point of $I\rightarrow T^{2}$, where $I$ is the set of $(x,t,u)$ with $x\in
C_{t}\cap C_{u}$. But $\hat{x}$ is generic in $C_{\hat{t}}$ over $\hat{t}$. By
the Baire category axiom, it follows that every neighborhood of $\hat{x}$
contains infinitely many points of $C_{\hat{t}}$. In particular, every
neighborhood of $(\hat{x},\hat{t},\hat{t})$ contains infinitely many points of
$I$ which map to $(\hat{t},\hat{t})\in T^{2}$. This is enough to prove the
claim. ∎

So $(\hat{x},\hat{t},\hat{t})$ is a generic multiple
$(\mathcal{C},\mathcal{C})$-intersection which is strongly approximable.
Moreover, $\mathcal{C}$ is a standard family over $\emptyset$ (since this is
weaker than being excellent). So, since $(\mathcal{K},\tau)$ has ramification
purity, there is a generic multiple $(\mathcal{C},\mathcal{C})$-intersection
over $\emptyset$ which has codimension at most 1 over $\emptyset$. Call this
intersection $(x,t,u)$. Note that, since $\mathcal{M}$ detects multiple
intersections, the parameters $t$ and $u$ are $\mathcal{M}$-dependent over
$\emptyset$.

We now have two bounds on $\dim(xtu)$ – a lower bound coming from the fact
that $\operatorname{codim}_{\emptyset}(x,t,u)\leq 1$, and an upper bound
coming from the $\mathcal{M}$-dependence of $t$ and $u$. The next two claims
make these explicit:

######  Claim 6.20.

$\dim(xtu)\geq 4\cdot\dim(M)-1$.

###### Proof.

Using that $(x,t)$ and $(x,u)$ are generic in $C$, it is straightforward to
see that $\dim(x)=2\cdot\dim(M)$, and $\dim(t/x)=\dim(u/x)=\dim(M)$. Since
$\operatorname{codim}_{\emptyset}(x,t,u)=1$, this implies $\dim(tu/x)\geq
2\cdot\dim(M)-1$. Thus, by additivity,

$\dim(xtu)=\dim(x)+\dim(tu/x)\geq 4\cdot\dim(M)-1.$

∎

######  Claim 6.21.

$\dim(xtu)\leq 3\cdot\dim(M)$.

###### Proof.

It is enough to show that $\operatorname{rk}(xtu)\leq 3$. Now since $t$ and
$u$ are $\mathcal{M}$-dependent over $\emptyset$, we have
$\operatorname{rk}(tu)\leq 3$. So if $\operatorname{rk}(x/tu)=0$, we are done.
Otherwise, we get that $C_{t}\cap C_{u}$ is infinite. By almost faithfulness,
$u$ is thus $\mathcal{M}$-algebraic over $t$, which gives that
$\operatorname{rk}(xtu)=\operatorname{rk}(xt)=3$. ∎

The theorem follows by combining the two bounds above. Namely, by the two
previous claims, we have $4\cdot\dim(M)-1\leq 3\cdot\dim(M)$. Rearranging
gives $\dim(M)\leq 1$, and since $M$ is infinite, we get $\dim(M)=1$. ∎

##  7\. Definable Slopes and Interpreting a Group

######  Assumption 7.1.

Throughout Section 7, in addition to Assumption 4.1, we assume that
$\dim(M)=1$.

Our goal in this section is to give conditions under which $\mathcal{M}$
interprets a strongly minimal group. Essentially, the conditions say that the
germ of a curve at a generic point is determined by a sequence of definable
approximations, each one of which adds one parameter over the previous one –
and that, moreover, these approximations can be ‘detected’ by $\mathcal{M}$ in
a certain sense. This setting is an abstraction of power series expansions,
where the definable approximations are Taylor polynomials.

Once we have our definable ‘Taylor expansions’, we then build a group in a
similar way to previous trichotomy papers (e.g. [HS24] and [Cas23]) – by
considering a collection of ‘slopes’ equipped with a composition operation. As
in [Cas23], we find it most convenient to work with groupoids rather than
groups, because this allows us to work solely with generic points of curves.
In the end, this is still sufficient to recover a group, by Hrushovski’s group
configuration theorem (see Fact 7.3).

###  7.1. Group Configurations

Our main tool for interpreting a group is Hrushovski’s group configuration
theorem, which is well-known in model theory. Let us now make explicit the
version of this theorem we will use:

######  Definition 7.2.

Let $A$ be a parameter set, and let

$\bar{s}=(s_{12},s_{13},s_{14},s_{23},s_{24},s_{34})$

be tuples in $\mathcal{M}^{\textrm{eq}}$. We say that $\bar{s}$ is a rank 1
group configuration in $\mathcal{M}$ over $A$ if the following hold:

  1. (1)

Each of the six points in $\bar{s}$ has rank 1 over $A$.

  2. (2)

Any two distinct points in $\bar{s}$ have rank 2 over $A$.

  3. (3)

$\operatorname{rk}(\bar{s}/A)=3$.

  4. (4)

If $i,j,k\in\\{1,2,3,4\\}$ with $i<j<k$ then
$\operatorname{rk}(s_{ij}s_{ik}s_{jk}/A)=2$.

The canonical example is as follows: suppose $\mathcal{M}$ is a strongly
minimal expansion of a group. Let $a,b,c\in M$ be independent generics, and
let $s_{12}=a$, $s_{23}=b$, $s_{34}=c$, $s_{13}=ba$, $s_{24}=cb$, and
$s_{14}=cba$. Then $\bar{s}$ is a rank 1 group configuration in $\mathcal{M}$
over $\emptyset$.

The main fact we need is:

######  Fact 7.3 (Hrushovski (see [Bou89])).

Suppose there is a rank 1 group configuration $\bar{s}$ in $\mathcal{M}$ over
$A$ as above. Then there are

  * •

$B\supset A$,

  * •

an $\mathcal{M}(B)$-interpretable group $G$ which is strongly minimal as an
interpretable set in $\mathcal{M}$, and

  * •

For each $i,j\in\\{1,2,3\\}$ with $i<j$, an $\mathcal{M}$-generic element
$g_{ij}$ over $B$,

such that:

  1. (1)

$g_{12}$ and $g_{23}$ are $\mathcal{M}$-independent over $B$,

  2. (2)

$g_{13}=g_{23}g_{12}$, and

  3. (3)

For each $i,j\in\\{1,2,3\\}$ with $i<j$, $g_{ij}$ and $s_{ij}$ are
$\mathcal{M}$-interalgebraic over $B$.

###  7.2. Definable Slopes

We now define our abstraction of Taylor expansions, as described above:
namely, we introduce the notion of $(\mathcal{K},\tau)$ having ‘definable
slopes’. This will involve several intermediate notions.

######  Definition 7.4.

Let $\mathcal{LHG}$ be the category of local homeomorphisms at generic points
in $(\mathcal{K},\tau)$, defined as follows:

  1. (1)

An object in $\mathcal{LHG}$ is a generic point of $K$ (over $\emptyset$).

  2. (2)

A morphism $x\rightarrow y$ is a germ of a local homeomorphism from $K$ to $K$
sending $x$ to $y$: that is, we consider homeomorphisms between neighorhoods
of $x$ and $y$, modulo agreement in a neighborhood of $x$.

######  Definition 7.5.

Let $f:x\rightarrow y$ be a morphism in $\mathcal{LHG}$. We call $f$ a basic
invertible arc if there are a set $X\subset K^{2}$ and a parameter set $A$
such that:

  1. (1)

$X$ is $\mathcal{K}(A)$-definable.

  2. (2)

$\dim(X)=1$, and each projection $X\rightarrow K$ is finite-to-one.

  3. (3)

$(x,y)$ is generic in $X$ over $A$.

  4. (4)

There are neighborhoods $U$ and $V$ of $x$ and $y$, respectively, so that
$X\cap(U\times V)$ is the graph of a homeomorphism of germ $f$.

######  Definition 7.6.

Let $f:x\rightarrow y$ be a morphism in $\mathcal{LHG}$. We call $f$ an
invertible arc if it is a composition of finitely many basic invertible arcs.

Suppose $X\subset K^{2}$ is $\mathcal{K}(A)$-definable and of dimension 1,
with each projection $X\rightarrow K$ finite-to-one, and let $(x,y)\in X$ be
generic over $A$. It follows from the generic local homeomorphism property
that $X$ determines a basic invertible arc $x\rightarrow y$.

######  Notation 7.7.

Let $x$, $y$, $X$, and $A$ be as above. We call the basic invertible arc
$x\rightarrow y$ determined by $X$ the arc of $X$ at $(x,y)$, denoted
$\alpha_{xy}(X)$.

It is easy to see that for each generic $x\in K$, the identity morphism
$\operatorname{id}:x\rightarrow x$ from $\mathcal{LHG}$ is a basic invertible
arc (witnessed by the diagonal in $K^{2}$). Moreover, by switching the roles
of the domain and target, one sees that basic invertible arcs are closed under
inverses. It follows that the invertible arcs form the morphisms of a groupoid
(in fact a subgroupoid of $\mathcal{LHG}$), with the same objects as
$\mathcal{LHG}$.

######  Definition 7.8.

We let $\mathcal{IA}_{\infty}$ be the groupoid of invertible arcs.

######  Remark 7.9.

In contrast, the basic invertible arcs typically do not form a groupoid (that
is, not every invertible arc is a basic invertible arc). For example, for
generic $x\in K$, the identity is the only basic invertible arc $x\rightarrow
x$ (as is easily checked from the definition); while other invertible arcs
$x\rightarrow x$ can typically be constructed as compositions of basic
invertible arcs $x\rightarrow y\rightarrow x$ where $y$ is generic over $x$.
In a sense, then, the reason we need compositions is to recover the full
collection of morphisms at points $(x,y)$ where $x$ and $y$ are dependent.

Note that $\mathcal{IA}_{\infty}$ is typically not a definable object in any
reasonable sense. In contrast, the main point of our setting of ‘Taylor
approximations’ is to require that $\mathcal{IA}_{\infty}$ can be approximated
by definable objects:

######  Definition 7.10.

We say that $(\mathcal{K},\tau)$ has definable slopes if there are groupoids
$\\{\mathcal{I}A_{n}\\}_{n\in\mathbb{Z}^{+}}$, and covariant functors
$T_{m,n}:\mathcal{IA}_{m}\rightarrow\mathcal{IA}_{n}$ for all
$n<m\in\mathbb{Z}^{+}\cup\\{\infty\\}$, such that:

  1. (1)

The objects of each $\mathcal{IA}_{n}$ are the same as in
$\mathcal{IA}_{\infty}$, and all $T_{m,n}$ are the identity on objects.

  2. (2)

The $T_{m,n}$ are compatible, and $\mathcal{IA}_{\infty}$ is the inverse limit
of the $\mathcal{IA}_{n}$ for $n<\infty$. More precisely, if $m>n>k$ then
$T_{mk}=T_{nk}\circ T_{nm}$, and for any two objects $x$ and $y$, the
$T_{m,n}$ turn the set of morphisms $x\rightarrow y$ in
$\mathcal{IA}_{\infty}$ into the inverse limit of the sets of morphisms
$x\rightarrow y$ in the $\mathcal{IA}_{n}$ for $n<\infty$.

  3. (3)

Let $f:x\rightarrow y$ be a morphism in $\mathcal{IA}_{n}$ for some
$n<\infty$. Then $f$ is a tuple in $\mathcal{K}^{eq}$, and $(x,y)$ is
$\mathcal{K}(f)$-definable.

  4. (4)

(Uniform Definability of Slopes) Let $T$ and $X\subset K^{2}\times T$ be
$\mathcal{K}(A)$-definable. Assume that for each $t\in T$, the fiber
$X_{t}\subset K^{2}$ has dimension 1 and projects finite-to-one to both copies
of $K$. Then for each $n$, the set of $(x,y,t,\alpha)$ such that $(x,y)$ is
generic in $X_{t}$ over $At$ and $T_{\infty,n}(\alpha_{xy}(X_{t}))=\alpha$, is
type-definable in $\mathcal{K}$ over $A$.

  5. (5)

(Definability of Composition and Inverse) If $f:x\rightarrow y$ and
$g:y\rightarrow z$ are morphisms in $\mathcal{IA}_{n}$ for some $n<\infty$,
then $g\circ f$ is $\mathcal{K}(f,g)$-definable, and $f^{-1}$ is
$\mathcal{K}(f)$-definable.

  6. (6)

(Coordinatization of Slopes) Suppose $f:x\rightarrow y$ is a morphism in
$\mathcal{IA}_{n+1}$ for some $n<\infty$, and let $g=T_{n+1,n}(f)$. Then $g$
is $\mathcal{K}(f)$-definable, and $\dim(f/g)\leq 1$.

######  Remark 7.11.

Definition 7.10(2) (particularly the clause about inverse limits) is a strong
assumption: it says that the germ of a curve at a generic point is determined
by its sequence of $n$th approximations (at that point) for every $n$. For
instance, this property holds in ACVF and in polynomially bounded o-minimal
structures, but we don’t know if it holds in arbitrary o-minimal structures
(because of the existence of ‘flat functions’ – non-constant functions with
all derivatives vanishing at some point).

On the other hand, we will only use the inverse limit property once (in the
proof of Lemma 7.22), to guarantee the existence of an infinite family of
$n$-slopes for some $n$. We note, then, that the statement of Lemma 7.22 can
be proven in many other settings (including all o-minimal fields) using a
version of Sard’s Theorem. For this to work, one just needs a differentiable
Hausdorff geometric structure whose differential structures is definable in an
appropriate sense.

Finally, we note that one could amend all of Section 7 accordingly in this
case – taking Lemma 7.22 as an axiom instead of the inverse limit clause of
(2) – and all the arguments would still work. The reason we don’t do this is
that Lemma 7.22 rather awkwardly involves the relic $\mathcal{M}$, and we
prefer Definition 7.10 to be purely a property of $(\mathcal{K},\tau)$.

######  Assumption 7.12.

For the rest of Section 7, we assume that $(\mathcal{K},\tau)$ has definable
slopes, witnessed by the fixed groupoids and functors $\mathcal{IA}_{n}$ and
$T_{m,n}$.

######  Remark 7.13.

We think of the functors $T_{m,n}$ as truncation maps. Given a morphism $f$ in
$\mathcal{IA}_{m}$ for some $m$, and given some $n<m$, we often call
$T_{m,n}(f)$ the $n$th truncation of $f$.

######  Definition 7.14.

Assume that $(\mathcal{K},\tau)$ has definable slopes, witnessed by
$\\{\mathcal{IA}_{n}\\}$ and $\\{T_{m,n}\\}$. Let $X\subset K^{2}$ be
$\mathcal{K}(A)$-definable of dimension 1, with both projections $X\rightarrow
K$ finite-to-one, and let $(x,y)\in X$ be generic over $A$. For $n<\infty$, we
call $T_{\infty,n}(\alpha_{xy}(X))$ the $n$-slope of $X$ at $(x,y)$, denoted
$\alpha_{xy}^{n}(X)$.

In this language, by an easy compactness argument, the uniform definability of
slopes (Definition 7.10(4)) has the following simpler consequence:

######  Lemma 7.15.

Let $X\subset K^{2}$ be $\mathcal{K}(A)$-definable of dimension 1, with both
projections finite-to-one. Let $(x,y)\in X$ be generic over $A$. Then for each
$n<\infty$, $\alpha_{xy}^{n}(X)\in\operatorname{dcl}_{\mathcal{K}}(Axy)$.

######  Remark 7.16.

Definition 7.10(4) is stronger than Lemma 7.15, because Lemma 7.15 does not
capture the uniformity of the way slopes are defined. This will matter in a
couple of places, but usually Lemma 7.15 will suffice.

###  7.3. Coherent Slopes and Codes

In this subsection, we give axioms for an encoding system of slopes into
$\mathcal{M}$. The idea is to assign each slope to an interalgebraic tuple in
$\mathcal{M}$, so that the model-theoretic properties of $\mathcal{K}$
appearing in Definition 7.10 can be replaced with the analogous properties in
$\mathcal{M}$. For this to have a chance of working, we first need to assign
slopes to plane curves in $M$ (rather than sets in $K^{2}$). We then need to
identify a class of coherent slopes – roughly, those appearing on coherently
defined plane curves.

Let us begin by ‘lifting’ slopes to $M$. Since $\dim(M)=1$, and $M$ is
$\mathcal{K}(\emptyset)$-definable, it follows that there is a finite-to-one
$\mathcal{K}(\emptyset)$-definable function $\rho:M\rightarrow K$ (one can
construct such a function piecewise by projections). For the rest of Section
7, we fix such a function $\rho$. Abusing notation, we also write
$\rho:M^{n}\rightarrow K^{n}$ for all $n$, applying coordinate-wise.

Suppose $C\subset M^{2}$ is $\mathcal{K}(A)$-definable of dimension 1 with
both projections $C\rightarrow M$ finite-to-one. Let $(x,y)\in C$ be generic
over $A$. Then $\rho(C)\subset K^{2}$ is $A$-definable and projects finite-to-
one in both directions, and $(\rho(x),\rho(y))$ is generic in $\rho(C)$ over
$A$. By the generic local homeomorphism property, $\rho$ induces a
homeomorphism between open subsets of $M^{2}$ and $K^{2}$ near $(x,y)$ and
$(\rho(x),\rho(y))$, and also between $C$ and $\rho(C)$ near $(x,y)$ and
$(\rho(x),\rho(y))$. This observation allows us, in a neighborhood of $(x,y)$,
to think of $C$ as a curve in $K^{2}$, and subsequently to extract slopes at
$(x,y)$:

######  Definition 7.17.

Let $C\subset M^{2}$, $A$, and $(x,y)$ be as in the above paragraph. For each
$n$, we define the $n$-slope of $C$ at $(x,y)$, denoted $\alpha_{xy}^{n}(X)$,
to be the $n$-slope of $\rho(C)$ at $(\rho(x),\rho(y))$.

We now proceed to discuss coherent slopes in $M$. Recall (see Remark 5.3) that
a plane curve $C\subset M^{2}$ is non-trivial if each projection $C\rightarrow
M$ is finite-to-one. Now we define:

######  Definition 7.18.

Let $n$ be a positive integer. By a coherent $n$-slope configuration, we mean
a 5-tuple $(x,y,C,c,\alpha)$ such that:

  1. (1)

$C\subset M^{2}$ is a stationary non-trivial plane curve in the sense of
$\mathcal{M}$ (so in particular $C$ is $\mathcal{M}$-definable), and
$c=\operatorname{Cb}(C)$.

  2. (2)

$c$ is coherent, and $(x,y)$ is generic in $C$ over $c$ (so $cxy$ is
coherent).

  3. (3)

$\alpha_{xy}^{n}(C)=\alpha$.

######  Definition 7.19.

Let $x,y\in M$, and let $\alpha:\rho(x)\rightarrow\rho(y)$ be a morphism in
$\mathcal{IA}_{n}$ for some $n<\infty$.

  1. (1)

We say that $\alpha$ is a coherent $n$-slope at $(x,y)$ if there is a coherent
$n$-slope configuration $(x,y,C,c,\alpha)$.

