00footnotetext: MSC Primary: 52C10; Secondary: 03C98, 14C17.

# Elekes-Szabó for collinearity on cubic surfaces

Martin Bays, Jan Dobrowolski, and Tingxiang Zou  Martin Bays, Tingxiang Zou,
Institut für Mathematische Logik und Grundlagenforschung, Fachbereich
Mathematik und Informatik, Universität Münster, Einsteinstrasse 62, 48149
Münster, Germany  Jan Dobrowolski, The University of Manchester, Alan Turing
Building, Oxford Rd, Manchester, M13 9PL, UK, and Instytut Matematyczny,
Uniwersystet Wrocławski, Wrocław, Poland<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study the orchard problem on cubic surfaces. We classify possibly reducible
cubic surfaces $X\subseteq\mathbb{P}^{3}({\mathbb{C}})$ with smooth components
on which there exist families of finite sets (of unbounded size) with
quadratically many 3-rich lines which do not concentrate (in a natural sense)
on any projective plane. Namely, we prove that such a family exists precisely
when $X$ is a union of three planes sharing a common line.

Along the way, we obtain a general result about nilpotency of groups admitting
an algebraic action satisfying an Elekes-Szabó condition, and we prove the
following purely algebrogeometric statement: if the composition of four Geiser
involutions through sufficiently generic points $a,b,c,d$ on a smooth
irreducible cubic surface has infinitely many fixed points, then a single
plane contains $a,b,c,d$ and all but finitely many of the fixed points.

Dobrowolski was supported by DFG project BA 6785/2-1 and by EPSRC Grant
EP/V03619X/1. All authors were partially supported by DFG EXC 2044–390685587
and ANR-DFG AAPG2019 (Geomod).

###### Contents

  1. 1 Introduction
    1. 1.1 Summary of results
    2. 1.2 Background and motivation
    3. 1.3 Overview
    4. 1.4 Outline of the paper
    5. 1.5 Acknowledgements
  2. 2 Preliminaries
    1. 2.1 Non-standard setup
    2. 2.2 Coarse pseudofinite independence
      1. 2.2.1 Weak general position
      2. 2.2.2 Collinearity on planar cubic curves
  3. 3 Elekes-Szabó for collinearity on a cubic surface
  4. 4 The reducible case
    1. 4.1 Coherent group actions
    2. 4.2 Non-collinear planes
    3. 4.3 Quadric surface
  5. 5 The irreducible case
    1. 5.1 Collinearity on an irreducible smooth cubic surface
      1. 5.1.1 Algebraic geometry preliminaries
      2. 5.1.2 Some geometry on the surface $S$
      3. 5.1.3 Geiser involutions
    2. 5.2 Fixed sets of compositions of four Geiser involutions
      1. 5.2.1 Symmetry of degrees and multiplicities
      2. 5.2.2 Nonsingularity of $a,b,c,d$
      3. 5.2.3 Planarity
    3. 5.3 A variant on Szemerédi-Trotter
      1. 5.3.1 Pseudofinite consequences
    4. 5.4 Planarity of Elekes-Szabó triangles

##  1\. Introduction

###  1.1. Summary of results

We prove the following result for the 3-dimensional orchard problem on a
complex cubic surface $S$, showing that if a large finite subset of $S$ has
many collinear triples, this must be due to a coplanar subset.

######  Theorem 1.1 (Corollary 3.8, symmetric case).

For any $\epsilon>0$ there exists $\eta>0$ and $N_{0}\in{\mathbb{N}}$ such
that if $S\subseteq\mathbb{P}^{3}({\mathbb{C}})$ is a cubic surface with
smooth irreducible components which is not the union of three planes sharing a
common line, and $A\subseteq S$ is a finite subset with $|A|\geq N_{0}$, and
$|C|\geq|A|^{2-\eta}$ where $C:=\\{(a_{1},a_{2},a_{3})\in
A^{3}:a_{1},a_{2},a_{3}\text{ are distinct, collinear, and not contained in a
line contained in $S$}\\}$, then for some plane
$\pi\subseteq\mathbb{P}^{3}({\mathbb{C}})$ we have
$|C\cap\pi^{3}|\geq|A|^{2-\epsilon}$.

