^1$ compatible with ${\mathbf {f}}(\mathbb R)\circ \sigma ({\underline{M}})$ such

$$\begin{aligned} U({\widetilde{T}},{\widetilde{\kappa }})\subset \prod _{x\in I}V_x\times \prod _{x\in C({\mathbb {R}})-I} {\mathbb {P}}^1({\mathbb {R}}(C)_x). \end{aligned}$$

Now let $R_+\subseteq R$ be an admissible semi-algebraic subset containing the image ${\mathbf {I}}$ of ${\mathbf {f}}({\mathbb {R}})\circ \sigma ({\underline{M}})$ and let $s:R_+\rightarrow {\mathcal {X}}({\mathbb {R}})$ be a butterfly extension of $\sigma ({\underline{M}})$. By removing every kissing point of $R_+$ which does not lie on ${\mathbf {I}}$ we may even assume that every kissing point of $R_+$ lies on ${\mathbf {I}}$. Let $\kappa :T\rightarrow \mathbb P^1$ be an interpolation condition subsuming ${\widetilde{\kappa }}:T\rightarrow {\mathbb {P}}^1$, compatible with $\mathbf{f}({\mathbb {R}})\circ \sigma ({\underline{M}})$ such that for every kissing point z of $R_+$ the point $\pi ({\mathbb {R}})(z)$ has coefficient at least 2 in T.

By Theorem 4.2 there is a regular map $\mathbf{c}:C\rightarrow {\mathbb {P}}^1$ compatible with $\kappa$ such that the graph $\Gamma _{{\mathbf {c}}}$ of ${\mathbf {c}}$ in $C\times {\mathbb {P}}^1(\mathbb R)$ lies in $R_+$. Let $c\in {\mathbb {P}}^1({\mathbb {R}}(C))$ be the generic point of ${\mathbf {c}}$. Let ${\mathcal {X}}_c$ denote the closed subscheme ${\mathbf {f}}^{-1}(\Gamma _{{\mathbf {c}}})\subset {\mathcal {X}}$ and let $\pi _c:{\mathcal {X}}_c\rightarrow C$ be the composition of ${\mathbf {f}}$ and $\pi$. Since all but finitely many points of $\Gamma _{{\mathbf {c}}}$ lie in $R_+^o\subset {\mathcal {U}}({\mathbb {R}})$, the generic fibre $X_c$ of ${\mathcal {X}}_c$ (with respect to $\pi _c$) is smooth. Let ${\mathcal {X}}_{c,sm}\subseteq {\mathcal {X}}_c$ be the smooth locus of $\pi _c$. By resolution of singularities there is a sequence of blow-ups $r:\widetilde{{\mathcal {X}}}_c\rightarrow {\mathcal {X}}_c$ such that $\widetilde{{\mathcal {X}}}_c$ is a model of $X_c$ over C and contains ${\mathcal {X}}_{c,sm}$ as an open sub C-scheme, i.e. the restriction $r|_{r^{-1}({\mathcal {X}}_{c,sm})}:r^{-1}({\mathcal {X}}_{c,sm})\rightarrow \mathcal X_{c,sm}$ is an isomorphism.

Lemma 6.7

The image of $s\circ {\mathbf {c}}({\mathbb {R}})$ lies in ${\mathcal {X}}_{c,sm}({\mathbb {R}})$ and the composition

$$\begin{aligned} s\circ {\mathbf {c}}({\mathbb {R}}):C({\mathbb {R}})\rightarrow {\mathcal {X}}_{c,sm}({\mathbb {R}})\subset \widetilde{{\mathcal {X}}}_c({\mathbb {R}}) \end{aligned}$$

is weakly continuous.

Proof

Since restriction of ${\mathbf {f}}$ onto ${\mathbf {f}}^{-1}({\mathcal {U}})$ is smooth, and $R^o_+$ lies in ${\mathcal {U}}({\mathbb {R}})$, for every $z\in \Gamma _c({\mathbb {R}})\cap R^o_+$ the point s(z) lies in ${\mathcal {X}}_{c,sm}({\mathbb {R}})$. If $z\in \Gamma _c({\mathbb {R}})-R^o_+$ then z is a kissing point of $R_+$ lying on ${\mathbf {I}}$. Therefore the point $\pi ({\mathbb {R}})(z)$ has coefficient at least 2 in T, so the tangent lines of $\Gamma _c$ and ${\mathbf {I}}$ at z are the same; let L denote this tangent line. Note that the fibre of ${\mathcal {X}}_{c,sm}$ above z is ${\mathcal {X}}_{L,sm}$. Since $\sigma ({\underline{M}})$ is a Nash section, its tangent vector at s(x) maps non-trivially into L, and we get that s(x) lies in ${\mathcal {X}}_{c,sm}$ in this case, too. Also note that the closure of the image of $s\circ {\mathbf {c}}(\mathbb R)$, considered as a map into $\widetilde{{\mathcal {X}}}_c({\mathbb {R}})$, lies in the pre-image of the closure of the image of $s\circ \mathbf{c}({\mathbb {R}})$, considered as a map into ${\mathcal {X}}_c({\mathbb {R}})$, with respect to $r({\mathbb {R}})$. Therefore the second half of the claim follows from Proposition 3.21. $\square$

Now we can complete the proof of Theorem 5.3. Since $c\in U(T,\kappa )$, for every $x\in I$ the set $f^{-1}(c)(\mathbb R(C)_x)\cap U_x$ is non-empty, and we may choose $N_x$ from this set. Using Hensel’s lemma for every $x\in C({\mathbb {R}})-I$ we get that there is an $N_x\in X_c({\mathbb {R}}(C)_x)$ such that the special fibre of the section ${\text {Spec}}({\mathcal {O}}_x)\rightarrow \mathcal X\times _C{\text {Spec}}({\mathcal {O}}_x)$ associated to $N_x$ is the point $s\circ {\mathbf {c}}({\mathbb {R}})(x)$. Set ${\underline{N}}=\prod _{x\in C({\mathbb {R}})}N_x$. Clearly ${\underline{N}}\in U$. Moreover $\sigma ({\underline{N}})=s\circ {\mathbf {c}}({\mathbb {R}})$, which is weakly continuous by Lemma 6.7, so ${\underline{N}}\in X_c({\mathbb {A}}_C)^{CT}$ by Theorem 3.6. $\square$