  2. (2)

If $(x,y,C,c,\alpha)$ is a coherent $n$-slope configuration, we say that $c$
is a coherent representative of $\alpha$.

  3. (3)

Suppose $\alpha$ is a coherent $n$-slope at $(x,y)$. If
$\alpha\in\operatorname{acl}(xy)$, we call $\alpha$ an algebraic coherent
$n$-slope at $(x,y)$. Otherwise, we call $\alpha$ a non-algebraic coherent
$n$-slope at $(x,y)$.

We now give various properties of coherent slopes. Lemma 7.20 is essentially a
restatement of Definition 7.10 (3) and (4), but is quite useful. Roughly, it
allows us to perform dimension computations with coherent slopes, while (in
the presence of any coherent representative) treating the slopes as elements
of $\mathcal{M}^{\textrm{eq}}$.

######  Lemma 7.20.

Let $\alpha$ be a coherent $n$-slope at $(x,y)$, and let $c$ be any coherent
representative of $\alpha$.

  1. (1)

$(x,y)\in\operatorname{acl}(\alpha)$.

  2. (2)

$\alpha$ is definable over $cxy$ (that is,
$\alpha\in\operatorname{dcl}_{\mathcal{K}}(cxy)$).

  3. (3)

$\alpha$ and $xy$ are interalgebraic over $c$.

###### Proof.

Clause (2) follows from Lemma 7.15 and the fact that $\rho$ is
$\mathcal{K}(\emptyset)$-definable. Clause (3) follows from (1) and (2). For
(1), since $\alpha$ is a morphism from $\rho(x)$ to $\rho(y)$, Definition
7.10(3) gives that $\rho(x)$ and $\rho(y)$ are
$\mathcal{K}(\alpha)$-definable. Meanwhile, since $\rho$ is finite-to-one, $x$
and $y$ are algebraic over $(\rho(x),\rho(y))$. ∎

It will also be convenient to know that, when working with non-algebraic
slopes, we can upgrade Definition 7.19 to include the genericity of $(x,y)$ in
$M^{2}$:

######  Lemma 7.21.

Let $\alpha$ be a c coherent $n$-slope at $(x,y)$. Then:

  1. (1)

If $\alpha\notin\operatorname{acl}(x,y)$, then $(x,y)$ is generic in $M^{2}$.

  2. (2)

If $(x,y)$ is generic in $M^{2}$, and $c$ is any coherent representative of
$\alpha$, then $\dim(c/xy)=\dim(c)-1$.

###### Proof.

For (1), suppose toward a contradiction that $(x,y)$ is not generic in
$M^{2}$. Then by coherence, $\operatorname{rk}(xy)\leq 1$. But
$\operatorname{rk}(xy/c)=1$ by assumption, so
$\operatorname{rk}(xy/c)=\operatorname{rk}(xy)=1$. So
$p=\operatorname{tp}_{\mathcal{M}}(xy/c)$ does not fork over $\emptyset$. This
implies that
$c=\operatorname{Cb}(p)\in\operatorname{acl}_{\mathcal{M}}(\emptyset)$. Then
by Lemma 7.20,
$\alpha\in\operatorname{acl}(cxy)\subset\operatorname{acl}(xy)$, so $\alpha$
is algebraic.

For (2), let $c$ be any coherent representative. Then $\dim(xy/c)=1$, so
$\dim(cxy)=\dim(c)+1$. On the other hand, $\dim(xy)=2$ by assumption. So

$\dim(c/xy)=(\dim(c)+1)-2=\dim(c)=1.$

∎

Later, under further assumptions, we will interpret a strongly minimal group
in $\mathcal{M}$ by finding codes for a one-dimensional family of coherent
slopes. Lemmas 7.22 and 7.25 guarantee that such one-dimensional families
exist.

######  Lemma 7.22.

There is a non-algebraic coherent $n$-slope for some $n$.

###### Proof.

Fix any stationary non-trivial plane curve $C$ with coherent canonical base
$c$, and fix a generic element $(x,y)\in C$ over $c$. So for each $n<\infty$,
$\alpha_{xy}^{n}(C)$ is a coherent $n$-slope at $(x,y)$.

Since the category $\mathcal{IA}_{\infty}$ is the inverse limit of the
$\mathcal{IA}_{n}$ for $n<\infty$ (see Definition 7.10(2)), the basic
invertible arc $\alpha_{\rho(x)\rho(y)}(\rho(C))$ (i.e. the germ of $\rho(C)$
at $(\rho(x),\rho(y))$) is determined by the truncations $\alpha_{xy}^{n}(C)$
for $n<\infty$. Since $\rho$ induces a local homeomorphism
$C\rightarrow\rho(C)$ near $(x,y)$, it follows that these truncations,
together with $(x,y)$, determine the germ of $C$ at $(x,y)$.

Let $A$ be the tuple of all $\alpha_{xy}^{n}(C)$ for $n<\infty$. It now
follows that $c=\operatorname{Cb}(c)$ is determined by $A$ – or more
precisely, $c\in\operatorname{dcl}_{\mathcal{K}}(A)$. Let us explain. By
$\aleph_{1}$-saturation, it is enough to show that for any
$c^{\prime}\models\operatorname{tp}_{\mathcal{K}}(c/A)$, we have
$c^{\prime}=c$. So fix such $c^{\prime}$. Then
$c^{\prime}=\operatorname{Cb}(C^{\prime})$ for some stationary non-trivial
plane curve $C^{\prime}$, also containing $(x,y)$ as a generic point. Now by
uniform definability of slopes, and since
$\operatorname{tp}_{\mathcal{K}}(c^{\prime}/A)=\operatorname{tp}_{\mathcal{K}}(c/A)$,
it follows that $\alpha_{xy}^{n}(C^{\prime})=\alpha_{xy}^{n}(C)$ for all
$n<\infty$. Then by the discussion from the above paragraph, $C^{\prime}$
agrees with $C$ in a neighborhood of $(x,y)$. But by the Baire category axiom
(see Remark 3.7), every neighborhood of $(x,y)$ in $C$ is infinite; so in fact
$C^{\prime}$ and $C$ have infinite intersection, and thus by strong minimality
they are almost equal. Thus, they have the same canonical base, i.e.
$c^{\prime}=c$.

Now assume that all coherent slopes are algebraic. Then, in the notation
above, we get $A\subset\operatorname{acl}(xy)$, and thus (by the previous
paragraph) $c\in\operatorname{acl}(xy)$. In particular, $\dim(c)\leq 2$. But,
by non-local modularity, $\dim(c)$ can be arbitrarily large, which is enough
to prove the lemma. ∎

######  Notation 7.23.

For the rest of section 7, we let $n_{0}$ be the smallest $n$ such that there
is a non-algebraic coherent $n$-slope.

######  Remark 7.24.

One could also define ‘non-algebraic coherent $n$-slopes’ relative to a
parameter set $A$ (so coherence is interpreted over $A$, and non-algebraicity
at $(x,y)$ means the slope is not algebraic over $Axy$). This would, a priori,
change the way that $n_{0}$ is defined. We note, however, that the value of
$n_{0}$ does not ultimately depend on parameters (which the reader can check
if desired). Thus, we find it simplest to stick to the parameter-free version
presented above.

######  Lemma 7.25.

Suppose $\alpha$ is a non-algebraic coherent $n_{0}$-slope at $(x,y)$.

  1. (1)

$\dim(\alpha/xy)=1$.

  2. (2)

If $c$ is any coherent representative of $\alpha$, then
$\dim(c/\alpha)=\dim(c/xy)-1$.

###### Proof.

Let $\beta$ be the $(n_{0}-1)$-th truncation of $\alpha$. By the minimality of
$n_{0}$, $\beta\in\operatorname{acl}(xy)$. By Definition 7.10(6),
$\dim(\alpha/\beta)\leq 1$. Thus $\dim(\alpha/xy)\leq 1$, and since $\alpha$
is non-algebraic, $\dim(\alpha/xy)=1$, proving (1).

For (2), let $c$ be any coherent representative of $\alpha$. Repeatedly using
Lemma 7.20 and (1), we compute:

$\dim(c/\alpha)=\dim(c/\alpha
xy)=\dim(c\alpha/xy)-\dim(\alpha/xy)=\dim(c/xy)-1.$

∎

Finally, we now end this subsection with our definition of codes of slopes in
$\mathcal{M}$. This is the key notion allowing us to recover composition of
slopes in $\mathcal{M}$.

######  Definition 7.26.

Let $\alpha$ be a coherent $n$-slope at $(x,y)$. A tuple $s$ from
$\mathcal{M}^{\textrm{eq}}$ is a code of $\alpha$ if the following hold:

  1. (1)

$(x,y)\in\operatorname{acl}_{\mathcal{M}}(s)$.

  2. (2)

For each coherent representative $c$ of $\alpha$,
$s\in\operatorname{acl}_{\mathcal{M}}(cxy)$.

  3. (3)

$\alpha$ and $s$ are $\mathcal{K}$-interalgebraic.

Note that codes might not exist – that is, if $\alpha$ is a coherent $n$-slope
at $(x,y)$, there is no reason in general to expect some
$s\in\mathcal{M}^{eq}$ to satisfy (1)-(3) in Definition 7.26. Moving forward,
one of our main jobs will be to find suitable topological conditions
guaranteeing that many slopes do have codes.

###  7.4. Detecting Composition when Codes Exist

Definition 7.10(5) says that slope composition is definable in an appropriate
sense. Our next goal is to transfer this property into $\mathcal{M}$, for
‘independent’ composition of coherent slopes that have codes. The idea is that
the composition of slopes is respected by the $\mathcal{M}$-definable
composition of plane curves.

The following notion will be useful in the next two sections:

######  Definition 7.27.

Let $a_{1},...,a_{n},b_{1},...,b_{n}\in\mathcal{K}^{\textrm{eq}}$. We say that
$a_{1},...,a_{n}$ are maximally independent over $b_{1},...,b_{n}$ if

$\dim(a_{1}...a_{n}/b_{1}...b_{n})=\dim(a_{1}/b_{1})+...+\dim(a_{n}/b_{n}).$

Note that if $b_{1}=...=b_{n}=b$, this is the same as saying that
$a_{1},...,a_{n}$ are independent over $b$.

The following transitivity property follows easily from additivity of
dimension:

######  Lemma 7.28.

$a_{1}b_{1},...,a_{n}b_{n}$ are maximally independent over $c_{1},...,c_{n}$
if and only if $a_{1},...,a_{n}$ are maximally independent over
$c_{1},...,c_{n}$ and $b_{1},...,b_{n}$ are maximally independent over
$a_{1}c_{1}...a_{n}c_{n}$.

######  Proposition 7.29.

Let $\alpha_{1}$ be a coherent $n$-slope at $(x_{0},y_{0})$, and let
$\alpha_{2}$ be a coherent $n$-slope at $(y_{0},z_{0})$. Assume that $x_{0}$
and $z_{0}$ are independent over $y_{0}$, and $\alpha_{1},\alpha_{2}$ are
maximally independent over $(x_{0},y_{0}),(y_{0},z_{0})$. Then
$\alpha_{3}=\alpha_{2}\circ\alpha_{1}$ is a coherent $n$-slope at
$(x_{0},z_{0})$. Moreover, if $s_{i}$ is a code of $\alpha_{i}$ for $i=1,2,3$,
then $s_{3}\in\operatorname{acl}_{\mathcal{M}}(s_{1}s_{2})$.

###### Proof.

Let $c_{1}$ and $c_{2}$ be coherent representatives of $\alpha_{1}$ and
$\alpha_{2}$, respectively. We may assume that $c_{1},c_{2}$ are maximally
independent over $\alpha_{1},\alpha_{2}$. Note also that
$(x_{0},y_{0}),(y_{0},z_{0})$ are maximally independent over $y_{0},y_{0}$ by
assumption. By repeated applications of Lemma 7.28, it then follows that
$c_{1},c_{2}$ are maximally independent over $y_{0},y_{0}$. Thus, $c_{1}$ and
$c_{2}$ are independent over $y_{0}$. But each $c_{i}$ is independent from
$y_{0}$ by assumption. It then follows from additivity of dimension that
$c_{1}$, $c_{2}$, and $y_{0}$ are independent over $\emptyset$. In particular,
$c_{1}c_{2}y_{0}$ is coherent, and $\dim(y_{0}/c_{1}c_{2})=1$. Clearly
$x_{0}$, $y_{0}$, and $z_{0}$ are pairwise $\mathcal{M}$-interalgebraic over
$c_{1}c_{2}$. Thus, we have:

######  Claim 7.30.

$c_{1}c_{2}x_{0}y_{0}z_{0}$ is coherent, and $\dim(x_{0}z_{0}/c_{1}c_{2})=1$.

Now let $C_{1}$ and $C_{2}$ be such that
$(x_{0},y_{0},C_{1},c_{1},\alpha_{1})$ and
$(y_{0},z_{0},C_{2},c_{2},\alpha_{2})$ are coherent $n$-slope configurations.
We may assume that each $C_{i}$ is $\mathcal{M}$-definable over $c_{i}$. Let
$C_{3}=C_{2}\circ C_{1}$ – that is, the set of $(x,z)$ such that for some $y$
we have $(x,y)\in C_{1}$ and $(y,z)\in C_{2}$. Then $C$ is a non-trivial plane
curve which is $\mathcal{M}(c_{1}c_{2})$-definable and (by Claim 7.30)
contains $(x_{0},z_{0})$ as a generic element over $c_{1}c_{2}$.

######  Claim 7.31.

$\alpha_{x_{0}z_{0}}^{n}(C_{3})=\alpha_{3}$.

###### Proof.

Let $q_{1}$ stand for $(x_{0},y_{0})$, $q_{2}$ stand for $(y_{0},z_{0})$, and
$q_{3}$ stand for $(x_{0},z_{0})$. By the generic local homeomorphism
property, for each $i=1,2,3$, the points of $C_{i}$ in a neighborhood of
$q_{i}$ form the graph of a local homeomorphism $f_{i}$. Shrinking these
neighborhoods if necessary, it is clear that $f_{3}=f_{2}\circ f_{1}$.

Now, for each $i$, let $\rho(f_{i})$ be the image of $f_{i}$ under $\rho$,
viewed as a local function on $K$ at $\rho(q_{i})$. It follows by definition
that each $\rho(f_{i})$ is an invertible arc, with $n$th truncation
$\alpha_{q_{i}}^{n}(C_{i})$. But since $\rho$ is a local homeomorphism
$M\rightarrow K$ near each of $x_{0}$, $y_{0}$, and $z_{0}$, it also follows
that $\rho(f_{3})=\rho(f_{2})\circ\rho(f_{1})$. So since the truncation maps
are functors, we get
$\alpha_{q_{3}}^{n}(C_{3})=\alpha_{q_{2}}^{n}(C_{2})\circ\alpha_{q_{1}}^{n}(C_{1})$.
Equivalently,
$\alpha_{x_{0}z_{0}}^{n}(C_{3})=\alpha_{2}\circ\alpha_{1}=\alpha_{3}$. ∎

Now since $(x_{0},z_{0})$ is generic in $C_{3}$, there is a strongly minimal
component $C$ of $C_{3}$, with canonical base $c_{3}$, such that
$(x_{0},z_{0})$ is generic in $C$ over $c_{3}$. Then
$c_{3}\in\operatorname{acl}_{\mathcal{M}}(c_{1}c_{2})$, so $c_{3}$ is
coherent. Moreover, by the strong frontier inequality, $C$ and $C_{3}$ agree
in a neighborhood of $(x_{0},z_{0})$, which shows that
$\alpha_{x_{0},z_{0}}^{n}(C)=\alpha_{x_{0},z_{0}}^{n}(C_{3})=\alpha_{3}$. It
follows that $(x_{0},z_{0},C,c_{3},\alpha_{3})$ is a coherent $n$-slope
configuration. So $\alpha_{3}$ is coherent, and $c_{3}$ is a coherent
representative of $\alpha_{3}$.

Now assume that $s_{i}$ is a code of $\alpha_{i}$ for each $i$. By the
definition of codes, we have
$s_{i}\in\operatorname{acl}_{\mathcal{M}}(c_{i}x_{0}y_{0}z_{0})$ for each $i$.
Since $c_{3}\in\operatorname{acl}_{\mathcal{M}}(c_{1}c_{2})$, this implies
$s_{1}s_{2}s_{3}\in\operatorname{acl}_{\mathcal{M}}(c_{1}c_{2}x_{0}y_{0}z_{0})$.
By Claim 7.30, $c_{1}c_{2}x_{0}y_{0}z_{0}$ is coherent, thus so is
$s_{1}s_{2}s_{3}$. But by the definability of composition (Definition
7.10(5)), $\alpha_{3}\in\operatorname{acl}(\alpha_{1}\alpha_{2})$. So by
interalgebraicity with the $s_{i}$ (Definition 7.26)(3),
$s_{3}\in\operatorname{acl}(s_{1}s_{2})$, and thus by coherence,
$s_{3}\in\operatorname{acl}_{\mathcal{M}}(s_{1}s_{2})$. ∎

######  Remark 7.32.

Note that the last clause of Proposition 7.29 requires that all three
$\alpha_{i}$ have codes. That is, we do not claim that if $\alpha_{1}$ and
$\alpha_{2}$ have codes, so does their composition.

###  7.5. Interpreting a Group when Codes Exist

Recall that $n_{0}$ is the smallest $n$ such that there is a non-algebraic
coherent $n$-slope. We now use Proposition 7.29 to interpret a group in
$\mathcal{M}$ assuming enough coherent $n_{0}$-slopes have codes. The
statement of Proposition 7.33 below is somewhat complicated. The reason is
that later on, in ACVF, we will need to analyze the specific structure of the
interpreted group – a task which will use the extra data in the proposition.
However, we stress the main point: if all coherent $n_{0}$-slopes have codes,
then $\mathcal{M}$ interprets a strongly minimal group.

######  Proposition 7.33.

Suppose every coherent $n_{0}$-slope has a code. Then there are

  * •

A parameter set $B$,

  * •

An $\mathcal{M}(B)$-interpretable group $(G,\cdot)$ which is strongly minimal
as in interpretable set in $\mathcal{M}$,

  * •

A generic element $y\in K$ over $\emptyset$,

  * •

For each $i,j\in\\{1,2,3\\}$ with $i<j$, an element $g_{ij}\in G$, and

  * •

For each $i,j\in\\{1,2,3\\}$ with $i<j$, a morphism $f_{ij}:y\rightarrow y$ in
$\mathcal{IA}_{n_{0}}$,

such that:

  1. (1)

$y$ is $\mathcal{K}(B)$-definable (that is,
$y\in\operatorname{dcl}_{\mathcal{K}}(B)$).

  2. (2)

The $(n_{0}-1)$-th truncation of each $f_{ij}$ is the identity at $y$.

  3. (3)

$g_{12}$ and $g_{23}$ are independent generics in $G$ over $B$.

  4. (4)

$g_{13}=g_{23}\cdot g_{12}$ and $f_{13}=f_{23}\circ f_{12}$.

  5. (5)

For each $i,j\in\\{1,2,3\\}$ with $i<j$, $g_{ij}$ and $f_{ij}$ are
interalgebraic over $B$.