###  1.2. Background and motivation

The (planar) _orchard problem_ asks for finite configurations of points in the
real plane such that many lines pass through exactly three of the points. Call
such lines _3-rich_. See [3] for details of the long history of this problem,
and the result that for any $n\geq 3$, taking a subgroup of order $n$ of an
elliptic curve yields a configuration with $\lfloor n(n-3)/6\rfloor+1$ 3-rich
lines. In [16, Theorem 1.3, §9], Green and Tao show that this is optimal for
large $n$, and moreover that taking finite subgroups of irreducible cubics is
the only source of such optimal configurations.

In [12], Elekes and Szabó consider a coarse asymptotic version of the problem:
for what configurations of $n$ points in ${\mathbb{R}}^{2}$ are there $\sim
n^{2}$ lines passing through at least three of the points? Call such lines
_triple lines_. In this context, any (possibly reducible) planar cubic curve
provides such configurations, again by considering group structures induced by
collinearity. If one restricts to the case that the configuration is to be a
subset of an algebraic curve, the problem becomes an instance of that
considered in [13], where algebraic groups are seen to be the only possible
way to achieve the quadratic bound; [12] uses this to show that only cubic
curves yield quadratically many triple lines. A subsequent strengthening by
Raz, Sharir, and de Zeeuw [20, Corollary 6.2]) gives a bound
$O_{d}(n^{2-\frac{1}{6}})$ on the number of such lines for $n$ points on an
irreducible curve of degree $d>3$.

The motivation behind this paper is the problem of extending [13] to higher
dimension, without the restrictive general position assumption made in that
paper (nor the only slightly weaker “coarse general position” assumption in
[1]). As a natural example of this, we consider the _spatial orchard problem_
, in which one now considers configurations of $n$ points in
${\mathbb{R}}^{3}$ and again considers the number of triple lines. See [11]
and its references for what little literature there is on these spatial
problems.

We can again formulate a problem of Elekes-Szabó-type by taking an algebraic
surface in ${\mathbb{R}}^{3}$ and asking for configurations of $n$ points on
the surface with $\sim n^{2}$ triple lines. We have uninteresting solutions to
this: if $S$ is a cubic surface, then the intersection of $S$ with a plane is
a cubic curve, which typically supports a configuration as above. Distributing
our points across a family of $m=m(n)$ such planes, we can get
$\geq\Omega(\frac{n^{2}}{m})$ triple lines with $\leq O((\frac{n}{m})^{2})$ of
them on any given plane. To rule out these unoriginal solutions, we may phrase
the question as follows (where we also pass to the more algebraically natural
generality of surfaces in complex projective space rather than
${\mathbb{R}}^{3}$):

######  Question 1.2.

For which algebraic surfaces $S\subseteq\mathbb{P}^{3}({\mathbb{C}})$ is there
$\epsilon>0$ and $c>0$ such that there are arbitrarily large finite subsets
$A\subseteq S$ supporting $\geq c|A|^{2}$ triple lines, no $|A|^{2-\epsilon}$
of which are coplanar?

We answer this question with additional conditions on $S$: we assume
$S\subseteq\mathbb{P}^{3}({\mathbb{C}})$ is a possibly reducible cubic
surface, i.e. the zero-set of a cubic homogeneous polynomial with no repeated
factors, and that each irreducible component is smooth. We find that such an
$S$ satisfies the conditions of Question 1.2 if and only if $S$ is the union
of three planes sharing a common line. This is a consequence of Corollary 3.8
stated above, which in fact has better and uniform (albeit inexplicit) bounds.

###  1.3. Overview

In proving the results sketched above, our first move will be to pass to
pseudofinite sets (ultraproducts of finite sets). This pseudofinite setting
converts asymptotic questions about arbitrarily large finite sets into more
convenient questions about individual pseudofinite sets. We make the further
move of working with complete types in a suitable language, in the sense of
first-order logic. This small abstraction leads to cleaner statements and
proofs; in particular it makes “popularity arguments” an invisible part of the
formalism. Furthermore, it brings into play useful tools from model theory,
particularly independence calculus via Hrushovski’s coarse pseudofinite
dimension, which converts asymptotic exponents in finite cardinalities into a
real-valued dimension with good properties. This leads us naturally to the
notion of _weak general position_ , which is a useful formalisation of the
idea of the smallest subvariety on which a configuration concentrates, and is
a key technical tool in our analysis.

Our main proofs on cubic surfaces split into two largely disjoint cases,
according to whether the surface is reducible or not. In the reducible case,
the surface is either the union of a smooth quadric surface $Q$ and a plane
$P$, or the union of three distinct planes. In the former case, for $z\in
Q\setminus P$ we obtain an automorphism
$\gamma_{z}\in{\operatorname{Aut}}(Q)\cong{\operatorname{PO}}_{4}$ of the
quadric, defined generically by $\gamma_{z}(x)=y$ if $(x,y,z)$ are distinct
and collinear. Considering this along with the assumption that we have
quadratically many collinear triples, we prove a general result (Theorem 4.4)
showing that in such a situation we obtain an approximate subgroup of the
automorphism group, of a kind which contradicts results of Breuillard-Green-
Tao [4] unless its Zariski closure is a nilpotent subgroup. Analysing the
possibilities for such subgroups yields the result in this case. The case of
three planes is similar (and easier).

In the case of a smooth irreducible cubic surface, the corresponding
birational maps $\gamma_{z}$, known as _Geiser involutions_ in this case, do
not generate an algebraic group, and so the techniques of the previous
paragraph do not apply. Instead, we argue as follows. If $A\subseteq S$
supports quadratically many 3-rich lines, then the “energy” relation

$\displaystyle E:=\\{$ $\displaystyle((x_{1},x_{2}),(y_{1},y_{2}))\in
S^{2}\times S^{2}:$ $\displaystyle\exists z\in S.\;((x_{1},y_{1},z)\text{ and
}(x_{2},y_{2},z)\text{ are distinct and collinear})\\}$

has $|E\cap A^{4}|\sim|A|^{3}$, which would contradict known bounds of
Szemerédi-Trotter type if $E$ omitted the bipartite complete graph $K_{2,s}$
for sufficiently small $s$. In fact $E$ does not have this property, but we
show (under some genericity assumptions) that any $K_{2,s}$ in $E$ is planar
after deleting a bounded finite subset. Adapting the proof of such Szemerédi-
Trotter type bounds (we use the version due to Chernikov-Galvin-Starchenko,
[7, 9]), we deduce that the energy of $A$ concentrates on planes. Using the
bounds in the Tóth-Zahl result on Szemerédi-Trotter in
$\mathbb{P}^{2}({\mathbb{C}})$, we reduce to the case that the energy
concentrates on a single plane, which yields our result. To show that each
$K_{2,s}$ is almost planar, we consider a curve $C$ of fixed points of the
composition of four (sufficiently generic) Geiser involutions
$(\gamma_{a_{i}})_{i<4}$, and use algebro-geometric techniques to show that
$C$ is planar, and coplanar with the $a_{i}$. The main part of this argument
is a careful analysis ruling out hypothetical exceptional cases where the
$a_{i}$ are singular points on $C$; for this we use divisor class calculations
and the arithmetic genus.