###### Proof.

Our strategy is as follows. First, we use the definability of composition and
inverse of slopes to find a ‘group configuration’ in the sense of dimension in
$\mathcal{K}$, whose elements are non-algebraic $n_{0}$-slopes. Then we use
Proposition 7.29 to check that the slopes in the configuration are coherent,
and subsequently extract codes. Finally, we check that the tuple of codes in
the entire configuration is coherent, and conclude that the codes form a group
configuration in the sense of rank in $\mathcal{M}$.

Let us proceed. We first want to find an appropriate collection of points and
slopes to form a group configuration. The first two lemmas below accomplish
this.

######  Lemma 7.34.

Suppose there is a non-algebraic coherent $n_{0}$-slope at $(x_{0},y_{0})$ for
some $(x_{0},y_{0})\in M^{2}$. Then there is a non-algebraic coherent
$n_{0}$-slope at $(y_{0},x_{0})$.

###### Proof.

Suppose that $(x_{0},y_{0},C,c,\alpha)$ is a non-algebraic coherent
$n_{0}$-slope configuration. It is then easy to check that
$(y_{0},x_{0},C^{-1},c,\alpha^{-1})$ is a non-algebraic coherent $n_{0}$-slope
configuration, where $C^{-1}$ is the set of $(y,x)$ with $(x,y)\in C$. ∎

######  Lemma 7.35.

There are independent generic elements $x_{1},x_{2},x_{3},x_{4}\in M$ such
that each of $(x_{1},x_{2})$, $(x_{2},x_{3})$, and $(x_{3},x_{4})$ admits a
non-algebraic coherent $n_{0}$-slope.

###### Proof.

Let $(x_{1},x_{2})$ be such that there is a non-algebraic coherent
$n_{0}$-slope at $(x_{1},x_{2})$. By Lemma 7.21, $(x_{1},x_{2})$ is generic in
$M^{2}$. Let $x_{3}$ be a realization of
$\operatorname{tp}_{\mathcal{K}}(x_{1}/x_{2})$ independent over $x_{1}$, and
let $x_{4}$ be a realization of $\operatorname{tp}_{\mathcal{K}}(x_{2}/x_{3})$
independent over $x_{1}x_{2}$. So $x_{1},x_{2},x_{3},x_{4}$ are independent
generics in $M$, and all of $(x_{1},x_{2})$, $(x_{3},x_{2})$, and
$(x_{3},x_{4})$ realize the same type in $\mathcal{K}$. Thus, each of
$(x_{1},x_{2})$, $(x_{3},x_{2})$, and $(x_{3},x_{4})$ admits a non-algebraic
coherent $n_{0}$-slope, and by Lemma 7.34, so does $(x_{2},x_{3})$. ∎

######  Notation 7.36.

For the rest of the proof of Proposition 7.33, we adopt the following:

  * •

We fix $x_{1},x_{2},x_{3},x_{4}$ as provided by Lemma 7.35.

  * •

We fix non-algebraic coherent $n_{0}$-slopes $\alpha_{12}$ at $(x_{1},x_{2})$,
$\alpha_{23}$ at $(x_{2},x_{3})$, and $\alpha_{34}$ at $(x_{3},x_{4})$, chosen
so that $\alpha_{12},\alpha_{23},\alpha_{34}$ are maximally independent over
$x_{1}x_{2},x_{2}x_{3},x_{3}x_{4}$.

  * •

Finally, we let $\alpha_{13}=\alpha_{23}\circ\alpha_{12}$,
$\alpha_{24}=\alpha_{34}\circ\alpha_{23}$, and
$\alpha_{14}=\alpha_{34}\circ\alpha_{23}\circ\alpha_{12}$, and we denote the
tuple
$(\alpha_{12},\alpha_{23},\alpha_{34},\alpha_{13},\alpha_{24},\alpha_{14})$ by
$\overline{\alpha}$.

Lemma 7.37 follows easily from Lemmas 7.25 and the definability of composition
and inverse (Definition 7.10(5)). We leave the details to the reader.

######  Lemma 7.37.

$\overline{\alpha}$ forms a ‘one-dimensional group configuration in
$\mathcal{K}$ over $x_{1}x_{2}x_{3}x_{4}$’. That is:

  1. (1)

Each point in $\overline{\alpha}$ has dimension 1 over $x_{1}x_{2}x_{3}x_{4}$.

  2. (2)

Any two distinct points in $\overline{\alpha}$ have dimension 2 over
$x_{1}x_{2}x_{3}x_{4}$.

  3. (3)

$\dim(\overline{\alpha}/x_{1}x_{2}x_{3}x_{4})=3$.

  4. (4)

If $i,j,k\in\\{1,2,3,4\\}$ with $i<j<k$, then
$\dim(\alpha_{ij}\alpha_{jk}\alpha_{ik}/x_{1}x_{2}x_{3}x_{4})=2$.

By Lemmas 7.25 and 7.37, any two elements
$\alpha_{ij},\alpha_{kl}\in\overline{\alpha}$ are maximally independent over
$x_{i}x_{j},x_{k}x_{l}$. This allows us to apply Proposition 7.29 to the
elements of $\overline{\alpha}$. We first conclude:

######  Lemma 7.38.

Each of the six elements of $\overline{\alpha}$ is a coherent $n_{0}$-slope.

###### Proof.

$\alpha_{12}$, $\alpha_{23}$, and $\alpha_{14}$ are coherent by the way they
are chosen. Now repeatedly apply Proposition 7.29. We get first that each of
$\alpha_{13}$ and $\alpha_{24}$ is a composition of two maximally independent
coherent slopes, so is coherent. Finally, $\alpha_{14}$ is now coherent by the
same reasoning, by writing it as $\alpha_{34}\circ\alpha_{13}$. ∎

By Lemma 7.38 and the assumption of the proposition, each $\alpha_{ij}$ has a
code.

######  Notation 7.39.

We fix a code $s_{ij}$ for each $\alpha_{ij}$, and denote the tuple of
$s_{ij}$ by $\overline{s}$.

Arguing similarly to Lemma 7.38, we now conclude:

######  Lemma 7.40.

$\overline{s}$ is coherent.

###### Proof.

By definition of codes, each $s_{ij}$ is interalgebraic with $\alpha_{ij}$. It
follows that $s_{12},s_{23},s_{34}$ are maximally independent over
$x_{1}x_{2},x_{2}x_{3},x_{3}x_{4}$. It then follows easily that
$s_{12}s_{23}s_{34}$ is coherent over $x_{1}x_{2}x_{3}x_{4}$, and since
$x_{1}x_{2}x_{3}x_{4}$ is coherent by construction, we get that
$s_{12}s_{23}s_{34}$ is coherent. But by repeated instances of Proposition
7.29 (exactly as in Lemma 7.38), we have
$\overline{s}\in\operatorname{acl}_{\mathcal{M}}(s_{12}s_{23}s_{34})$. So
$\overline{s}$ is coherent. ∎

Now, since each $s_{ij}$ is interalgebraic with $\alpha_{ij}$, Lemma 7.37
holds with each $\overline{\alpha}$ replaced with $\overline{s}$. Since
$\overline{s}$ is coherent, all of the statements in Lemma 7.37 hold with rank
instead of dimension. Thus, $\overline{s}$ is a rank 1 group configuration in
$\mathcal{M}$ over $x_{1}x_{2}x_{3}x_{4}$, and thus, by Fact 7.3,
$\mathcal{M}$ interprets a strongly minimal group.

If we only wanted to find a group, we could stop here. However, let us now
apply Fact 7.3 more carefully in order to verify the extra conditions of
Proposition 7.33. First, we define:

  * •

Let $\overline{\beta}\overline{t}$ be an independent realization of
$\operatorname{tp}_{\mathcal{K}}(\overline{\alpha}\overline{s}/\operatorname{acl}(x_{1}x_{2}x_{3}x_{4}))$
over $\overline{\alpha}\overline{s}$ (by this we mean that for $i<j$ we have
distinguished tuples $\beta_{ij}$ and $t_{ij}$).

  * •

Let $A=\overline{t}x_{1}x_{2}x_{3}x_{4}$.

  * •

Let $y=\rho(x_{4})$.

  * •

For $i,j\in\\{1,2,3\\}$ with $i<j$, let
$f_{ij}=\beta_{j4}\circ\alpha_{ij}\circ\beta_{i4}^{-1}$.

We now check:

######  Lemma 7.41.

  1. (1)

$y$ is generic in $K$ over $\emptyset$.

  2. (2)

$y$ is $\mathcal{K}(A)$-definable.

  3. (3)

Each $\alpha_{ij}$ has the same $(n_{0}-1)$-th truncation as $\beta_{ij}$

  4. (4)

Each $f_{ij}$ is a morphism from $y$ to $y$ in $\mathcal{IA}_{n_{0}}$.

  5. (5)

The $(n_{0}-1)$-th truncation of each $f_{ij}$ is the identity at $y$.

  6. (6)

Each $f_{ij}$ is interalgebraic with $s_{ij}$ over $A$.

  7. (7)

$f_{13}=f_{23}\circ f_{12}$.

  8. (8)

$\overline{s}$ is a rank 1 group configuration in $\mathcal{M}$ over $A$.

###### Proof.

  1. (1)

Clear, because $y$ and $x_{4}$ are interalgebraic.

  2. (2)

Clear because $\rho$-is $\emptyset$-definable.

  3. (3)

By the minimality of $n_{0}$, the $(n_{0}-1)$-th truncation of $\alpha_{ij}$
is contained in $\operatorname{acl}(x_{1}x_{2}x_{3}x_{4})$, and $\alpha_{ij}$
and $\beta_{ij}$ realize the same type over
$\operatorname{acl}(x_{1}x_{2}x_{3}x_{4})$.

  4. (4)

Clear, because $f_{ij}$ is the composition of
$\beta_{i4}^{-1}:\rho(x_{4})\rightarrow\rho(x_{i})$,
$\alpha_{ij}:\rho(x_{i})\rightarrow\rho(x_{j})$, and
$\beta_{j4}:\rho(x_{j})\rightarrow\rho(x_{4})$.

  5. (5)

By (3), $\beta_{i4}$ and $\beta_{j4}$ have the same $(n_{0}-1)$-th truncations
as $\alpha_{i4}$ and $\alpha_{j4}$, respectively. So, the $(n_{0}-1)$-th
truncation of $f_{ij}=\beta_{j4}\circ\alpha_{ij}\circ\beta_{i4}^{-1}$ is the
same as that of

$\alpha_{j4}\circ\alpha_{ij}\circ\alpha_{i4}^{-1}=\alpha_{j4}\circ\alpha_{ij}\circ(\alpha_{j4}\circ\alpha_{ij})^{-1}$
$=\alpha_{j4}\circ\alpha_{ij}\circ\alpha_{ij}^{-1}\alpha_{j4}^{-1}=\mathrm{id}:y\rightarrow
y.$

  6. (6)

Clearly, $f_{ij}$ is interdefinable with $\alpha_{ij}$ over
$\overline{\beta}$. Since $\overline{\beta}\overline{t}$ and
$\overline{\alpha}\overline{s}$ realize the same $\mathcal{K}$-type over
$x_{1}x_{2}x_{3}x_{4}$ (and by definition of codes), it follows that
$\overline{\beta}$ is interalgebraic with
$\overline{t}x_{1}x_{2}x_{3}x_{4}=A$. Thus $f_{ij}$ is interalgebraic with
$\alpha_{ij}$ over $A$. Finally, again by definition of codes, $\alpha_{ij}$
is interalgebraic with $s_{ij}$ over $A$. Thus, $f_{ij}$ is interalgebraic
with $s_{ij}$ over $A$.

  7. (7)

We have

$f_{23}\circ
f_{12}=(\beta_{34}\circ\alpha_{23}\circ\beta_{24}^{-1})\circ(\beta_{24}\circ\alpha_{12}\circ\beta_{14}^{-1})$
$=\beta_{34}\circ(\alpha_{23}\circ\alpha_{12})\circ\beta_{14}^{-1}=\beta_{34}\circ\alpha_{13}\circ\beta_{14}^{-1}=f_{13}.$

  8. (8)

By construction (see Lemma 4.13(4)), $A\overline{s}$ is coherent, and $A$ is
$\mathcal{M}$-independent from $\overline{s}$ over $x_{1}x_{2}x_{3}x_{4}$. So
since $\overline{s}$ is a rank 1 group configuration in $\mathcal{M}$ over
$x_{1}x_{2}x_{3}x_{4}$, the same holds over $A$.

∎

Now using Lemma 7.41(8), we can apply Fact 7.3 to $\overline{s}$ and $A$. We
obtain:

######  Lemma 7.42.

There are

  * •

A parameter set $B\supset A$,

  * •

An $\mathcal{M}(B)$-interpretable group $G$ which is strongly minimal as an
interpretable set in $\mathcal{M}$, and

  * •

For each $i,j\in\\{1,2,3\\}$ with $i<j$, an $\mathcal{M}$-generic element
$g_{ij}\in G$ over $B$,

such that:

  1. (1)

$g_{12}$ and $g_{23}$ are $\mathcal{M}$-independent over $B$,

  2. (2)

$g_{13}=g_{23}\cdot g_{12}$, and

  3. (3)

For each $i,j\in\\{1,2,3\\}$ with $i<j$, $g_{ij}$ and $s_{ij}$ are
$\mathcal{M}$-interalgebraic over $B$.

We can thus add to our above list of conclusions:

######  Lemma 7.43.

Each $g_{ij}$ is interalgebraic with $f_{ij}$ over $B$.

###### Proof.

Note that $s_{ij}$ is interalgebraic with each of them over $B$ (by Lemma
7.42(3) and Lemma 7.41(6)). ∎

At this point, Lemmas 7.41 and 7.43 cover all of the requirements of
Proposition 7.33, with one exception: we need $g_{12}$ and $g_{23}$ to be
independent generics in $G$ over $B$, and we only know they are
$\mathcal{M}$-independent $\mathcal{M}$-generics. To remedy this, note that
all of our initial assumptions on $B$ and the $g_{ij}$ (i.e. those in Lemma
7.42) are expressed solely in terms of the structure $\mathcal{M}$. So Lemmas
7.42 and 7.43 will still hold after replacing $Bg_{12}g_{23}g_{13}$ with any
tuple realizing the same $\mathcal{M}$-type over $A$. Now by Lemma 4.13(6),
there is one such tuple which is coherent over $A$ (in case $B$ is infinite,
we interpret coherence as saying that all finite subsets are coherent; in this
case, enumerate $B$ and apply Lemma 4.13(6) inductively). Then, by coherence,
we can transfer $\mathcal{M}$-genericity to full genericity (i.e. Lemma
4.13(1)). In particular, $g_{12}$ and $g_{23}$ are now independent generics in
$G$ over $B$, as desired. ∎

###  7.6. Detecting Tangency and Finding Codes

We now turn to the task of coding $n_{0}$-slopes. The following is an analog
of the notion of ‘detecting generic non-transversalities’ from [Cas23]:

######  Definition 7.44.

$\mathcal{M}$ detects tangency if whenever $(x,y)\in M^{2}$ is generic, and
$\alpha$ is a non-algebraic coherent $n$-slope at $(x,y)$ for some $n$, then
any two coherent representatives of $\alpha$ are $\mathcal{M}$-dependent over
$(x,y)$.

We will show that if $\mathcal{M}$ detects tangency, every coherent
$n_{0}$-slope has a code, and thus by Proposition 7.33,
$\mathcal{M}$-interprets a strongly minimal group. In [Cas23], one has a clear
construction of codes: indeed, in ACF0, a rank 2 family of plane curves
realizes every non-algebraic coherent $n_{0}$-slope finitely many times, and
one can code a slope using its finite set of occurrences in any such family.
However, in the current setting, there is no reason that a non-algebraic
coherent slope must occur in a rank 2 family. So we might have an infinite set
of occurrences in a larger family, any two of which are
$\mathcal{M}$-dependent by assumption, and we want to ‘code’ this infinite
set. The proper tool for doing such a thing is canonical bases. As it turns
out, we will code each slope using the canonical base of the dependence
between two independent occurrences of that slope in a family.

######  Proposition 7.45.

If $\mathcal{M}$ detects tangency, then every coherent $n_{0}$-slope has a
code.

###### Proof.

The idea is to use the detection of tangency to $\mathcal{M}$-definably
organize the curves in a family by slope, and then take the ‘equivalence
classes’ (i.e. canonical bases) to be the codes. For this to work, we need to
know that the $\mathcal{M}$-dependence between coherent representatives in
Definition 7.44 is not ‘coarser’ than the tangency relation, so that the
canonical bases we take are genuinely capturing curves up to tangency. This
will work because we are only considering one-dimensional families of slopes
(i.e. by the minimality of $n_{0}$). The precise statements we need are given
in Lemmas 7.46 and 7.47 below:

######  Lemma 7.46.

Let $\alpha$ be a coherent $n_{0}$-slope at $(x,y)$. Let $c_{1}$ and $c_{2}$
be coherent representatives of $\alpha$, and assume that $c_{1}$ and $c_{2}$
are independent over $\alpha$. Then $c_{2}$ is coherent over $c_{1}xy$.

###### Proof.

If $\alpha\in\operatorname{acl}(xy)$, then $c_{1}$ and $c_{2}$ are independent
over $xy$. By assumption, each $c_{i}$ is coherent over $xy$, so it follows
that $c_{1}c_{2}$ is coherent over $xy$, making the lemma obvious. Thus, we
assume $\alpha$ is not algebraic.

By Lemma 4.11, $\dim(c_{2}/c_{1}xy)\leq\operatorname{rk}(c_{2}/c_{1}xy)$. We
prove the reverse inequality.

Since $\mathcal{M}$ detects tangency,
$\operatorname{rk}(c_{2}/c_{1}xy)\leq\operatorname{rk}(c_{2}/xy)-1$. By
assumption $c_{2}xy$ is coherent, so
$\operatorname{rk}(c_{2}/xy)-1=\dim(c_{2}/xy)-1$. By Lemma 7.25,
$\dim(c_{2}/xy)-1=\dim(c_{2}/\alpha)$. Since the $c_{i}$ are independent over
$\alpha$, $\dim(c_{2}/\alpha)=\dim(c_{2}/c_{1}\alpha)$. By Lemma 7.20,
$\dim(c_{2}/c_{1}\alpha)=\dim(c_{2}/c_{1}xy)$.

Putting together everything in the last paragraph gives $\
rk(c_{2}/c_{1}xy)\leq\dim(c_{2}/c_{1}xy)$, which proves the lemma. ∎

######  Lemma 7.47.

Let $\alpha$ be a coherent $n_{0}$-slope at $(x,y)$. Let $c$, $c_{1}$, and
$c_{2}$ be coherent representatives of $\alpha$, and assume $c$ is independent
from $c_{1}c_{2}$ over $\alpha$. Then
$\operatorname{stp}_{\mathcal{M}}(c/c_{1}xy)$ and
$\operatorname{stp}_{\mathcal{M}}(c/c_{2}xy)$ are parallel (i.e. have a common
non-forking extension).