Our methods may be seen as pointing a way towards a general analysis of
Elekes-Szabó problems in higher dimension. One formulation of this for
surfaces would be as follows. Suppose $R\subseteq S_{1}\times S_{2}\times
S_{3}$ is a constructible ternary relation between algebraic surfaces, such
that the projection to any pair $S_{i}\times S_{j}$ is dominant with
generically finite fibres. Suppose that for any $n>0$, we can find $N\geq n$
and $A_{i,n}\subseteq S_{i}$ with $|A_{i,n}|=N$ such that
$|R\cap\prod_{i}A_{i,n}|\geq N^{2-\frac{1}{n}}$. Then what can we say about
$R$ or the $A_{i,n}$? In [13, 1] it is shown that if $R$ is not the image
under algebraic correspondences on the co-ordinates of the graph of a
commutative group operation, then for some $i$ each $A_{i,n}$ has
$|A_{i,n}|^{O(1)}$ intersection with a subcurve of $S_{i}$. But the
possibilities once one allows such non-negligible intersections with curves
are not at all clear. The present work analyses this in the particular case
that $R$ is the collinearity relation on $S_{1}=S_{2}=S_{3}$ a cubic surface,
and we can consider what our methods and results have to say about the general
problem. On one hand we can take it as a warning that a straightforward
solution is unlikely, since even in this special case the analysis is
difficult and intricate. More positively, we can expect that at least some of
the ideas which arose in this case, in particular the role of infinite
complete bipartite subgraphs of the “energy” relation, will translate to the
general problem.

###  1.4. Outline of the paper

§2 contains preliminary definitions and results required for the rest of the
paper, in particular developing what we need of the calculus of pseudo-finite
dimensions.

In §3, we state our main results and reduce them to a certain condition that
“wES-triangles are planar”. The following sections then prove this separately
in the cases mentioned above. §4 deals with the reducible case, beginning with
an abstract result Theorem 4.4 on nilpotence of groups with a “coherent
action”, then applying this first in the warm-up case §4.2 of three non-
collinear planes, then in the case §4.3 of a quadric union a plane. Next we
turn in §5 to the case of an irreducible smooth cubic surface. We start by
developing in §5.1-§5.2 what we need of the algebraic geometry of Geiser
involutions, in particular our key result Theorem 5.13 on fixed curves of
compositions of four Geiser involutions. Then in §5.3 we give our revised
version of the Szemerédi-Trotter bounds, and finally we bring everything
together in §5.4 to prove planarity of wES-triangles in this case.

###  1.5. Acknowledgements

Thanks to Emmanuel Breuillard for suggesting the problem, and for helpful
initial discussion. Thanks to Yifan Jing for helpful discussion around the
topic of Remark 4.5.

##  2\. Preliminaries

###  2.1. Non-standard setup

We give an abbreviated version of the necessary setup for working with
pseudofinite dimensions. For a more leisurely account of the same approach,
see [1, Section 2].

Let ${\mathcal{U}}$ be a non-principal ultrafilter on ${\mathbb{N}}$. For a
set $Z$ the internal subsets of the ultrapower $Z^{\mathcal{U}}$ are the
elements of $\mathbb{P}(Z)^{\mathcal{U}}\subseteq\mathbb{P}(Z^{\mathcal{U}})$.
The finite cardinality function
$|\cdot|:\mathbb{P}(Z)\to{\mathbb{N}}\cup\\{\infty\\}$ induces a function
$|\cdot|:\mathbb{P}(Z)^{\mathcal{U}}\to{\mathbb{N}}^{\mathcal{U}}\cup\\{\infty\\}$
defining the non-standard cardinality $|X|$ of an internal subset $X$ of
$Z^{\mathcal{U}}$. We fix some $\xi\in{\mathbb{R}}^{\mathcal{U}}$ greater than
any standard real, and for $\alpha\in{\mathbb{R}}^{\mathcal{U}}_{\geq 0}$ we
set
$\boldsymbol{\delta}(\alpha)=\boldsymbol{\delta}_{\xi}(\alpha):={\operatorname{st}}\log_{\xi}\alpha\in{\mathbb{R}}\cup\\{-\infty,\infty\\}$,
where
${\operatorname{st}}:{\mathbb{R}}^{\mathcal{U}}\cup\\{-\infty,\infty\\}\to{\mathbb{R}}\cup\\{-\infty,\infty\\}$
is the standard part map, and we set
$\boldsymbol{\delta}(X):=\boldsymbol{\delta}(|X|)$. In other words,
$\boldsymbol{\delta}(\prod_{i\rightarrow{\mathcal{U}}}X_{i})$ is (when finite)
the metric ultralimit in ${\mathbb{R}}$ along ${\mathcal{U}}$ of
$\log_{\xi_{i}}(|X_{i}|)$, where $\xi=(\xi_{i})_{i}/{\mathcal{U}}$.

A $\bigwedge$-internal subset of $Z^{\mathcal{U}}$ is an intersection
$\bigcap_{i\in\omega}X_{i}$ of countably many internal subsets $X_{i}$. We
define
$\boldsymbol{\delta}(\bigcap_{i\in\omega}X_{i}):=\inf_{n\in\omega}\boldsymbol{\delta}(\bigcap_{i<n}X_{i})$.

A $\bigwedge$-internal set $X$ is broad if $0<\boldsymbol{\delta}(X)<\infty$.

Recall that $Z^{\mathcal{U}}$ is $\aleph_{1}$-compact, meaning that a
$\bigwedge$-internal set $\bigcap_{i\in\omega}X_{i}$ is non-empty as long as
each finite intersection $\bigcap_{i<n}X_{i}$ is non-empty.

######  Lemma 2.1.

Any $\bigwedge$-internal set $X$ contains an internal subset
$X^{\prime}\subseteq X$ with
$\boldsymbol{\delta}(X^{\prime})=\boldsymbol{\delta}(X)$.

###### Proof.

Say $X=\bigwedge_{i}X_{i}$ with each $X_{i}$ internal. Suppose
$a:=\boldsymbol{\delta}(X)\in{\mathbb{R}}$ (the case $a=\infty$ is similar,
and in the case $a=-\infty$ we can take $X^{\prime}:=\emptyset$). Then finite
intersections of

$\\{\\{Z\in\mathbb{P}(S)^{\mathcal{U}}:|Z|\geq\xi^{a-\frac{1}{m}}\\}:m>0\\}\cup\\{\\{Z\in\mathbb{P}(S)^{\mathcal{U}}:Z\subseteq
X_{i}\\}:i\in\omega\\}$

are non-empty, as witnessed by $\bigcap_{i<n}X_{i}$, and we conclude by
$\aleph_{1}$-compactness. ∎

Let ${\mathbb{K}}:={\mathbb{C}}^{\mathcal{U}}$.111We use ${\mathbb{C}}$ here
only for concreteness; all we will actually use about ${\mathbb{K}}$ is that
it is an ultraproduct across ${\mathcal{U}}$ of fields of characteristic 0.

###  2.2. Coarse pseudofinite independence

Here we define $\boldsymbol{\delta}$ for tuples and recall its basic
properties.

Consider ${\mathbb{K}}$ as a structure in a countable language ${\mathcal{L}}$
expanding the ring language, and suppose that every ${\mathcal{L}}$-definable
set of ${\mathbb{K}}$ is internal.

Then for $C\subseteq{\mathbb{K}}$ countable and $a$ a tuple from
${\mathbb{K}}$, let
$\boldsymbol{\delta}(a/C)=\boldsymbol{\delta}^{\mathcal{L}}(a/C):=\boldsymbol{\delta}({\operatorname{tp}}(a/C)({\mathbb{K}}))$
where
${\operatorname{tp}}(a/C)({\mathbb{K}})={\operatorname{tp}}^{\mathcal{L}}(a/C)({\mathbb{K}})=\bigcap_{\phi\in{\operatorname{tp}}^{\mathcal{L}}(a/C)}\phi({\mathbb{K}})$
is considered as a $\bigwedge$-internal set. We include ${\mathcal{L}}$ in the
notation only when we need to make it explicit. We write
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}b$ to mean
$\boldsymbol{\delta}(a/C)=\boldsymbol{\delta}(a/Cb)$.

We assume that $\boldsymbol{\delta}$ is continuous; see [1, Section 2.1.6] for
the definition of continuity and the fact that it is harmless to assume this,
since one can “close off” the language to a countable language in which
$\boldsymbol{\delta}$ is continuous and definability still implies
internality.

We have the following basic properties, which we will use throughout without
comment. We write $a^{\prime}\equiv_{C}a$ to mean
${\operatorname{tp}}(a^{\prime}/C)={\operatorname{tp}}(a/C)$.