###### Proof.

By transitivity, we have (1) $c$ is independent from $c_{1}$ over $\alpha$,
and (2) $c$ is independent from $c_{2}$ over $c_{1}\alpha$ – equivalently, by
Lemma 7.20, over $c_{1}xy$. By (1) and Lemma 7.46, $c$ is coherent over
$c_{1}xy$. So, by (2) and Lemma 4.11(4), $c$ is $\mathcal{M}$-independent from
$c_{2}$ over $c_{1}xy$. Thus,
$\operatorname{stp}_{\mathcal{M}}(c/c_{1}c_{2}xy)$ is a non-forking extension
of $\operatorname{stp}_{\mathcal{M}}(c/c_{1}xy)$. By a symmetric argument,
$\operatorname{stp}_{\mathcal{M}}(c/c_{1}c_{2}xy)$ is also a non-forking
extension of $\operatorname{stp}_{\mathcal{M}}(c/c_{2}xy)$. ∎

We now prove the proposition. Let $\alpha$ be a coherent $n_{0}$-slope at
$(x,y)$. Let $c_{1}$ be a coherent representative of $\alpha$. Let $c_{2}$ be
an independent realization of $\operatorname{tp}_{\mathcal{K}}(c_{1}/\alpha)$
over $c_{1}$. Let $p=\operatorname{stp}_{\mathcal{M}}(c_{2}/c_{1}xy)$, and let
$b=\operatorname{Cb}(p)$. We show that $s=bxy$ is a code of $\alpha$, by
verifying (1), (2), and (3) in Definition 7.26:

  1. (1)

It is obvious that $(x,y)\in\operatorname{acl}_{\mathcal{M}}(s)$, since
$(x,y)$ is included in $s$.

  2. (2)

Let $c$ be a coherent representative of $\alpha$. We show that
$s\in\operatorname{acl}_{\mathcal{M}}(cxy)$. It is enough to show that
$b\in\operatorname{acl}_{\mathcal{M}}(cxy)$.

By replacing $c_{2}$ with an independent realization if necessary, we may
assume that $c_{2}$ is independent from $c$ over $c_{1}xy$ (note that this
does not change $b$). In particular, this easily implies that $c_{2}$ is
independent from $c_{1}c$ over $\alpha$. Then by Lemma 7.47,
$q=\operatorname{stp}_{\mathcal{M}}(c_{2}/cxy)$ is parallel to $p$, so has
canonical base $b$. Since $q$ is a type over $cxy$, this implies that
$b\in\operatorname{acl}_{\mathcal{M}}(cxy)$.

  3. (3)

We now have to show that $s$ and $\alpha$ are interalgebraic. First, by (2)
and Lemma 7.20, we have

$s\in\operatorname{acl}(c_{1}\alpha)\cap\operatorname{acl}(c_{2}\alpha).$

Since the $c_{i}$ are independent over $\alpha$, this implies
$s\in\operatorname{acl}(\alpha)$.

Next, we show that $s$ is coherent. Indeed, since $p$ is a type over
$c_{1}xy$, we have $b\in\operatorname{acl}_{\mathcal{M}}(c_{1}xy)$. But
$c_{1}xy$ is coherent by assumption, thus so is $c_{1}bxy$, and thus so is
$s$.

Now toward a contradiction, assume that
$\alpha\notin\operatorname{acl}(s)=\operatorname{acl}(bxy)$. By Lemma 7.25,
$\dim(\alpha/xy)=1$. So if $s\in\operatorname{acl}(\alpha)$ but
$\alpha\notin\operatorname{acl}(s)$, then $b\in\operatorname{acl}(xy)$. Since
$s$ is coherent, this implies $b\in\operatorname{acl}_{\mathcal{M}}(xy)$. Thus
$p$ does not fork over $xy$, and thus $c_{1}$ and $c_{2}$ are
$\mathcal{M}$-independent over $xy$. This contradicts that $\mathcal{M}$
detects tangency.

∎

###  7.7. Tangent and Multiple Intersections

We now give conditions under which $\mathcal{M}$ detects tangency. The idea is
to require that an ‘unusually high slope agreement’ between plane curves is
always a ‘topological multiple intersection.’ This is analogous to similar
facts regarding intersection multiplicity that were utilized in past
trichotomy papers (e.g. Bezout’s theorem in [Rab93] and the argument
principle/Rouche’s theorem in [MP90]).

In characteristic $p$ environments, it is possible for any two curves in a
family to be first-order tangent at all of their intersection points. However,
under mild assumptions, there is always a larger $n$ so that no two
independent generic curves are $n$th order tangent at a point. For this
reason, the notion of ‘unusually high’ slope agreement is slightly
complicated. The informal idea is that for some $n$, the two curves should
have the same $n$-slope even though most nearby pairs of curves do not. To
make this precise, we will use the machinery of weakly generic intersections
developed in section 6. We will specifically work with families of
correspondences in $K$:

######  Definition 7.48.

A $\mathcal{K}(A)$-definable family of correspondences in $K$ is a
$\mathcal{K}(A)$-definable family $\mathcal{X}=\\{X_{t}:t\in T\\}$ of subsets
of $K^{2}$, such that each $X_{t}\subset K^{2}$ has dimension 1 and projects
finite-to-one to both copies of $K$.

In addition to the terminology on weakly generic intersections developed in
section 6, we need the following notions involving slopes:

######  Definition 7.49.

Let $\mathcal{X}=\\{X_{t}:t\in T\\}$ and $\mathcal{Y}=\\{Y_{u}:u\in U\\}$ be
$\mathcal{K}(A)$-definable families of correspondences in $K$.

  1. (1)

If $n\in\mathbb{Z}^{+}$ and $V\subset K^{2}\times T\times U$ is any open set,
we say that $\mathcal{X}$ and $\mathcal{Y}$ have $n$-branching in $V$ over $A$
if for all strongly generic $(\mathcal{X},\mathcal{Y})$-intersections
$(x,y,t,u)$ over $A$, if $(x,y,t,u)\in V$ then
$\alpha_{xy}^{n}(X_{t})\neq\alpha_{xy}^{n}(Y_{u})$.

  2. (2)

A generic $(\mathcal{X},\mathcal{Y})$-tangency over $A$ is a tuple
$(x,y,t,u,n)$ such that $(x,y,t,u)$ is a weakly generic
$(\mathcal{X},\mathcal{Y})$-intersection, $n\in\mathbb{Z}^{+}$,
$\alpha_{xy}^{n}(X_{t})=\alpha_{xy}^{n}(Y_{u})$, and $X$ and $Y$ have
$n$-branching in a neighborhood of $(x,y,t,u)$.

######  Definition 7.50.

$(\mathcal{K},\tau,\\{\mathcal{IA}_{n}\\},\\{T_{mn}\\})$ satisfies TIMI
(‘tangent intersections are multiple intersections’) if the following holds:
suppose $\mathcal{X}=\\{X_{t}:t\in T\\}$ and $\mathcal{Y}=\\{Y_{u}:u\in U\\}$
are $\mathcal{K}(A)$-definable families of correspondences in $K$. If
$(x,y,t,u,n)$ is a generic $(\mathcal{X},\mathcal{Y})$-tangency over $A$, then
$(x,y,t,u)$ is a generic multiple $(\mathcal{X},\mathcal{Y})$-intersection
over $A$.

######  Remark 7.51.

Recall that at this moment we are working with fixed definable slopes on
$(\mathcal{K},\tau)$. In the future, after dropping this assumption, we will
use the phrase ‘$(\mathcal{K},\tau)$ has definable slopes satisfying TIMI’ to
mean that $(\mathcal{K},\tau)$ has definable slopes $\\{\mathcal{IA}_{n}\\}$
and $\\{T_{mn}\\}$ so that
$(\mathcal{K},\tau,\\{\mathcal{IA}_{n}\\},\mathcal{T}_{mn})$ satisfies TIMI.

Recall the notion of detection of multiple intersections, Definition 6.9. In
those terms, our main result is:

######  Theorem 7.52.

Assume that $(\mathcal{K},\tau,\\{\mathcal{IA}n\\},\\{T_{mn}\\})$ satisfies
TIMI, and $\mathcal{M}$ detects multiple intersections. Then $\mathcal{M}$
detects tangency. In particular, $\mathcal{M}$ interprets a strongly minimal
group.

###### Proof.

Let $\alpha_{0}$ be a non-algebraic coherent $n$-slope at $(x_{0},y_{0})$, and
let $(x_{0},y_{0},C_{1},c_{1},\alpha_{0})$ and
$(x_{0},y_{0},C_{1},c_{1},\alpha_{0})$ be coherent $n$-slope configurations.
We want to show that $c_{1}$ and $c_{2}$ are $\mathcal{M}$-dependent over
$(x_{0},y_{0})$. We do this by realizing the $C_{i}$ as generic members of
$\mathcal{M}$-definable families of plane curves, $Z_{t_{0}}\in\mathcal{Z}$
and $W_{u_{0}}\in\mathcal{W}$; we then apply TIMI to these families (or
rather, their images through our fixed map $\rho:M\rightarrow K$, which we
call $\mathcal{X}$ and $\mathcal{Y}$, respectively). The idea is that the non-
algebraicity of $\alpha_{0}$ forces each of $\mathcal{X}$ and $\mathcal{Y}$ to
realize infinitely many slopes at $(x_{0},y_{0})$, which forces
$(x_{0},y_{0},t_{0},u_{0},n)$ to be a generic $(\mathcal{X},\mathcal{Y})$
tangency. By TIMI, we conclude that $(x_{0},y_{0},t_{0},u_{0})$ is a generic
multiple $(\mathcal{X},\mathcal{Y})$-intersection; then since $\mathcal{M}$
detects multiple intersections, we get the desired dependence.

Let us proceed. Let $d_{i}=\operatorname{rk}(c_{i})$ for $i=1,2$. By Lemma
7.21, $(x_{0},y_{0})$ is generic in $M^{2}$, and each
$\dim(c_{i}/x_{0}y_{0})=d_{i}-1$. Thus $d_{i}\geq 1$. Now by Lemma 6.8, we can
realize each $C_{i}$ on a generic curve in a standard family (see Definition
2.4 and Lemma 6.8). That is, we can find standard families of non-trivial
plane curves $\mathcal{Z}=\\{Z_{t}:t\in T\\}$ and $\mathcal{W}=\\{W_{u}:u\in
U\\}$ over $\emptyset$, and generic $t_{0}\in T$ and $u_{0}\in U$, so that
$C_{1}$ is almost contained in $Z_{t_{0}}$ and $C_{2}$ is almost contained in
$W_{u_{0}}$. For each $t\in T$, let $X_{t}=\rho(Z_{t})$. Similarly, for each
$u\in U$, let $Y_{u}=\rho(W_{u})$. Then $\mathcal{X}=\\{X_{t}:t\in T\\}$ and
$\mathcal{Y}=\\{Y_{u}:u\in U\\}$ are $\mathcal{K}(\emptyset)$-definable
families of correspondences in $K$. Note that $\dim(T)=d_{1}$ and
$\dim(U)=d_{2}$.

We want to apply TIMI to $\mathcal{X}$ and $\mathcal{Y}$, at the tuple
$(\rho(x_{0}),\rho(y_{0}),t_{0},u_{0},n)$. First, note that
$(x_{0},y_{0},t_{0},u_{0})$ is a weakly generic
$(\mathcal{Z},\mathcal{W})$-intersection over $\emptyset$. It follows easily
that $(\rho(x_{0}),\rho(y_{0}),t_{0},u_{0})$ is a weakly generic
$(\mathcal{X},\mathcal{Y})$-intersection over $\emptyset$.

Next, by the strong frontier inequality, $Z_{t_{0}}$ and $C_{1}$ agree in a
neighborhood of $(x_{0},y_{0})$, as do $W_{u_{0}}$ and $C_{2}$. So
$\alpha_{x_{0}y_{0}}^{n}(Z_{t_{0}})=\alpha_{x_{0}y_{0}}^{n}(W_{u_{0}})=\alpha_{0}$.
Finally, we check:

######  Claim 7.53.

$\mathcal{X}$ and $\mathcal{Y}$ have $n$-branching in a neighborhood of
$(\rho(x_{0}),\rho(y_{0}),t_{0},u_{0})$ over $\emptyset$.

###### Proof.

By uniform definability of slopes,
$\alpha_{0}\in\operatorname{acl}(t_{0}x_{0}y_{0})$. Since $\alpha_{0}$ is not
algebraic, this implies
$\dim(t_{0}/\alpha_{0})<\dim(t_{0}/x_{0}y_{0})=\dim(c_{1}/x_{0}y_{0})=d_{1}-1$.
So $\dim(t_{0}/x_{0}y_{0})=d_{1}-1$ and $\dim(t_{0}/\alpha_{0})\leq d_{1}-2$.

By uniform definability of slopes, and the definability of dimension, we can
find a formula
$\phi(x,y,t)\in\operatorname{tp}_{\mathcal{K}}(x_{0},y_{0},t_{0})$ such that
whenever $(x,t,y)\in Z$ is generic and $\phi(x,y,t)$ holds, then for
$\alpha=\alpha_{xy}^{n}(Z_{t})$, we have $\dim(t/xy)\leq d_{1}-1$ and
$\dim(t/\alpha)\leq d_{1}-2$. By the genericity of $(x_{0},y_{0},t_{0})$,
$\phi$ holds in a neighborhood of $(x_{0},y_{0},t_{0})$, which we may assume
has the form $V_{1}\times V_{2}$, where $V_{1}\subset M^{2}$ and $V_{2}\subset
T$. Shrinking if necessary, we may assume that $\rho$ is a local homeomorphism
on $V_{1}$, so $\rho(V_{1})\subset K^{2}$ is open. Now let
$V=\rho(V_{1})\times V_{2}\times U\subset K^{2}\times T\times U$.

To prove the claim, we show that $\mathcal{X}$ and $\mathcal{Y}$ have
$n$-branching in $V$ over $\emptyset$. To do this, let
$(\rho(x),\rho(y),t,u)\in V$ be a strongly generic
$(\mathcal{X},\mathcal{Y})$-intersection over $\emptyset$, where $(x,y)\in
V_{1}$. Let
$\alpha=\alpha_{\rho(x)\rho(y)}^{n}(X_{t})=\alpha_{xy}^{n}(Z_{t})$.

Since $\rho$ is finite-to-one, it follows that $(x,y,t)\in Z$ is generic. Thus
$\dim(xyt)=\dim(Z)=d_{1}+1$. But by the choice of $V_{1}\times V_{2}$,
$\dim(t/xy)\leq d_{1}-1$, thus $\dim(xy)=2$.

Now by uniform definability of slopes,

$\dim(txy\alpha)=\dim(txy)=d_{1}+1.$

Since $\dim(xy)=2$, this implies $\dim(t\alpha/xy)=d_{1}-1$. But by the choice
of $V_{1}\times V_{2}$ again, $\dim(t/\alpha xy)\leq d_{1}-2$. Thus
$\dim(\alpha/xy)\geq 1$. In particular, $\alpha$ is not algebraic.

Now let $\beta=\alpha_{xy}^{n}(W_{u})$. So $\alpha\in\operatorname{acl}(txy)$
and $\beta\in\operatorname{acl}(uxy)$. Since $t$ and $u$ are independent over
$\rho(x)\rho(y)$, they are clearly also independent over $xy$. Thus, $\alpha$
and $\beta$ are independent over $xy$. In particular,
$\alpha\notin\operatorname{acl}(\beta)$, and thus $\alpha\neq\beta$. This
proves the claim. ∎

Now, by the claim, we have verified that
$(\rho(x_{0}),\rho(y_{0}),t_{0},u_{0},n)$ is a generic
$(\mathcal{X},\mathcal{Y})$-tangency over $\emptyset$. By TIMI, we conclude
that $(\rho(x_{0}),\rho(y_{0}),t_{0},u_{0})$ is a generic multiple
$(\mathcal{X},\mathcal{Y})$-intersection over $\emptyset$. Since $\rho$ is a
local homeomorphism near $(x_{0},y_{0})$, it follows that
$(x_{0},y_{0},t_{0},u_{0})$ is a generic multiple
$(\mathcal{Z},\mathcal{Y})$-intersection over $\emptyset$. But then since
$\mathcal{M}$ detects multiple intersections, we conclude that $t_{0}$ and
$u_{0}$ are $\mathcal{M}$-dependent over $\emptyset$. By
$\mathcal{M}$-interalgebraicity, so are $c_{1}$ and $c_{2}$.

Finally, by coherence, we have
$\operatorname{rk}(c_{1}/x_{0}y_{0})=\dim(c_{1}/x_{0}y_{0})=d_{1}-1$, and
similarly $\operatorname{rk}(c_{2}/x_{0}y_{0})=d_{2}-1$. So if $c_{1}$ and
$c_{2}$ are $\mathcal{M}$-independent over $x_{0}y_{0}$, then
$\operatorname{rk}(c_{1}c_{2}/x_{0}y_{0})=d_{1}+d_{2}-2$. Since
$(x_{0},y_{0})$ is generic in $M^{2}$, this implies
$\operatorname{rk}(c_{1}c_{2}x_{0}y_{0})=d_{1}+d_{2}$. Since $c_{1}$ and
$c_{2}$ are $\mathcal{M}$-dependent over $\emptyset$,
$\operatorname{rk}(c_{1}c_{2})<d_{1}+d_{2}$, so
$\operatorname{rk}(x_{0}y_{0}/c_{1}c_{2})>0$. Since $c_{1}$ and $c_{2}$ are
canonical bases of strongly minimal sets, this implies $c_{1}=c_{2}$. But then
by the assumed $\mathcal{M}$-independence of the $c_{i}$ over $x_{0}y_{0}$,

$0=\operatorname{rk}(c_{1}/c_{2}x_{0}y_{0})=\operatorname{rk}(c_{1}/x_{0}y_{0})=d_{1}-1.$

So $d_{1}=1$. This contradicts that $\dim(t_{0}/\alpha_{0}x_{0}y_{0})\leq
d_{1}-2$. We conclude that the $c_{i}$ are independent over $x_{0}y_{0}$,
proving that $\mathcal{M}$ detects tangency.

Finally, since $\mathcal{M}$ detects tangency, Proposition 7.45 implies that
every coherent $n_{0}$-slope has a code. Then Proposition 7.33 implies that
$\mathcal{M}$ interprets a strongly minimal group. ∎

##  8\. Summing Up

We are now done with the abstract setting, and we drop all data that we have
fixed up until now. The rest of the paper concerns concrete examples. Before
moving on, let us summarize our work to this point:

######  Theorem 8.1.

Let $(\mathcal{K},\tau)$ be a Hausdorff geometric structure with enough open
maps (in particular, it suffices to assume $(\mathcal{K},\tau)$ is either
differentiable or has the open mapping property). Assume that
$(\mathcal{K},\tau)$ has ramification purity, and definable slopes satisfying
TIMI. Let $\mathcal{M}=(M,...)$ be a non-locally modular strongly minimal
definable $\mathcal{K}$-relic. Then $\dim(M)=1$, and $\mathcal{M}$ interprets
a strongly minimal group.