######  Fact 2.2.

Let $a,b,d$ be tuples from ${\mathbb{K}}$ and $C\subseteq{\mathbb{K}}$ a
countable subset, and suppose
$\boldsymbol{\delta}(a/C),\boldsymbol{\delta}(b/C),\boldsymbol{\delta}(d/C)<\infty$

  * •

$a\equiv_{C}b\Rightarrow\boldsymbol{\delta}(a/C)=\boldsymbol{\delta}(b/C)$
(Invariance)

  * •

$\boldsymbol{\delta}(ab/C)=\boldsymbol{\delta}(a/Cb)+\boldsymbol{\delta}(b/C)$
(Additivity)

  * •

$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}b\Leftrightarrow
b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}a$ (Symmetry)

  * •

$ab\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}d\Leftrightarrow(b\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}d\wedge
a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{Cb}d)$ (Monotonicity and Transitivity)

  * •

For any tuple $e$ from ${\mathbb{K}}$, there exists $a^{\prime}\equiv_{C}a$
such that $a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}e$; (Extension)
more generally, any broad $\bigwedge$-definable set $X$ over $C$ contains an
element $x$ with $\boldsymbol{\delta}(x/C)=\boldsymbol{\delta}(X)$.

###### Proof.

Invariance is immediate from the definition. For Additivity, see e.g. [1, Fact
2.8]. Symmetry and Monotonicity and Transitivity follow. The second clause of
Extension is a consequence of $\aleph_{1}$-compactness and the countability of
${\mathcal{L}}$ and $C$, and the first clause follows, since it yields
$a^{\prime}\in{\operatorname{tp}}(a/C)({\mathbb{K}})$ with
$\boldsymbol{\delta}(a^{\prime}/Ce)=\boldsymbol{\delta}({\operatorname{tp}}(a/C)({\mathbb{K}}))=\boldsymbol{\delta}(a/C)$.
∎

We will often apply these definitions and properties to tuples of elements of
projective algebraic varieties or of arbitrary algebraic groups, rather than
just to tuples of elements of ${\mathbb{K}}$. This can be understood by
extending the above definitions and arguments to such elements, or by using
elimination of imaginaries in $\operatorname{ACF}$ to reduce to the case of
tuples, as in [1, Section 2.1.10]; in the case of a projective variety, this
just comes down to taking co-ordinates in one of the standard affine patches
$\mathbb{A}^{n}\subseteq\mathbb{P}^{n}$.

By _constructible_ (over $C\subseteq{\mathbb{K}}$) we mean: definable (over
$C$) in the field $({\mathbb{K}};+,\cdot)$. By quantifier elimination in the
theory of algebraically closed fields, the constructible subsets of a variety
are precisely the boolean combinations of Zariski closed subsets. The
dimension $\dim(X)$ of a constructible set is the dimension of its Zariski
closure.

######  Lemma 2.3.

Suppose $f:X\rightarrow Y$ is a constructible map of constructible sets, and
$A\subseteq X$ is a broad $\bigwedge$-definable subset, all over
$C\subseteq{\mathbb{K}}$, and $a\in A$ satisfies
$\boldsymbol{\delta}(a/C)=\boldsymbol{\delta}(A)$.

Then $a$ is also $\boldsymbol{\delta}$-generic in its fibre of $f$, namely
$\boldsymbol{\delta}(a/Cf(a))=\boldsymbol{\delta}(f^{-1}(f(a))\cap A)$.

###### Proof.

Let $F:=f^{-1}(f(a))\cap A$, and let $a^{\prime}\in F$ with
$\boldsymbol{\delta}(a^{\prime}/Cf(a))=\boldsymbol{\delta}(F)$.

Then

$\displaystyle\boldsymbol{\delta}(a/Cf(a))$
$\displaystyle=\boldsymbol{\delta}(a/C)-\boldsymbol{\delta}(f(a)/C)$
$\displaystyle=\boldsymbol{\delta}(A)-\boldsymbol{\delta}(f(a)/C)$
$\displaystyle\geq\boldsymbol{\delta}(a^{\prime}/C)-\boldsymbol{\delta}(f(a)/C)$
$\displaystyle=\boldsymbol{\delta}(a^{\prime}/Cf(a))=\boldsymbol{\delta}(F),$

and the converse inequality is immediate. ∎

######  Assumption 2.4.

The set of interpretations of the constants in our language ${\mathcal{L}}$ is
an algebraically closed subfield of ${\mathbb{K}}$.