###### Proof.

Since non-local modularity is witnessed by a single rank 2 family of plane
curves, we may assume the language of $\mathcal{M}$ is finite. If we then add
countably many constants, we can assume the language of $\mathcal{M}$ is
countable and $\operatorname{acl}_{\mathcal{M}}(\emptyset)$ is infinite.

Next, all of the assumptions on $(\mathcal{K},\tau)$ are invariant under
adding a countable set of parameters. Since the language of $\mathcal{M}$ is
countable, we may thus assume that every $\emptyset$-definable set in
$\mathcal{M}$ is $\emptyset$-definable in $\mathcal{K}$. Thus, we have
satisfied all requirements in Assumption 4.1, and so we can apply all the
results of sections 4-7.

Now by assumption (or by Proposition 3.33 or Proposition 3.40, depending on
which property is assumed), $(\mathcal{K},\tau)$ has enough open maps. So, by
Theorem 4.10, $\mathcal{M}$ weakly detects closures. Then, by Theorem 6.10,
$\mathcal{M}$ detects multiple intersections. So by Theorem 6.18, $\dim(M)=1$.
Then, by Theorem 7.52, $\mathcal{M}$ interprets a strongly minimal group. ∎

######  Remark 8.2.

One also gets the various additional properties of the group in the statement
of Proposition 7.33. We omit them here in order to not overcomplicate the
statement. Note, in particular, that changing the language of $\mathcal{M}$ in
the proof above does not affect the additional data in Proposition 7.33,
because that data only involves the background structure $\mathcal{K}$.

##  9\. Examples of Hausdorff Geometric Structures

In this section, we give various examples of Hausdorff geometric structures
satisfying some or all of the additional properties we have studied. The most
successful example is algebraically closed valued fields, which satisfy all
properties (except differentiability in positive characteristic) we have
defined.

###  9.1. Visceral structrues

In [DG22] Dolich and Goodrick introduce _visceral structures_ as a common
generalisation of o-minimality and P-minimality, that – when restricted to the
dp-minimal setting – is very similar to the tame uniform structures of Simon
and Walsberg ([SW19]) called SW-uniformities in [HHP22]. This formalism also
covers the context of 1-h-minimal structures (as pointed out in [CHRV21,
Example 2.2.2]). It follows immediately from the results of [DG22] that
certain strengthenings of visceral structures considered in that paper are
Hausdorff geometric structures.

By [DG22, Proposition 3.10] any visceral structure has uniform finiteness, so
any visceral structure satisfying the exchange principle is geometric. A
visceral structure is said to have definable finite choice (DFC) if any
definable function with finite fibres has a definable section. In particular,
any ordered visceral structure has DFC, and so does any visceral structure
with definable Skolem functions. We claim that ($\aleph_{1}$-saturated)
visceral structures with the exchange property and DFC are Hausdorff geometric
structures (see Remark 9.1 below for a discussion of removing the DFC
hypothesis).

First, such structures satisfy the strong frontier inequality by Remark 3.9
and [DG22, Corollary 3.35]. The Baire property, as explained in Remark 3.8,
would follow from the generic local homeomorphism property and the fact that
the topology in Hausdorff visceral structures has no isolated points. So it
remains to show the generic local homeomorphism property. So let $Z\subset
X\times Y$, all of the same dimension, with $Z\rightarrow X$ and $Z\rightarrow
Y$ finite-to-one, and let $(x,y)\in Z$ be generic. By visceral cell
decomposition ([DG22], Theorem 3.23, Definition 3.20), we can assume after
shrinking that $X$ and $Y$ are open subsets of $K^{n}$, where $n=\dim(X)$.
Moreover, by either directly applying DFC or by using the frontier inequality
(see the proof of Claim 9.34), we can assume after shrinking again that
$Z\rightarrow X$ and $Z\rightarrow Y$ are injective, and thus $Z$ defines a
bijection $X\leftrightarrow Y$. The result then follows by generic continuity
([DG22, Theorem 3.19]) applied to each coordinate component of
$X\leftrightarrow Y$ in each direction.

######  Remark 9.1.

In fact, it seems to follow from new results of Johnson ([Joh24b]) that one
can remove the DFC assumption above and conclude that all
$\aleph_{1}$-saturated visceral structures with exchange are Hausdorff
geometric structures. For this, one first deduces the strong frontier
inequality from Johnson’s new version ([Joh24b], Theorem 1.10). Now given
$Z\subset X\times Y$ as above, one similarly uses the frontier inequality to
assume $Z\rightarrow X$ and $Z\rightarrow Y$ are injective (again, as in Claim
9.34), and thus $Z$ defines a bijection $X\leftrightarrow Y$. One then applies
the generic continuity clause of Johnson’s cell decomposition ([Joh24b],
Theorem 1.14(2), which is enough by the frontier inequality) to each
coordinate component of $X\rightarrow Y$ and $Y\rightarrow X$ as above to get
the desired homeomorphism.

Hensel minimal structures, introduced in [CHRK22] (for residue characteristic
$0$ and in [CHRV21] for positive residue characteristic) cover all pure non-
trivially valued henselian fields (of characteristic $0$) as well as many
interesting expansions (covering the $V$-minimal algebraically closed valued
fields of [HK06] and the $T$-convex expansions of power-bounded o-minimal
fields, [vdDL95]). One sub-class of Hensel minimal structures of special
interest (covering all the above examples) is the class of 1-h-minimal fields.
As already mentioned, 1-h-minimal structures are visceral, and it is not hard
to verify that they are geometric structures (e.g., [AH24, Proposition 2.11]).
While 1-h-minimal fields do not necessarily have DFC, every 1-h-minimal field
has a 1-h-minimal expansion which does ([CHRV21, Proposition 3.2.3]). Thus, we
have:

######  Corollary 9.2.

Every $\aleph_{1}$-saturated 1-h-minimal field is a Hausdorff geometric
structure.

###### Proof.

By what we have just said, any 1-h-minimal field $\mathcal{K}$ has a DFC
expansion $\mathcal{K}^{\prime}$ that is a Hausdorff Geometric Structure.

In both $\mathcal{K}$ and $\mathcal{K}^{\prime}$, the topology is the
valuation topology, and it is definable. Since the dimension in both
structures is determined by the topology (see, e.g., [AH24, Proposition
2.11]), it follows that for a $\mathcal{K}$-definable set $X$ we have
$\dim_{\mathcal{K}}(X)=\dim_{\mathcal{K}^{\prime}}(X)$. Now consider $X$ a
$\mathcal{K}$-definable set. Then $\mathrm{Fr}(X)$ is also
$\mathcal{K}$-definable, and by the frontier inequality in
$\mathcal{K}^{\prime}$ we have that

$\dim_{\mathcal{K}}\mathrm{Fr}(X)=\dim_{\mathcal{K}^{\prime}}\mathrm{Fr}(X)<\dim_{\mathcal{K}^{\prime}}(X)=\dim_{\mathcal{K}}(X),$

implying the frontier inequality in $\mathcal{K}$.

So we only have to verify the Generic Local Homeomorphism Property. If
$Z\subseteq X\times Y$ are $\mathcal{K}$-definable of the same dimension,
projecting finite-to-one one both components, then for every
$\mathcal{K}^{\prime}$-generic $(x,y)\in Z$ we know that $Z$ is, in a
neighbourhood of $(x,y)$ the graph of a homeomorphism. As this is a
$\mathcal{K}$-definable property of $(x,y)$ (and since dimension in
$\mathcal{K}$ and in $\mathcal{K}^{\prime}$ coincide), it must also hold of
every $\mathcal{K}$-generic $(x,y)\in Z$. ∎

######  Remark 9.3.

As stated before, if one uses the new results of Johnson on visceral
structures [Joh24b], one does not need DFC, so the above corollary is
automatic from the discussion preceding it.

Both o-minimal and 1-h-minimal expansions of fields have a well-developed (and
rather similar) basic differential geometry, turning them into differential
Hausdorff geometric structures. Let us give the details in the 1-h-minimal
case. The situation in the o-minimal setting is analogous and better known.

First, we have to define a notion of smoothness meeting Definition 3.27. We
declare the smooth locus of a definable set $X$ to be the set of points $x\in
X$ where $X$ is, locally near $x$, a $\dim(X)$-dimensional weak strictly
differentiable manifold (see [AH24, Definition 5.3]). By visceral cell
decomposition, every definable set is generically locally a topological
manifold, and by Proposition 3.12 _loc. cit._ it is, in fact, generically, a
strictly differentiable manifold. This gives all the properties of Definition
3.27 except maybe (5), the preservation of smoothness under preimages of “nice
enough” projections. But recall (see the remark following Definition 3.27)
that this clause is a generalization of, and would follow from, the
‘submersion theorem’ of differential geometry. Moreover, there is a submersion
theorem for 1-h-minimal field ([AH24, Proposition 4.10]). So we simply apply
that proposition exactly as in the earlier remark.

So now we have to show that the above notion of smoothness admits a
differential structure, as in Definition 3.34. Naturally, in both the
1-h-minimal and the o-minimal settings, the differential structure is given by
the tangent space. As before, we focus on the 1-h-minimal setting, as the
o-minimal case is similar and more familiar.

We use the usual notion of a tangent space associated with the notion of
(weak) strictly differentiable manifolds ([AH24, Definition 5.4]). All
properties (1)-(7) of the definition are automatic, and we don’t dwell on
them. Clause (8) of the definition, asserting that a definable morphism
inducing an isomorphism on the level of tangent spaces is open (locally, near
the point in question) is an immediate consequence of the Inverse Mapping
Theorem ([AH24, Proposition 4.4]). So it only remains to check that if
$f:(X,x)\to(Y,y)$ is such that $x\in X$ and $y\in Y$ are generic, then the
induced map between the tangent spaces is surjective. This, in turn, is an
immediate result of the corresponding version of Sard’s Theorem [AH24,
Proposition 5.21], asserting that the set of singular values of $f$ is non-
generic.

So we conclude:

######  Theorem 9.4.

Every $\aleph_{1}$-saturated o-minimal expansion of a real closed field, as
well as every $\aleph_{1}$-saturated 1-h-minimal valued field, is a
differential Hausdorff geometric structure.

###  9.2. éz Expansions of Fields

We turn now toward ACVF in all characteristics. We will work with a general
setting of topological fields. In fact, the ensuing results could serve as an
alternate proof of Theorem 9.4 for RCVF and for pure characteristic zero
Henselian fields.

Our abstract setting is inspired by éz fields. We briefly recall this notion.
First, recall that a field $K$ is large if, whenever $V$ is a smooth variety
over $K$ and $V(K)\neq\emptyset$, then $V(K)$ is infinite (equivalently,
$V(K)$ is Zariski dense in some irreducible component). In [JTWY23], the
authors place a canonical non-discrete topology on the set $V(K)$ for every
(irreducible) variety $V$ over a large field $K$. The topology is called the
étale open topology, and is defined as the weakest topology where étale maps
are open (so it is generated by the images $f(W(K))$ of all étale morphisms of
$K$-varieties $f:W\rightarrow V$). It is shown in [JTWY23, Section 6] and
[JWY23, DWY22] that the étale open topology often coincides with existing
topologies of interest when the field is close to being henselian
(particularly, the order topology in real closed fields, and the valuation
topology in characteristic zero Henselian fields).

One says that the étale open topology on $K$ is induced by a field topology if
there is a field topology $\tau$ on $K$ such that, for every affine variety
$V$ over $K$, the étale open topology agrees on $V(K)$ with the topology
induced on $V(K)$ by $\tau$ (via the product and subspace topologies). In
general, the étale open topology is induced by a field topology if it respects
products (i.e. the étale open topology on $K^{n}$ is the product topology
induced by the étale open topology on $K$). This is not always the case (the
main counterexample being pseudofinite fields, or more generally PAC fields).
See [JTWY23, Proposition 4.9 and Section 8] or [DWY22] for more details.

The follow-up work [WY23] then defined éz fields as large fields $K$ such that
every definable set $X\subset K^{n}$ (in the pure field language) is a union
of finitely many definable étale-open subsets of Zariski closed sets. The idea
is that éz fields have ‘local quantifier elimination’ – where the word ’local’
is in the sense of the étale open topology. Éz fields include algebraically
closed fields, real closed fields, pseudofinite fields, and characteristic
zero Henselian fields (e.g. $p$-adically closed fields). It is shown in [WY23]
that many tameness properties of definable sets in these examples (e.g.
generic continuity of functions) can be adapted to the general éz setting.

It is well-known that algebraically closed valued fields and real closed
valued fields satisfy a similar local quantifier elimination: every definable
set is a union of finitely many valuation-open subsets of Zariski closed sets.
In fact, this description of definable sets holds in all Henselian valued
fields of characteristic 0 [vdD89]. Our goal is to introduce a single notion
capturing such a structural decomposition and incorporating éz fields. Since
we have used the product topology freely throughout the paper, we will
restrict to those éz fields whose étale open topology is induced by a field
topology.

Before making the general definition, we establish some conventions:

######  Notation 9.5.

Suppose $\mathcal{K}=(K,+,\cdot,...)$ is an expansion of a field. Throughout
the rest of Section 9:

  1. (1)

By a variety over $K$, we mean a reduced, separate scheme of finite type over
$K$. Varieties are denoted with letters such as $V$, $W$, etc. Note that
varieties are not definable objects in a first-order sense.

  2. (2)

If $V$ is a quasi-projective variety over $K$, we use $V(K)$ to denote the
$\mathcal{K}$-interpretable set of $K$-rational points of $V$.

  3. (3)

If $X\subset K^{n}$ is definable, the Zariski closure of $X$ is the smallest
affine $K$-variety $V$ such that $X\subset V(K)$.

  4. (4)

On the other hand, we still use the term Zariski closed for subsets of
$K^{n}$: $X\subset K^{n}$ is Zariski closed if $X=V(K)$ for some affine
variety $V$ over $K$.

  5. (5)

Similarly, if $V$ is a variety over $K$, and $X\subset V(K)$, we say that $X$
is Zariski dense in $V$ if for every proper closed subvariety $W\subset V$,
there is some $x\in X\cap V(K)-W(K)$.

  6. (6)

If $V$ is a quasi-affine variety over $K$, and $A\subset K$, we distinguish
between $V$ being defined over $A$ (i.e. defined by polynomials with
coefficients in $A$) and $V(K)$ being definable over $A$ (definable in the
structure $\mathcal{K}$ with parameters $A$).

  7. (7)

If $V$ is a quasi-affine variety over $K$, the notation $\dim(V)$ refers to
the dimension of $V$ as an algebraic variety. The notation $\dim(V(K))$ will
only be used if $\mathcal{K}$ is a geometric structure; and in this case, it
refers to dimension in the sense of geometric structures.

We will use freely the following:

######  Lemma 9.6.

Let $\mathcal{K}=(K,+,\cdot,...)$ be an expansion of a field. Let $X\subset
K^{n}$ be definable over $A$, and let $Z$ be its Zariski closure.

  1. (1)

$Z(K)$ is definable over $A$ (regardless of whether the variety $Z$ is over
$A$).

  2. (2)

If $Z^{S}$ is the smooth locus of $Z$, then $Z^{S}(K)$ is definable over $A$.

###### Proof.

These statements are unaffected by passing to an elementary extension of
$\mathcal{K}$. In particular, as opposed to topological statements about
Hausdorff geometric structures, the current lemma only references definable
objects.

So, let us assume that $\mathcal{K}$ is a ‘monster model’ – i.e.
$\kappa$-saturated and $\kappa$-strongly homogenous for some large $\kappa$.
In this case, one can detect the parameters in a definition using automorphism
invariance. Thus, to prove the lemma, it suffices to note that $Z(K)$ and
$Z^{S}(K)$ are invariant under field automorphisms fixing $X$ setwise. ∎

We also need a general notion of an invariant topology on a structure:

######  Definition 9.7.

Let $\mathcal{K}=(K,...)$ be a structure, and $\tau$ a topology on $K$. We
call $\mathcal{K}$ invariant if there is a basis $\mathcal{B}$ for $\tau$ such
that:

  1. (1)

Each $X\in\mathcal{B}$ is definable.

  2. (2)

Suppose $\phi(x,y)$ is a formula, where $x$ is a single variable and $y$ is a
tuple. Let $a$ and $b$ be tuples in the arity of $y$ with
$\operatorname{tp}(a)=\operatorname{tp}(b)$. Let $X$ and $Y$ be the sets
defined by $\phi(x,a)$ and $\phi(x,b)$, respectively. Then $X\in\mathcal{B}$
if and only if $Y\in\mathcal{B}$.

Note that, if $(K,+,\cdot,\tau)$ is a topological field, then $\tau$ induces a
natural topology on $V(K)$ for every variety $V$ over $K$ (the affine case is
given by the subspace topology from $K^{n}$, and the general case is obtained
by gluing; see [Mum88, Page 57, Chapter I Section 10], for example). Using
this, and inspired by the above discussion, we now define:

######  Definition 9.8.

An éz topological field expansion is an $\aleph_{1}$-saturated structure
$\mathcal{K}=(K,+,\cdot,...)$ over a countable language, equipped with an
invariant Hausdorff topology $\tau$ on $K$ (extended canonically to all
subsets of powers of $K$), such that:

  1. (1)

$(K,+,\cdot,\tau)$ is a non-discrete topological field.

  2. (2)

Every definable set $X\subset K^{n}$ is a finite union of definable
$\tau$-open subsets of Zariski closed sets.

  3. (3)

For each étale morphism $f:V\rightarrow W$ of varieties over $K$, the induced
map $V(K)\rightarrow W(K)$ is $\tau$-open.

######  Remark 9.9.

Suppose $(\mathcal{K},\tau)$ is an éz topological field expansion. Note that
since $\tau$ is invariant, every elementary extension of $\mathcal{K}$ is
canonically a field with an invariant topology satisfying (1) and (3) of
Definition 9.8. However, (2) is not first-order unless an additional
uniformity condition is assumed. Thus, it is not clear that elementary
extensions of $\mathcal{K}$ are also éz topological field expansions (though
this holds in most natural examples).

Note that if $(K,+,\cdot)$ is an $\aleph_{1}$-saturated éz field whose étale
open topology is induced by a field topology, then $K$ is (trivially) an éz
topological field expansion when endowed with the étale open topology.

It also follows from well-known facts that $\aleph_{1}$-saturated models of
ACVF and RCVF are éz topological field expansions. We do not elaborate on
RCVF, because it is 1-h-minimal (thus covered by the previous subsection). On
the other hand, let us sketch the argument for ACVF.

The main fact we need is:

######  Fact 9.10 ([MB12a]).

Let $(K,v)$ be an algebraically closed valued field. Let $f:V\rightarrow W$ be
a universally open morphism of $K$-varieties. Then the induced map
$V(K)\rightarrow W(K)$ is open in the valuation topology.

######  Lemma 9.11.

Let $\mathcal{K}=(K,v)$ be an $\aleph_{1}$-saturated model of ACVF. Then
$(K,v)$ is an éz topological field expansion.

###### Proof.

Let $\tau$ be the valuation topology. As is well-known, $\tau$ is a non-
discrete Hausdorff field topology. Moreover, as the collection of balls is a
$\emptyset$-definable basis, $\tau$ is invariant.