We can always ensure this assumption holds by adding parameters.

######  Definition 2.5.

Let $E\leq{\mathbb{K}}$ be the algebraically closed field of interpretations
of the constants in our language ${\mathcal{L}}$. Then we use superscript-0 to
refer to the ring language expanded by constants for $E$, as follows. Let $C$
be a subset of ${\mathbb{K}}$.

  * •

${\operatorname{trd}}^{0}(a/C):={\operatorname{trd}}(a/CE)$ is transcendence
degree relativised to $E$.

  * •

$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{0}$ means
algebraic independence over $E$: $a\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{0}_{C}b$ if and
only if ${\operatorname{trd}}^{0}(a/Cb)={\operatorname{trd}}^{0}(a/C)$.

  * •

${\operatorname{acl}}^{0}(C):={\operatorname{acl}}^{\operatorname{ACF}}(EC)$
is the smallest algebraically closed subfield of ${\mathbb{K}}$ containing
$EC$.

  * •

$a\equiv^{0}_{C}b$ means that $a$ and $b$ have the same field type over $CE$,
i.e. they are in the same constructible sets over $CE$.

  * •

If $V$ is a variety over $CE$ and $a\in V({\mathbb{K}})$, then
${\operatorname{locus}}^{0}_{V}(a/C)$ is the smallest algebraic subvariety $W$
of $V$ over $CE$ which contains $a$.

  * •

A $C$-constructible0 set is a set definable in the ring language with
constants for $E$ and $C$, in other words a constructible set over $CE$.

  * •

A constructible0 set is a $\emptyset$-constructible0 set.

When we want to make the choice of language (and hence the choice of $E$)
explicit, we write ${\operatorname{trd}}^{0,{\mathcal{L}}}$ etc.

Note that ${\operatorname{trd}}^{0}$ and
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{0}$ satisfy the
properties of Fact 2.2. We will use these properties without comment.

If a variety $V$ is constructible over a countable $C\subseteq{\mathbb{K}}$, a
generic point of $V$ over $C$ is an $a\in V$ such that
${\operatorname{trd}}(a/C)=\dim(V)$, i.e. $a$ is contained in no lower-
dimensional subvariety of $V$ over $C$. Such an $a$ exists by
$\aleph_{1}$-compactness of ${\mathbb{K}}$. We say $a$ is generic in $V$ if it
is generic in $V$ over some $C$ (over which $V$ is constructible).

####  2.2.1. Weak general position

######  Definition 2.6.

For a countable set $C\subseteq{\mathbb{K}}$ and a tuple $a\in{\mathbb{K}}$,
the complete type ${\operatorname{tp}}(a/C)$ is in weak general position (or
is wgp) if $\boldsymbol{\delta}(a/C)<\infty$ and for any tuple
$b\in{\mathbb{K}}$ such that
${\operatorname{trd}}^{0}(a/Cb)<{\operatorname{trd}}^{0}(a/C)$, we have
$\boldsymbol{\delta}(a/Cb)<\boldsymbol{\delta}(a/C)$.

Equivalently: for any $b$, if $a\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}b$ then
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{0}_{C}b$.

######  Remark 2.7.

It follows from invariance of $\boldsymbol{\delta}$ that wgp is indeed a
property of the type rather than its realisation.

######  Definition 2.8.

A broad $\bigwedge$-internal set $A$ is wgp in a constructible set $X\supseteq
A$ if $\boldsymbol{\delta}(A\cap Y)<\boldsymbol{\delta}(A)$ for any
constructible $Y\subseteq X$ over ${\mathbb{K}}$ with $\dim(Y)<\dim(X)$.

######  Remark 2.9.

Compactness yields uniformity in this definition: if $A$ is wgp in $X$, and
$(V_{b})_{b}$ is a constructible family of lower dimensional constructible
subsets of $X$, then there exists $\eta$ such that $\boldsymbol{\delta}(A\cap
V_{b})\leq\boldsymbol{\delta}(A)-\eta$ for all $b$.