So it remains to check (2) and (3) in Definition 9.8.$(\mathcal{K},\tau)$. (2)
is a well-known consequence of quantifier elimination (see also [vdD89]). For
(3), recall that étale morphisms are universally open ([Sta24, Lemmas 03WT and
02GO]), and thus apply Fact 9.10. ∎

######  Assumption 9.12.

For the rest of section 9, we fix an éz topological field expansion
$(\mathcal{K},\tau)$, with underlying field $(K,+,\cdot)$.

######  Remark 9.13.

Note that, since $\tau$ is a field topology, it refines the Zariski topology:
for every affine variety $V$ over $K$, the set $V(K)$ is $\tau$-closed.
Similarly, every morphism of $K$-varieties induces a $\tau$-continuous map on
$K$-points. We will use these facts throughout.

###  9.3. Basic Properties

Our general goal is to show that $(\mathcal{K},\tau)$ is a Hausdorff geometric
structure. First, we note that many of the facts from [WY23] transfer (that
is, there is no dependence on the particular language or topology). Thus, we
now prove a series of basic lemmas about $(\mathcal{K},\tau)$.

We begin with:

######  Lemma 9.14.

Let $X\subset K^{n}$ be definable and Zariski dense in $\mathbb{A}^{n}$. Then
$X$ has non-empty interior in the sense of $\tau$. In particular, every
infinite definable subset of $K$ has non-empty interior.

###### Proof.

By Definition 9.8(2), we can write $X=\bigcup_{i=1}^{m}U_{i}$, where each
$U_{i}$ is a definable open subset of $Z_{i}(K)$ for some closed subvariety
$Z_{i}$ of $\mathbb{A}^{n}$. Since $X$ is Zariski dense in $\mathbb{A}^{n}$,
there is some $i$ such that $U_{i}$ is Zariski dense in $\mathbb{A}^{n}$. It
follows that $Z_{i}=\mathbb{A}^{n}$, so that $Z_{i}(K)=K^{n}$. But then
$U_{i}$ is open in $K^{n}$ (and since it is Zariski dense, it is non-empty). ∎

The next two lemmas are consequences of Lemma 9.14:

######  Lemma 9.15.

  1. (1)

$K$ is perfect.

  2. (2)

If $\operatorname{char}(K)=p>0$ then the Frobenius map $x\mapsto x^{p}$ is a
homeomorphism.

###### Proof.

  1. (1)

Suppose $\operatorname{char}(K)=p>0$ and let $K^{p}$ be the set of $p$th
powers. By Lemma 9.14, $K^{p}$ has non-empty interior. Let $U\subset K^{p}$ be
a non-empty definable open set. Since $K^{p}$ is a subfield, we may translate
and assume $0\in U$.

Now let $a\in K$. We show $a\in K^{p}$. If $a=0$ this is clear. If $a\neq 0$,
then $U\cap\frac{1}{a}U$ is a neighborhood of 0, and since $\tau$ is non-
discrete, there is some $b\in U\cap\frac{1}{a}U$ with $b\neq 0$. By the choice
of $U$, there are $x,y\in K$ with $x^{p}=b$ and $y^{p}=ab$. Then
$(\frac{y}{x})^{p}=a$.

  2. (2)

By (1), $x\mapsto x^{p}$ is a continuous bijection. We show it is open. Since
$\tau$ is a field topology, and $x\mapsto x^{p}$ respects addition, it will
suffice to show that if $X\subset K$ is open and contains 0, then $X^{p}$
contains a neighborhood of 0. So, fix such an $X$. We may assume $X$ is
definable. Since subtraction is continuous, there is an open definable $Y$
containing 0 such that $Y-Y\subset X$. Since $\tau$ is not discrete, $Y$ is
infinite, and thus so is $Y^{p}$. So by Lemma 9.14, $Y^{p}$ contains an
infinite open set $Z$. Let $a\in Z$. Then $Z-a$ is a neighborhood of 0, and

$Z-a\subset Z-Z\subset Y^{p}-Y^{p}\subset X^{p},$

as desired.

∎

######  Lemma 9.16.

$\mathcal{K}$ is geometric.

###### Proof.

It was shown in [JY23] that if an expansion of a field satisfies exchange, it
also eliminates $\exists^{\infty}$. Thus, we only need to show that
$\mathcal{K}$ satisfies exchange. But if not, then there is a definable
$X\subset K^{2}$ so that (1) the left projection $X\rightarrow K$ is finite-
to-one and (2) the right projection $X\rightarrow K$ has infinitely many
infinite fibers. By (2), $X$ is Zariski dense in $\mathbb{A}^{2}$, so by Lemma
9.14, $X$ has non-empty interior. Thus $X$ contains a set $B_{1}\times B_{2}$,
where each $B_{i}\subset K$ is non-empty and open. Since $\tau$ is non-
discrete, each $B_{i}$ is infinite. This contradicts (1). ∎

Now that we know $\mathcal{K}$ is geometric, we will use the notation $\dim$
on definable sets. We next show:

######  Lemma 9.17.

Let $V$ be a smooth variety over $K$. Let $X$ be a non-empty open definable
subset of $V(K)$. Then $\dim(X)=\dim(V)$. In particular, $K$ is a large field.

###### Proof.

It is easy to see that $\dim(X)\leq\dim(V)$. We show the reverse inequality.
Let $a\in X$, and $n=\dim(V)$. Using that $a$ is a smooth point of $V$, one
can show there is a rational map $f:V\rightarrow\mathbb{A}^{n}$ which is
defined and étale at $a$. Let $U\subset V$ be the (necessarily relatively
Zariski open) étale locus of $f$. Without loss of generality, we may assume
$X\subset U(K)$. Now by Definition 9.8(3), the image $\pi(X)\subset K^{n}$ is
non-empty and open, so contains a non-empty definable box
$B_{1}\times...\times B_{n}$, where each $B_{i}\subset K$ is open. Since
$\tau$ is non-discrete, each $B_{i}$ is infinite. Thus
$\dim(B_{1}\times...\times B_{n})=n$. Finally, since $\pi$ is étale on $U$ it
is finite-to-one, and thus preserves dimension. So

$\dim(X)=\dim(f(X))\geq\dim(B_{1}\times...\times B_{n})=n.$

∎

The next three lemmas give the main technical tools we will need to show that
$(\mathcal{K},\tau)$ is a Hausdorff geometric structure. We will use the
following notion:

######  Definition 9.18.

Let $X\subset K^{n}$ be definable, with Zariski closure $Z$. Let $Z^{S}$ be
the smooth locus of $Z$. Let $x\in Z(K)$. We call $x$ polished for $X$ if
there is an open neighborhood $U$ of $x$ in $Z(K)$ such that:

  1. (1)

$U\subset Z^{S}(K)$.

  2. (2)

$U$ is either contained in $X$ or disjoint from $X$.

######  Lemma 9.19.

Let $X\subset K^{n}$ be definable, with Zariski closure $Z$. Then there is a
closed subvariety $T\subset Z$ with $\dim(T)<\dim(Z)$, such that every $x\in
Z(K)-T(K)$ is polished for $X$.

###### Proof.

By Definition 9.8(2), we can write $X=\bigcup_{i=1}^{m}U_{i}$ and
$Z(K)=\bigcup_{j=1}^{k}V_{j}$, where each $U_{i}$ (resp. $V_{j}$) is a
definable open subset of $Z_{i}(K)$ (resp. $W_{j}(K)$) for some affine
varieties $Z_{i}$ and $W_{j}$ over $K$. Refining the decomposition if
necessary, one shows easily that we may assume each $Z_{i}$ and $W_{j}$ is
irreducible over $K$ and contained in $Z$ (note that irreducibility is only
ensured over $K$, so we cannot assume absolute irreducibility).

Now, since $K$ is perfect, varieties over $K$ are generically smooth. That is,
if $Z^{S}$ is the (Zariski open) smooth locus of $Z$, then
$\dim(Z-Z^{S})<\dim(Z)$. We now define $T$ to be the union of $Z-Z^{S}$ with
all those $Z_{i}$ and $W_{j}$ of dimension less than $\dim(Z)$. Clearly, then,
$\dim(T)<\dim(Z)$.

To see this works, let $x\in Z(K)-T(K)$. We show that $x$ is polished for $X$.
Without loss of generality, assume $x\in X$ (the case $x\in Z-X$ is
symmetric). So $x\in U_{i}$ for some $i$. By the choice of $T$,
$\dim(Z_{i})\geq\dim(Z)$. But by assumption, $Z_{i}$ is an irreducible closed
subvariety of $Z$. It follows that $Z_{i}$ is an irreducible component of $Z$.

Recall that to show $x$ is polished, we need to find a neighborhood of $x$ (in
$Z(K)$) which is contained in $X\cap Z^{S}(K)$. We claim that $U=U_{i}\cap
Z^{S}(K)$ works. Clearly $U\subset X\cap Z^{S}(K)$, so all we need to show is
that $U$ is relatively open in $Z(K)$.

For this, we establish the following three claims:

######  Claim 9.20.

$U$ is relatively open in $Z_{i}(K)\cap Z^{S}(K)$.

###### Proof.

By assumption, $U_{i}$ is open in $Z_{i}(K)$. Thus, automatically,
$U=U_{i}\cap Z^{S}(K)$ is open in $Z_{i}(K)\cap Z^{S}(K)$. ∎

######  Claim 9.21.

$Z_{i}(K)\cap Z^{S}(K)$ is relatively open in $Z^{S}(K)$.

###### Proof.

Note that (by generic smoothness and the fact that $\dim(Z_{i})=\dim(Z)$)
$Z_{i}\cap Z^{S}$ is an irreducible component of the smooth variety $Z^{S}$.
Since smooth Zariski-connected varieties are irreducible, the irreducible
components of $Z^{S}$ are relatively Zariski open. Thus $Z_{i}\cap Z^{S}$ is
Zariski open in $Z^{S}$; and since $\tau$ refines the Zariski topology,
$Z_{i}(K)\cap Z^{S}(K)$ is relatively open in $Z^{S}(K)$. ∎

######  Claim 9.22.

$Z^{S}(K)$ is relatively open in $Z(K)$.

###### Proof.

Since $\tau$ refines the Zariski topology, it is enough to note that $Z^{S}$
is automatically Zariski open in $Z$. ∎

By the three claims, $U$ is relatively open in $Z(K)$, completing the proof of
Lemma 9.19. ∎

We now use Lemma 9.19 to characterize the dimension function of $\mathcal{K}$:

######  Lemma 9.23.

Let $X\subset K^{n}$ be definable, with Zariski closure $Z$. Then
$\dim(X)=\dim Z(K)=\dim Z$.

###### Proof.

It is easy to see that $\dim(X)\leq\dim(Z(X))\leq\dim(Z)$. We show that
$\dim(X)\geq\dim(Z)$.

Let $T$ be a relatively closed subvariety as in Lemma 9.19. By the choice of
$Z$, $X$ is Zariski dense in $Z$. In particular, there is some $x\in X-T(K)$.
So, by the choice of $T$, $x$ is polished for $X$. It follows by definition
that $X$ contains a non-empty open subset of $Z^{S}(K)$, where $Z^{S}$ is the
smooth locus of $Z$. So by Lemma 9.17, $\dim(X)\geq\dim(Z^{S})$. But by
generic smoothness over perfect fields, $\dim(Z^{S})=\dim(Z)$, completing the
proof. ∎

Finally, we deduce our main technical lemma:

######  Lemma 9.24.

Let $X\subset K^{n}$ be definable over a countable set $A$. Let $Z$ be the
Zariski closure of $X$. If $x\in Z(K)$ is generic over $A$, then $x$ is
polished for $X$.

###### Proof.

Let $T$ be a closed subvariety of $Z$ as in Lemma 9.19. Let $B\supset A$ be
countable so that $T(K)$ is definable over $B$.

Let $y$ be an independent realization of $\operatorname{tp}(x/A)$ over $B$. So
$y\in Z(K)$, and by Lemma 9.23, we have

$\dim(y/B)=\dim(x/A)=\dim(Z)>\dim(T)\geq\dim(T(K)),$

so that $y\notin T(K)$. Thus $y$ is polished for $X$. Let $\phi(z,b)$ (for $z$
in the arity of $Z(K)$) be a formula defining a neighborhood of $y$ witnessing
that $y$ is polished for $X$. Since
$\operatorname{tp}(x/A)=\operatorname{tp}(y/A)$, there is $a$ with
$\operatorname{tp}(xa/A)=\operatorname{tp}(yb/A)$. Then by the invariance of
$\tau$ (Definition 9.7), it follows that $\phi(z,a)$ defines a neighborhood of
$x$ witnessing that $x$ is polished for $X$. ∎

######  Remark 9.25.

Lemma 9.24 says that, near generic points, we can treat definable sets as
smooth varieties. More precisely, if $x\in X$ is generic over $A$, then $x$ is
also generic in $S(K)$ for some smooth variety $S$ having the same dimension
and germ as $X$ at $x$. Thus, if we are only interested in d-local properties
of $(X,x)$, we may replace $X$ with $S(K)$. If we are also willing to pass
from $A$ to $\operatorname{acl}(A)$ (which does not affect the genericity of
$x$), then we may further assume $S$ is irreducible over $A$. This will be our
main strategy moving forward.

###  9.4. Homeomorphisms and Reduction to Generically Smooth Maps

One of the difficulties of the trichotomy in the positive characteristic case
is the existence of finite everywhere-ramified morphisms. As has become
standard, we will get around this problem by decomposing an arbitrary map into
a universal homeomorphism composed with a generically smooth map. Then, in
studying topological properties, we can pay attention only to the generically
smooth part.

Such a decomposition was given explicitly in [WY23], using the reduced
relative Frobenius construction. This is not quite good enough for us, because
we need the universal homeomorphism to preserve normality. We thus give a more
intricate argument:

######  Lemma 9.26.

Let $V\rightarrow W$ be a dominant morphism of irreducible varieties over $K$.
Then there are an irreducible variety $V^{\prime}$, and a factorization
$V\rightarrow V^{\prime}\rightarrow W$, such that:

  1. (1)

$V\rightarrow V^{\prime}$ is a universal homeomorphism, and $V(K)\rightarrow
V^{\prime}(K)$ is a $\tau$-homeomorphism.

  2. (2)

$V^{\prime}\rightarrow W$ is generically smooth.

  3. (3)

If $V$ is normal then so is $V^{\prime}$.

###### Proof.

It is shown in [WY23, Lemma 1.12, Fact 3.3 and 3.5] that there is such a
factorization satisfying (1) and (2), given by a sufficient iterate of the
reduced relative Frobenius of $V\rightarrow W$ (the fact that this
construction gives a $\tau$-homeomorphism follows from Lemma 9.15(2)). Call
this factorization $V\rightarrow V_{1}\rightarrow W$. Now suppose $V$ is
normal (and note that $V_{1}$ might not be). We show how to refine the
construction to satisfy (3).

Let $N\rightarrow V_{1}$ be the normalization of $V_{1}$. So $N\rightarrow
V_{1}$ is surjective and birational. Since $V$ is normal, and by the universal
property of normalizations [Sta24, Lemma 035Q(4)], $V\rightarrow V_{1}$
factors as $V\rightarrow N\rightarrow V_{1}$. Since dominant quasi-finite
morphisms to normal varieties are universally open [Sta24, Lemma 0F32], the
image of $V$ in $N$ is an open (so also normal) subvariety, say $U$.

We now have $V\rightarrow U\rightarrow N\rightarrow V_{1}\rightarrow W$, where

  1. (1)

$V\rightarrow U$ is surjective and universally open.

  2. (2)

$U\rightarrow N$ is an open immersion.

  3. (3)

$N\rightarrow V_{1}$ is surjective and birational.

  4. (4)

$V_{1}\rightarrow W$ is generically smooth.

Since $V\rightarrow V_{1}$ is a universal homeomorphism, it is universally
injective. Thus $V\rightarrow U$ is also universally injective. Note that base
surjections are automatically universally surjective [Sta24, Lemma 01S1].
Thus, $V\rightarrow U$ is universally open, universally injective, and
universally surjective, and so is a universal homeomorphism.

We now apply the reduced relative Frobenius again, this time to $V\rightarrow
U$. We obtain a factorization $V\rightarrow V^{\prime}\rightarrow U$ again
satisfying (1) and (2) of the lemma (for $V\rightarrow U$). Thus, we have
dominant morphisms

$V\rightarrow V^{\prime}\rightarrow U\rightarrow N\rightarrow V_{1}\rightarrow
W.$

To complete the proof of the lemma, we note:

######  Claim 9.27.

$V\rightarrow V^{\prime}$ is a universal homeomorphism and a
$\tau$-homeomorphism.

###### Proof.

This is because $V\rightarrow V^{\prime}\rightarrow U$ satisfies (1) and (2).
∎

######  Claim 9.28.

$V^{\prime}\rightarrow W$ is generically smooth.

###### Proof.

Because $V^{\prime}\rightarrow U\rightarrow N\rightarrow V_{1}\rightarrow W$
is a chain of generically smooth maps. ∎

And, the main point:

######  Claim 9.29.

$V^{\prime}$ is normal.

###### Proof.

We have that $V\rightarrow U$ and $V\rightarrow V^{\prime}$ are both universal
homeomorphisms, which implies that $V^{\prime}\rightarrow U$ is also a
universal homeomorphism [Sta24, Lemma 0H2M], and thus (by generic smoothness)
a birational universal homeomorphism. But then, as a birational universal
homeomorphism to a normal variety, $V^{\prime}\rightarrow U$ is an isomorphism
[Sta24, Lemma 0AB1]. Thus $V^{\prime}$ is normal because $U$ is. ∎

∎

######  Remark 9.30.

Suppose, in the setup of Lemma 9.26, the morphism $V(K)\rightarrow W(K)$ is
defined over $A$. We note that each step in the proof of the lemma was given
by a functorial construction. It then follows that the whole sequence
$V(K)\rightarrow V^{\prime}(K)\rightarrow W(K)$ is still defined over $A$. We
will use this implicitly.

###  9.5. Proofs of the Axioms

Armed with the machinery from the previous subsection, we now show that
$(\mathcal{K},\tau)$ is a Hausdorff geometric structure. We also give
conditions under which $(\mathcal{K},\tau)$ has enough open maps.

######  Theorem 9.31.

$(\mathcal{K},\tau)$ is a Hausdorff geometric structure.

###### Proof.

We already showed that $\mathcal{K}$ is geometric. Thus, it suffices to
establish the strong frontier inequality, the Baire category axiom, and the
generic local homeomorphism property. These are given in the next three
claims.

######  Claim 9.32.

Let $X\subset K^{n}$ be definable over $A$, and let
$a\in\overline{\operatorname{Fr}(X)}$. Then $\dim(a/A)<\dim(X)$.

###### Proof.

Let $Z$ be the Zariski closure of $X$. Note that $a\in Z(K)$, since
$a\in\overline{X}$ and $\tau$ refines the Zariski topology. By Lemma 9.23,
$\dim(Z(K))=\dim(X)$. Now assume toward a contradiction that
$\dim(a/A)=\dim(X)$. Then $\dim(a/A)=\dim(Z(K))$, so $a$ is generic in $Z(K)$.
By Lemma 9.24, $a$ is thus polished for $X$. But the definition of being
polished for $X$ implies that $a\notin\overline{\operatorname{Fr}(X)}$, a
contradiction. ∎

######  Claim 9.33.

Let $X\subset K^{n}$ be definable over $A$, and let $a\in X$ be generic over
$A$. Let $B\supset A$ be countable. Then every neighborhood of $a$ contains a
generic of $X$ over $B$.

###### Proof.

Let $Z$ be the Zariski closure of $X$, and let $Z^{S}$ be its smooth locus. By
Lemma 9.23, $\dim(X)=\dim(Z(K))$. So since $a$ is generic in $X$ over $A$, it
is also generic in $Z(K)$ over $A$. So by Lemma 9.24, $a$ is polished for $X$.
Let $U\subset Z(K)$ be a neighborhood of $a$ witnessing this. That is, $U$ is
open in $Z(K)$, and $U$ is contained in $X\cap Z^{S}(K)$.

To prove the claim, let $V$ be any neighborhood of $a$ in $X$. Shrinking if
necessary, we may assume $V$ is definable and contained in $U$, so in
particular $V$ is relatively open in $Z^{S}(K)$. Adding to $B$ if necessary,
we may assume $V$ is definable over $B$.

Now by Lemmas 9.17 and 9.23, and generic smoothness over perfect fields, we
conclude that

$\dim(V)=\dim(Z^{S})=\dim(Z)=\dim(X).$

In particular, if we choose any $b\in V$ generic over $B$, then $b$ is also
generic in $X$ over $B$, proving the claim. ∎

######  Claim 9.34.

Let $Z\subset X\times Y$ be definable over $A$ and all of dimension $d$, with
$Z\rightarrow X$ and $Z\rightarrow Y$ finite-to-one. Let $(x,y)\in Z$ be
generic over $A$. Then $Z$ restricts to a homeomorphism between neighborhoods
of $x$ and $y$ in $X$ and $Y$, respectively.

###### Proof.

It is enough to show the projection $Z\rightarrow X$ is locally a
homeomorphism near $(x,y)$, since by symmetry the same will apply to
$Z\rightarrow Y$. For this, it suffices to show separately that $Z\rightarrow
X$ is locally injective, locally continuous, and locally open at $(x,y)$.
Moreover, local continuity is automatic, since $Z\rightarrow X$ is a
projection. So we show local injectivity and local openness.

First, suppose $Z\rightarrow X$ is not locally injective at $(x,y)$. Then
$(x,y,y)\in\operatorname{Fr}(T)$, where $T$ is the set of $(u,v,w)$ with
$(v\neq w)$ and $(u,v),(u,w)\in Z$. Since $Z\rightarrow X$ is finite-to-one,
$\dim(T)\leq d$. Then by the strong frontier inequality, we get
$\dim(xy/A)<d$, a contradiction. So $Z\rightarrow X$ is locally injective near
$(x,y)$.

Now we show local openness at $(x,y)$. The idea is that, as outlined in the
beginning of this subsection, we can reduce to the case that $Z\rightarrow X$
is generically smooth – thus étale – and then apply Definition 9.8(3). More
precisely, by Lemma 9.24, there are smooth d-dimensional varieties $W$ and
$V\subset\mathbb{A}^{n}\times W$ so that:

  1. (1)

$V(K)$ and $W(K)$ are definable over $A$.

  2. (2)

$V(K)$ and $W(K)$ have the same germs as $Z$ and $X$ at $(x,y)$ and $x$,
respectively.

We may assume, without loss of generality, that $A=\operatorname{acl}(A)$.
Thus, we may also assume $V$ and $W$ are irreducible. Since $Z\rightarrow X$
is finite-to-one, it follows that $V\rightarrow W$ is dominant.

Now let $V\rightarrow V^{\prime}\rightarrow W$ be a factorization as in Lemma
9.26, and let $z$ be the image of $(x,y)$ in $V^{\prime}(K)$. It follows from
Remark 9.30 that $z$ is generic in $V^{\prime}(K)$. Moreover, since
$V(K)\rightarrow V^{\prime}(K)$ is a homeomorphism, it suffices to show that
$V^{\prime}(K)\rightarrow W(K)$ is open near $z$. But by the choice of
$V^{\prime}$ (and the fact that $\dim(V^{\prime})=\dim(W)$,
$V^{\prime}\rightarrow W$ is generically étale; and since $z\in V^{\prime}$ is
generic, it follows that $V^{\prime}\rightarrow W$ is étale in a Zariski
neighborhood of $z$. Local openness then follows as promised from Definition
9.8(3). ∎

∎

###  9.6. Smoothness and Enough Open Maps

We now proceed to give conditions under which $(\mathcal{K},\tau)$ is either
differentiable or has the open mapping property. First, we define a canonical
notion of smoothness.

######  Definition 9.35.

For definable $X\subset K^{n}$, we let $X^{S}$ be the set of $x\in X$ which
are polished for $X$ – that is, the relative interior of $X$ in the smooth
locus of its Zariski closure.

In other words, the smooth points of $X$ are those points where we can view
$X$ d-locally as a smooth variety. We now show:

######  Lemma 9.36.

The map $X\mapsto X^{S}$ is a notion of smoothness on $(\mathcal{K},\tau)$.

###### Proof.

Most of the properties in Definition 3.27 follow from standard facts about
smooth varieties (e.g. closure under products and isomorphisms). The fact that
generic points are smooth follows from Lemma 9.24. We show only Definition
3.27(5) (smoothness of preimages under appropriate maps).

To recall the setting: we have a projection $f:X\rightarrow Y$ of definable
sets over $A$, and a point $x\in X$ generic over $A$, such that $y=f(x)$ is
generic in $Y$ over $A$. We then have a set $Z$, definable over $B$, so that
(1) $y$ is generic in $Z$ over $B$ and (2) the germ of $Z$ at $y$ is contained
in the germ of $Y$ at $y$. Our goal is to show that $x\in(f^{-1}(Z\cap
Y))^{S}$.

To do this, one follows exactly the same procedure as in the proof of Claim
9.34 to reduce everything d-locally to smooth varieties and smooth morphisms.
In the end, one only needs to show the following:

######  Claim 9.37.

Let $g:V\rightarrow W$ be a smooth morphism of smooth varieties over $K$, and
let $U\subset W$ be a smooth subvariety. Then $g^{-1}(U)$ is smooth.

###### Proof.

Recall that smooth morphisms are stable under composition and base change
[Sta24, Lemma 01VA,01VB]. Moreover, since $K$ is perfect, a variety over $K$
is smooth if and only if its structure morphism to $\operatorname{Spec}(K)$ is
smooth. We proceed to repeatedly use these facts.

First, the map $g^{-1}(U)\rightarrow U$ is isomorphic to the base change of
$V\rightarrow W$ by $U\rightarrow W$. Since $V\rightarrow W$ is smooth, so is
$g^{-1}(U)\rightarrow U$. But $U\rightarrow\operatorname{Spec}(K)$ is smooth
by assumption, so the composition $g^{-1}(U)\rightarrow\operatorname{Spec}(K)$
is smooth, and thus $g^{-1}(U)$ is smooth. ∎

∎

We now, automatically, have:

######  Lemma 9.38.

Suppose that whenever $g:V\rightarrow W$ is a quasi-finite morphism of smooth
$d$-dimensional varieties over $K$, the induced map $g:V(K)\rightarrow W(K)$
is open. Then $(\mathcal{K},\tau)$ has the open mapping property.

###### Proof.

Let $f:X\rightarrow Y$ be a finite-to-one definable projection, where
$\dim(X)=\dim(Y)$, and let $x\in X^{S}$ with $y=f(x)\in Y^{S}$. We need to
show that $f$ is open near $x$. But this is automatic: the assumption that
$x\in X^{S}$ and $y\in Y^{S}$ lets us reduce d-locally to a projection of
smooth varieties, and we apply the hypothesis of the lemma. ∎

######  Remark 9.39.

We note, for example, that the hypothesis of Lemma 9.38 holds of the complex
field with the analytic topology (though technically, this structure is not an
éz topological field expansion, because it has no invariant basis). Indeed, in
this case, the induced map $g:V(K)\rightarrow W(K)$ is a holomorphic map of
$d$-dimensional complex manifolds with all fibers discrete, and one can apply
a usual open mapping theorem from complex analysis (e.g. [GR84], p. 107). This
was one of the key geometric properties used in [Cas23]), and is the
inspiration for Lemma 9.38. Indeed, we will see later (Corollary 9.60) that
the hypothesis of the lemma holds in ACVF in all characteristics.

We now move on to differentiability. In fact, quite generally, we have:

######  Lemma 9.40.

If $K$ has characteristic zero, then $(\mathcal{K},\tau)$ is differentiable.

###### Proof.

Given a definable $X$ and $x\in X^{S}$, the germ of $X^{S}$ at $x$ agrees with
a smooth variety $S(K)$, where $\dim(S)=\dim(X)$. We then set $T_{x}(X)$ to be
the Zariski tangent space $T_{x}(S)$. All of the required properties of
Definition 3.34 follow easily. Note that the weak inverse function theorem
follows since étale maps are open (Definition 9.8(3)), and Sard’s theorem
follows since dominant morphisms in characteristic zero are generically
smooth. ∎

###  9.7. The Algebraically Closed Case and Purity of Ramification

We now address the special case that the underlying field $K$ is algebraically
closed. In this case, we will prove a ‘topologized’ purity of ramification
statement for morphisms of varieties, and subsequently deduce that
$(\mathcal{K},\tau)$ has ramification purity in the sense of Definition 6.16.

######  Assumption 9.41.

Throughout this subsection, we assume that $K$ is algebraically closed.

The main advantages of the algebraically closed case are summarized by the
following fact and corollary:

######  Fact 9.42.

Let $V$ be an irreducible variety over $K$, and let $U\subset V(K)$ be
definable, relatively open, and non-empty. Then $U$ is Zariski dense in $V$.
In particular, $\dim(U)=\dim(V)$.

###### Proof.

A proof can be found in [MB12b, Proposition 2.1.1]. ∎

Fact 9.42 implies that $\tau$ behaves identically to the Zariski topology on
the level of constructible sets:

######  Corollary 9.43.

Let $V$ be a variety over $K$ and let $X\subset V$ be constructible
(equivalently, definable in the pure field language). Then the relative
Zariski and $\tau$-closures of $X$ in $V(K)$ coincide.

We will also use the following fact about rings:

######  Proposition 9.44.

Let $A\leq B$ be a flat extension of integral domains, and suppose the
extension of fraction fields
$\operatorname{Frac}(A)\leq\operatorname{Frac}(B)$ is finite and separable.
Then the tensor product $B\otimes_{A}B$ is a reduced ring.

###### Proof.

For convenience, let $K_{A}$ and $K_{B}$ denote the respective fraction
fields. The idea of the proof is to use flatness to embed $B\otimes_{A}B$ into
$K_{B}\otimes_{K_{A}}K_{B}$, so that we can assume $A$ and $B$ are already
fields. The next claim follows easily from flatness:

######  Claim 9.45.

$B\otimes_{A}B$ embeds into $K_{B}\otimes_{A}K_{B}$.

Let $S=A-\\{0\\}$, so that $K_{A}$ is the localization $S^{-1}A$. Note that
$K_{B}$ is already a $K_{A}$-module, so $S^{-1}K_{B}\cong K_{B}$. Moreover,
the following is standard:

######  Claim 9.46.

$S^{-1}(K_{B}\otimes_{A}K_{B})\cong K_{B}\otimes_{A}K_{B}$.

Finally, we conclude:

######  Claim 9.47.

$B\otimes_{A}B$ embeds into $K_{B}\otimes_{K_{A}}K_{B}$. In particular, in
proving Proposition 9.44, we may assume $A$ and $B$ are fields.

###### Proof.

Recall [Sta24, Lemma 00DL] that we have a canonical isomorphism

$S^{-1}K_{B}\otimes_{S^{-1}A}S^{-1}K_{B}\cong S^{-1}(K_{B}\otimes_{A}K_{B}),$

which simplifies to

$K_{B}\otimes_{K_{A}}K_{B}\cong S^{-1}(K_{B}\otimes_{A}K_{B}).$

By Claim 9.46, we moreover obtain an isomorphism
$K_{B}\otimes_{K_{A}}K_{B}\cong K_{B}\otimes_{A}K_{B}$. Then combining with
Claim 9.45 gives an embedding of $B\otimes_{A}B$ into
$K_{B}\otimes_{K_{A}}K_{B}$.

Now to reduce to the case that $A$ and $B$ are fields, simply note that if
$B\otimes_{A}B$ had nilpotents, then so would $K_{B}\otimes_{K_{A}}K_{B}$. ∎

Now assume that $A$ and $B$ are fields, so that $A\leq B$ is a finite
separable extension. By the primitive element theorem, we have $B=A(b)$ for
some $b$. Let $p(x)$ be the minimal polynomial of $b$ over $A$. One then sees
easily that

$B\otimes_{A}B\cong A[x,y]/(p(x),p(y))\cong B[y]/(p(y)).$

By separability, $p$ factors over $B$ into a product of distinct irreducibles
$p_{i}$. Then by the Chinese Remainder Theorem, our tensor product is now
$\Pi_{i}B[y]/p_{i}(y)$, which is a product of fields, and thus reduced. ∎

We now move toward our version of purity of ramification. First, recall the
classical purity of ramification in algebraic geometry (see, e.g., [Sta24,
Lemma 0EA4]):

######  Fact 9.48.

Let $f:V\rightarrow W$ be a dominant morphism of irreducible varieties over
$K$. Assume that $V$ is normal and $W$ is smooth. Then every irreducible
component of the non-étale locus of $f$ has dimension at least $\dim(V)-1$.

Recall that a function $f:X\rightarrow Y$ between topological spaces is
topologically unramified at $x\in X$ if $f$ is injective on some neighborhood
of $x$. Our goal is to prove an analog of Fact 9.48 for $(\mathcal{K},\tau)$,
where we replace ‘étale’ with ‘topologically unramified’ (for the topology
$\tau$).

The proof will consist, essentially, of two steps. First, we use Fact 9.26 to
reduce to the case that $f$ is generically smooth. Then we show that, in the
generically smooth case, the topological ramification locus coincides with the
non-étale locus up to a codimension 2 error. Using these two steps, one can
reduce our statement to Fact 9.48.

The first of the two steps above is automatic. For the second step, we use the
following technical fact:

######  Proposition 9.49.

Let $V$ and $W$ be smooth irreducible varieties over $K$, and let
$f:V\rightarrow W$ be a dominant, generically smooth morphism. Then the
following are equivalent:

  1. (1)

$f$ is étale.

  2. (2)

The restricted map $f:V(K)\rightarrow W(K)$ is topologically unramified.

###### Proof.

Throughout, let $Z$ be the fiber product $V\times_{W}V$ (as schemes). Let
$Z_{red}$ be the reduced scheme associated to $Z$ (i.e. the fiber product as
varieties). So $Z_{red}(K)$ is the set of $(x,y)\in V(K)^{2}$ with
$f(x)=f(y)$. Moreover, let $\Delta$ be the diagonal (viewed as a subvariety of
$Z_{red}$ – equivalently the image of $V$ in $Z_{red}$). Finally, let
$\delta:V\rightarrow Z$ be the diagonal morphism. Since $V$ is a variety (thus
reduced), $\delta$ factors as $V\rightarrow Z_{red}\rightarrow Z$.

First suppose $f$ is étale. Then $f$ is unramified, so $\delta$ is an open
immersion, and thus $\Delta$ is Zariski open in $Z_{red}$. By Corollary 9.43,
$\Delta(K)$ is $\tau$-open in $Z_{red}(K)$. This is a restatement of the
assertion that $f:V(K)\rightarrow W(K)$ is topologically unramified.

Now assume $f:V(K)\rightarrow W(K)$ is topologically unramified. As above, we
get that $\Delta(K)$ is $\tau$-open in $Z_{red}(K)$, and thus $\Delta$ is
Zariski open in $Z_{red}$. We first conclude:

######  Lemma 9.50.

$f$ is quasifinite and flat.

###### Proof.

Since $\Delta$ is Zariski open in $Z_{red}$, it follows that for each $x\in
V(K)$, $\\{(x,x)\\}$ is open in $(Z_{red})_{x}(K)=\\{y:(x,y)\in
Z_{red}(K)\\}$. Equivalently, $\\{x\\}$ is open in the fiber $f^{-1}(f(x))$.
Thus $f$ has discrete fibers, and so is quasifinite. Flatness then follows by
Miracle Flatness [Sta24, Lemma 00R4] by looking at the induced maps of local
rings between points. ∎

By Lemma 9.50, we have:

######  Lemma 9.51.

$Z$ is reduced, and thus $Z_{red}\rightarrow Z$ is an isomorphism.

###### Proof.

After reducing to affine opens, this reduces exactly to Proposition 9.44. In
particular, the fact that $V=\mathrm{Spec}B\rightarrow\mathrm{Spec}A=W$ is
quasifinite and generically smooth guarantees that the field extension
$\operatorname{Frac}(A)\leq\operatorname{Frac}(B)$ appearing in Proposition
9.44 is finite separable. ∎

It remains to show that $f$ is unramified, that is, that $\delta$ is an open
immersion. But $\delta$ is automatically a (locally closed) immersion [Sta24,
Lemma 01KJ], namely it factorizes as $j\circ i$ where $i$ is a closed
immersion and $j$ is an open immersion. And we have shown it has open image;
By the irreducibility and the fact that the only surjective closed immersion
to a reduced scheme is an isomorphism, we have that we can assume $i$ is an
isomorphism. Thus, by Lemma 9.51, we are done. ∎

Finally, we are ready to ‘topologize’ Fact 9.48. In what follows, it will be
convenient to extend the notion of local dimension to non-definable sets:

######  Definition 9.52.

Let $V$ be a variety over $K$, and $X\subset V(K)$ arbitrary. Let $x\in X$,
and $d$ a non-negative integer. We say that $X$ has local dimension at least
$d$ at $x$ if every relative neighborhood of $x$ in $X$ contains a definable
set of dimension $d$.

We now show:

######  Theorem 9.53.

Let $f:V\rightarrow W$ be a dominant morphism of irreducible varieties over
$K$. Assume that $V$ is normal and $W$ is smooth. Let $R$ be the topological
ramification locus of $f$. Then $R$ has local dimension at least $\dim(V)-1$
at every point.

###### Proof.

First note that, by Lemma 9.26, we may assume $f$ is generically smooth (since
homeomorphisms preserve the topological ramification locus). In particular,
the variety $V^{\prime}$ obtained by Lemma 9.26 is still normal, so the
hypotheses of the theorem still hold.

Thus, moving forward, we assume $f$ is generically smooth. As in Proposition
9.49, let $Z$ be the fiber product $V\times_{W}V$ (as schemes), and let
$Z_{red}$ be the associated reduced scheme. Then let $\Delta$ be the diagonal
subvariety of $Z_{red}$, and $\delta:V\rightarrow Z$ the diagonal morphism,
which factors as $V\rightarrow Z_{red}\rightarrow Z$.

Toward a proof of the theorem, let $x\in R$, and let $U_{1}$ be any relative
neighborhood of $x$ in $R$. So $U_{1}=U\cap R$ for some open $U\subset V(K)$.
Shrinking $U_{1}$ and $U$ if necessary, we may assume $U$ is definable (in
fact so are $U_{1}$ and $R$, but this won’t be directly used). Our goal is to
find a definable $X\subset U_{1}$ with $\dim(X)=\dim(V)-1$.

First, since $f$ topologically ramifies at $x$, one sees that $(x,x)$ is not
in the $\tau$-interior of $\Delta(K)$ in $Z_{red}(K)$. By Corollary 9.43,
$(x,x)$ is not in the Zariski interior of $\Delta$ in $Z_{red}$. It follows
easily that $V\rightarrow Z_{red}$ is not an open map in any neighborhood of
$x$. But $Z_{red}\rightarrow Z$ is a Zariski homeomorphism, so $\delta$ is
also not an open map in any neighborhood of $x$. Thus, $\delta$ is not locally
an open immersion at $x$, and thus $f$ ramifies at $x$ [Sta24, Lemma 02GE].

So $f$ is not étale at $x$. By purity of ramification (Fact 9.48), there is an
irreducible closed codimension 1 subvariety $T\subset V$ such that $x\in T$
and $f$ is not étale on any point of $T$. Let $V^{S}$ be the smooth locus of
$V$. Let $T^{\prime}$ be the open subvariety $T\cap V^{S}\subset T$. Finally,
let $X=U\cap T^{\prime}(K)$. So $X$ is definable because $U$ is. To conclude,
we need the following two claims:

######  Claim 9.54.

$f$ topologically ramifies at every point of $X$, and thus $X\subset U_{1}$.

###### Proof.

By Proposition 9.49 applied to the restricted map $V^{S}(K)\rightarrow W(K)$,
it suffices to observe (by the choice of $T$) that $f$ is non-étale at every
point of $T$. ∎

######  Claim 9.55.

$\dim(X)=\dim(V)-1$.

###### Proof.

Since normal varieties are smooth outside codimension 2, we have

$\dim(T-V^{S})\leq\dim(V-V^{S})\leq\dim(V)-2<\dim(V)-1=\dim(T),$

and thus $\dim(T^{\prime})=\dim(T)$. So $T^{\prime}$ is an open subvariety of
$T$ of the same dimension; since $T$ is irreducible, this implies $T^{\prime}$
is Zariski dense in $T$. Then, by Corollary 9.43, $T^{\prime}(K)$ is also
$\tau$-dense in $T(K)$. In particular, since $U\cap T(K)$ is non-empty
(witnessed by $x$), it follows that $U\cap T^{\prime}(K)$ is also non-empty,
so that (by Fact 9.42)

$\dim(X)=\dim(U\cap T^{\prime}(K))=\dim(T^{\prime})=\dim(V)-1.$

∎

∎

Finally, we proceed to the proof of ramification purity. The first step is the
following approximation of the full statement:

######  Lemma 9.56.

Let $\mathcal{Y}=\\{Y_{t}:t\in U\\}$ and $\mathcal{Z}=\\{Z_{u}:u\in U\\}$ be
$\mathcal{K}(A)$-definable families of subsets of a $\mathcal{K}(A)$-definable
set $X$, with graphs $Y\subset X\times T$ and $Z\subset X\times U$. Let $I$ be
the set of $(x,t,u)$ with $(x,t)\in Y$ and $(x,u)\in Z$, and let $R$ be the
topological ramification locus of $I\rightarrow T\times U$. Suppose
$(x_{0},t_{0},u_{0})$ is a weakly generic $(\mathcal{Y},\mathcal{Z})$-multiple
intersection over $A$ which is strongly approximable over $A$. Then $R$ has
local dimension at least $d-1$ at $(x_{0},t_{0},u_{0})$, where
$d=\dim(Y)+\dim(Z)-\dim(X)$.

###### Proof.

The idea is to reduce all sets and maps in the statement to smooth varieties
and smooth morphisms, and then apply Theorem 9.53.

First, note that $x_{0}\in X^{S}$, $t_{0}\in T^{S}$, $u_{0}\in U^{S}$,
$(x_{0},t_{0})\in Y^{S}$, and $(x_{0},u_{0})\in Z^{S}$ (since each point is
generic in the corresponding set). Moreover, we are only interested in d-local
data of $X$, $Y$, $Z$, and $I$ near $(x_{0},t_{0},u_{0})$. So it is harmless
to assume each of $X$, $Y$, $Z$, $T$, and $U$ is the set of $K$-points of a
smooth variety. It is also harmless to replace $A$ with
$\operatorname{acl}(A)$, so we may assume these smooth varieties are all
irreducible. For ease of notation, let us call these varieties $X$, $Y$, $Z$,
$T$, and $U$, and rename the original definable sets as $X(K)$, $Y(K)$,
$Z(K)$, $T(K)$, and $U(K)$.

The assumption that $(x,t)\in Y$ and $x\in X$ are generic over $A$ gives that
$Y\rightarrow X$ is dominant. Similarly, $Z\rightarrow X$ is dominant. Now, it
is harmless to apply a Frobenius power to each of $T$ and $U$, since we are
only interested in topological data. So by Lemma 9.26, we may assume the
morphisms $Y\rightarrow X$ and $Z\rightarrow X$ are generically smooth. In
particular, by the genericity of $(x,t)\in Y$ and $(x,u)\in Z$, each of
$Y\rightarrow X$ and $Z\rightarrow X$ is smooth in a Zariski neighborhood of
the relevant point. It is harmless to shrink $Y$ and $Z$ so that $Y\rightarrow
X$ and $Z\rightarrow X$ are each smooth.

In this case, $I\rightarrow X$ is the fiber product of the two smooth
morphisms $Y\rightarrow X$ and $Z\rightarrow X$. Thus $I\rightarrow X$ is also
smooth. Since $X$ is smooth, this implies $I$ is also smooth. Let $I^{\prime}$
be an irreducible component of $I$ containing $(x_{0},t_{0},u_{0})$. Since we
are assuming $A=\operatorname{acl}(A)$, $I^{\prime}(K)$ is definable over $A$.

We now consider the morphism $I^{\prime}\rightarrow T\times U$ of irreducible
smooth varieties. We claim this morphism is dominant. To see this, note that
since $(x_{0},t_{0},u_{0})$ is strongly approximable, $I^{\prime}(K)$ contains
a strongly generic $(\mathcal{Y},\mathcal{Z})$-intersection $(x,t,u)$ over
$A$. By definition of strong genericity, $(t,u)$ is generic in $T(K)\times
U(K)$ over $A$. This shows that $I^{\prime}(K)\rightarrow T(K)\times U(K)$ has
generic image, and this implies dominance.

We have now set up Theorem 9.53 for the morphism $I^{\prime}\rightarrow
T\times U$. Let $R^{\prime}$ be the topological ramification locus of
$I^{\prime}\rightarrow T\times U$ (and note that $R^{\prime}\subset R$).
Theorem 9.53 tells us that $R^{\prime}$ has local dimension at least
$\dim(I^{\prime})-1$ at $(x_{0},t_{0},u_{0})$. Since $R^{\prime}\subset R$, we
conclude that $R$ also has local dimension at least $\dim(I^{\prime})-1$ at
$(x_{0},t_{0},u_{0})$.

Finally, to prove the lemma, we show that $\dim(I^{\prime})\geq
d=\dim(Y)+\dim(Z)-\dim(X)$. For this, we again use our strongly generic
intersection $(x,t,u)$. Indeed, an easy computation, using strong genericity,
yields that $\dim(xtu/A)=d$. Meanwhile, since $(x,t,u)\in I^{\prime}(K)$, we
have $\dim(xtu/A)\leq\dim(I^{\prime})$, and thus $\dim(I^{\prime})\geq d$ as
desired. ∎

We next deduce a slightly stronger statement:

######  Lemma 9.57.

Let $\mathcal{Y}=\\{Y_{t}:t\in T\\}$ and $\mathcal{Z}=\\{Z_{u}:u\in U\\}$ be
$\mathcal{K}(A)$-definable families of subset of a $\mathcal{K}(A)$-definable
set $X$, with graphs $Y\subset X\times T$ and $Z\subset X\times U$. Let
$(x_{0},t_{0},u_{0})$ be a generic $(\mathcal{Y},\mathcal{Z})$-multiple
intersection over $A$ which is strongly approximable over $A$. Then there is a
generic $(\mathcal{Y},\mathcal{Z})$-multiple intersection $(x,t,u)$ satisfying
$\dim(xtu/A)\geq d-1$, where $d=\dim(Y)+\dim(Z)-\dim(X)$.

###### Proof.

Let $I$ be the set of $(x,t,u)$ with $(x,t)\in Y$ and $(x,u)\in Z$, and let
$R$ be the topological ramification locus of $I\rightarrow T\times U$. We are
asked to find $(x,t,u)$ satisfying:

  1. (1)

$(x,t,u)\in R$.

  2. (2)

$x$, $t$, $u$, $(x,t)$, and $(x,u)$ are generic in $X$, $T$, $U$, $Y$, and $Z$
over $A$, respectively.

  3. (3)

$\dim(xtu/A)\geq d-1$.

Since $\mathcal{K}$ is $\aleph_{1}$-saturated, it suffices to realize any
finite fragment of (1)-(3) above. In particular, we will find $(x,t,u)$
realizing (1), (3), and a finite fragment of (2).

Let us briefly elaborate. For the purposes of this proof, given
$\mathcal{K}$-definable sets $D_{1}\subset D_{2}$, let us call $D_{1}$ large
in $D_{2}$ if $\dim(D_{2}-D_{1})<\dim(D_{2})$. It is easy to see that, if $D$
is $A$-definable, then the set of $A$-generic elements of $D$ is the
intersection of all $A$-definable large subsets of $D$. In particular, to show
some partial type is consistent with the $A$-generic locus of $D$, one can
show that it is consistent with each $A$-definable large subset of $D$. So our
plan is to satisfy (1), (3), and an approximation of (2) replacing each
$\mathcal{K}(A)$-definable set in the statement with a
$\mathcal{K}(A)$-definable large subset.

Now let us give the details. Suppose $X^{\prime}$, $Y^{\prime}$, $Z^{\prime}$,
$T^{\prime}$, and $U^{\prime}$ are any $\mathcal{K}(A)$-definable large
subsets of $X$, $Y$, $Z$, $T$, and $U$, respectively. Then, precisely, it
suffices to find $(x,t,u)$ satisfying (1), (2’), and (3), where (2’) is the
conjunction of $x\in X^{\prime}$, $(x,t)\in Y^{\prime}$, $(x,u)\in
Z^{\prime}$, $t\in T^{\prime}$, and $u\in U^{\prime}$.

We now construct such $(x,t,u)$. First, let $I^{\prime}$ be the set of
$(x,t,u)\in X^{\prime}\times T^{\prime}\times U^{\prime}$ with $(x,t)\in
Y^{\prime}$ and $(x,u)\in Z^{\prime}$. Let $R^{\prime}$ be the topological
ramification locus of the projection $I^{\prime}\rightarrow T^{\prime}\times
U^{\prime}$. Note that any $(x,t,u)\in R^{\prime}$ automatically satisfies (1)
and (2’). Thus, to additionally satisfy (3), it suffices to find an element
$(x,t,u)\in R^{\prime}$ with $\dim(xtu/A)\geq d-1$.

Now by Lemma 3.20, the sets $X$ and $X^{\prime}$ have the same germ at $x_{0}$
(that is, by genericity of $x_{0}$, they both realize the germ of
$\operatorname{tp}_{\mathcal{K}}(x_{0}/A)$); and the analogous statements hold
for $Y$, $Z$, $T$, and $U$. Since they are defined analogously, it follows
that $I$ and $I^{\prime}$ have the same germ at $(x_{0},t_{0},u_{0})$ and thus
that $R$ and $R^{\prime}$ have the same germ at $(x_{0},t_{0},u_{0})$. So
$R^{\prime}$ contains some relative neighborhood of $(x_{0},t_{0},u_{0})$ in
$R$. But by Lemma 9.56, every relative neighborhood of $(x_{0},t_{0},u_{0})$
in $R$ contains a definable set of dimension $d-1$. Thus, in particular,
$R^{\prime}$ contains a definable set – say $D$ – of dimension $d-1$. Let $D$
be definable over $B\supset A$, and let $(x,t,u)\in D$ be generic over $B$.
Then $(x,t,u)\in R^{\prime}$, and

$\dim(xtu/A)\geq\dim(xtu/B)=\dim(D)=d-1,$

as desired. ∎

######  Remark 9.58.

Note that in the above lemma, we cannot assume $R$ and $R^{\prime}$ are
definable (this would require a uniformly definable basis for $\tau$, and we
do not assume uniformity). This is the reason we needed the more general
notion of local dimension in Definition 9.52.

Finally, we show:

######  Theorem 9.59.

$(\mathcal{K},\tau)$ has ramification purity.

###### Proof.

Let $\mathcal{Y}=\\{Y_{t}:t\in T\\}$ and $\mathcal{Z}=\\{Z_{u}:u\in U\\}$ be
$\mathcal{K}(A)$-definable subsets of a $\mathcal{K}(A)$-definable set $X$,
with graphs $Y\subset X\times T$ and $Z\subset X\times U$. Let
$(x_{0},t_{0},u_{0})$ be a generic $(\mathcal{Y},\mathcal{Z})$-multiple
intersection over $A$ which is strongly approximable over $A$. We want to find
a generic $(\mathcal{Y},\mathcal{Z})$-multiple intersection over $A$ which has
codimension at most 1 over $A$.

By Lemma 9.57, there is a generic $(\mathcal{Y},\mathcal{Z})$-multiple
intersection $(x,t,u)$ over $A$ with $\dim(xtu/A)\geq d-1$, where
$d=\dim(Y)+\dim(Z)-\dim(X)$. We claim that $\operatorname{codim}_{A}(xtu)\leq
1$, which will complete the proof. Indeed, we have

  * •

$\dim(x/A)=\dim(X)$

  * •

$\dim(xt/A)=\dim(Y)$,

and thus by additivity, $\dim(t/Ax)=\dim(Y)-\dim(X)$. Similarly, we get
$\dim(u/Ax)=\dim(Z)-\dim(X)$.

Meanwhile, we have

  * •

$\dim(x/A)=\dim(X)$

  * •

$\dim(xtu/A)\geq d-1$,

and thus by additivity, $\dim(tu/Ax)\geq d-1-\dim(X)$. So, putting everything
together, we have

$\operatorname{codim}_{A}(xtu)=\dim(t/Ax)+\dim(u/Ax)-\dim(tu/Ax)$
$\leq\dim(Y)-\dim(X)+\dim(Z)-\dim(X)-(d-1-\dim(X))$
$\dim(Y)+\dim(Z)-\dim(X)-(d-1)=1.$

∎

Let us now drop the ambient assumptions on $(\mathcal{K},\tau)$. We end
Section 9 by collecting our results for ACVF and pure éz fields:

######  Corollary 9.60.

Every $\aleph_{1}$-saturated algebraically closed valued field, equipped with
the valuation topology, is a Hausdorff geometric structure with the open
mapping property and ramification purity.

###### Proof.

Let $(\mathcal{K},\tau)$ be such a field. By Lemma 9.11, $(\mathcal{K},\tau)$
is a Hausdorff geometric structure. By Theorem 9.59, $(\mathcal{K},\tau)$ has
ramification purity. Finally, to prove the open mapping property, it suffices
to verify the hypothesis of Lemma 9.38. So, let $f:V\rightarrow W$ be a quasi-
finite morphism of smooth $d$-dimensional $K$-varieties. Then by [Sta24, Lemma
0F32], $f$ is universally open, and thus by Fact 9.10, $V(K)\rightarrow W(K)$
is open. ∎

######  Corollary 9.61.

Let $(K,+,\cdot)$ be an $\aleph_{1}$-saturated éz field, whose étale open
topology is induced by a field topology. Then $(K,+,\cdot)$, equipped with the
étale open topology, is a Hausdorff geometric structure, and if $K$ has
characteristic zero then $(K,\tau)$ is differentiable.

###### Proof.

By Theorem 9.31 and Lemma 9.40. ∎

##  10\. Definable Slopes in ACVF

Our goal now is to show that algebraically closed valued fields have definable
slopes satisfying TIMI (Definition 7.10 and Definition 7.50). Throughout this
section, we fix an $\aleph_{1}$-saturated algebraically closed valued field
$(K,v)$. In particular, unlike the rest of the paper to this point, our
topology now has a uniformly definable basis. This means that we can definably
speak about germs of functions at a point (because the equivalence relation of
having the same germ is definable). We will do this throughout.

Otherwise, we use very little from the theory ACVF, so we do not elaborate
here. Interested readers may refer to [HHM06, §2.1], [HK06, §2.1] and
references therein for more background. Section 2.1 of the more recent [HOP24]
contains also some background (and relevant references) on the analytic theory
of functions definable in complete models of ACVF that is needed in a couple
of points below.

###  10.1. Taylor Groupoids

We will work with approximate versions of TIMI, successively generalizing
until we get the full statement. To streamline the presentation, we work with
ind-definable sets. For us, an $A$-ind-definable set is a countable union of
$A$-definable sets; a set is ind-definable if it is $A$-ind-definable for some
$A$; and a function is ($A$)-ind-definable if its graph is. Thus, an $A$-ind-
definable group is a group whose underlying set, composition operation, and
inverse operation, are all $A$-indefinable.

We want to work more generally with ind-definable groupoids:

######  Definition 10.1.

A $\emptyset$-ind-definable groupoid is a groupoid $\mathcal{C}$ such that (i)
the set of objects of $\mathcal{C}$, (ii) the set of morphisms of
$\mathcal{C}$, (iii) the composition operation on morphisms of $\mathcal{C}$,
and (iv) the inverse operation on morphisms of $\mathcal{C}$, are all
$\emptyset$-ind-definable.

######  Example 10.2.

Define the groupoid $\mathcal{LDH}$ (local definable homeomorphisms) whose
objects are the elements of $K$, and whose morphisms $x\rightarrow y$ are
germs of definable homeomorphisms between neighborhoods of $x$ and $y$. Then
$\mathcal{LDH}$ is a $\emptyset$-ind-definable groupoid.

Our first goal is to introduce, in the language of ind-definable groupoids, an
abstract notion of Taylor series.

######  Definition 10.3.

A weak Taylor groupoid consists of the following data and requirements:

  1. (1)

A $\emptyset$-ind-definable groupoid $\mathcal{C}$, which is a sub-groupoid of
$\mathcal{L}DH$.

  2. (2)

A $\emptyset$-ind-definable group $G$.

  3. (3)

A $\emptyset$-ind-definable homomorphism $a$ from morphisms in $\mathcal{C}$
to $G$. This means that if $g\circ f=h$ as morphisms in $\mathcal{C}$, then
$a(g)a(f)=a(h)$ in $G$.

  4. (4)

For each $n\geq 1$, a $\emptyset$-ind-definable map $f\mapsto c_{n}(f)$ from
morphisms in $\mathcal{C}$ to $K$, such that $c_{1}(f)\neq 0$ for all $f$.