######  Lemma 2.10.

Let $V$ be a variety and let $a\in V({\mathbb{K}})$. Then
${\operatorname{tp}}(a/C)$ is wgp if and only if the $\bigwedge$-internal set
${\operatorname{tp}}(a/C)({\mathbb{K}})$ is wgp in
${\operatorname{locus}}^{0}_{V}(a/C)$.

###### Proof.

Suppose ${\operatorname{tp}}(a/C)({\mathbb{K}})$ is not wgp in
${\operatorname{locus}}^{0}_{V}(a/C)$, so say
$V_{b}\subseteq{\operatorname{locus}}^{0}_{V}(a/C)$ is a constructible subset
of lower dimension, but
$\boldsymbol{\delta}({\operatorname{tp}}(a/C)({\mathbb{K}})\cap
V_{b}({\mathbb{K}}))=\boldsymbol{\delta}(a/C)$. Then say $a^{\prime}$ realises
a completion of the partial type ${\operatorname{tp}}(a/C)\cup\\{x\in
V_{b}\\}$ with
$\boldsymbol{\delta}(a^{\prime}/Cb)=\boldsymbol{\delta}(a/C)=\boldsymbol{\delta}(a^{\prime}/C)$.
Then $a^{\prime}\in V_{b}$, so
${\operatorname{trd}}^{0}(a^{\prime}/Cb)<{\operatorname{trd}}^{0}(a^{\prime}/C)$,
so ${\operatorname{tp}}(a/C)={\operatorname{tp}}(a^{\prime}/C)$ is not wgp.

Conversely, if ${\operatorname{tp}}(a/C)$ is not wgp, then say
${\operatorname{trd}}^{0}(a/Cb)<{\operatorname{trd}}^{0}(a/C)$ but
$\boldsymbol{\delta}(a/Cb)=\boldsymbol{\delta}(a/C)$. Let
$V:={\operatorname{locus}}^{0}_{V}(a/Cb)\subseteq{\operatorname{locus}}^{0}_{V}(a/C)$.
Then $\dim(V)<\dim({\operatorname{locus}}^{0}_{V}(a/C))$ but
$\boldsymbol{\delta}({\operatorname{tp}}(a/C)({\mathbb{K}})\cap
V({\mathbb{K}}))=\boldsymbol{\delta}(a/C)$, so
$\boldsymbol{\delta}(a/C)({\mathbb{K}})$ is not wgp in
${\operatorname{locus}}^{0}_{V}(a/C)$. ∎

The following two lemmas will allow us to assume wgp in the situations we will
consider, by adding suitable parameters to the language.

######  Lemma 2.11.

For any tuple $a\in{\mathbb{K}}$ and countable set $C\subseteq{\mathbb{K}}$
such that $\boldsymbol{\delta}(a/C)<\infty$, there exists a tuple
$b\in{\mathbb{K}}$ such that $b\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}a$ and ${\operatorname{tp}}(a/Cb)$ is
wgp.

###### Proof.

By induction on ${\operatorname{trd}}^{0}(a/C)$. If ${\operatorname{tp}}(a/C)$
is not wgp, then there exists $b$ such that
${\operatorname{trd}}^{0}(a/Cb)<{\operatorname{trd}}^{0}(a/C)$ but
$\boldsymbol{\delta}(a/Cb)=\boldsymbol{\delta}(a/C)$. The latter implies by
Symmetry that $b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}a$. By the inductive hypothesis, there
exists $b^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{Cb}a$ with
${\operatorname{tp}}(a/Cbb^{\prime})$ wgp. By Transitivity,
$bb^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}a$, as required. ∎

######  Remark 2.12.

Although we won’t use it, it actually follows from [18, Lemma 2.13] that $b$
can be taken with $\boldsymbol{\delta}(b/C)=0$.

######  Lemma 2.13.

Suppose ${\operatorname{tp}}(a/C)$ is wgp. Let
${\mathcal{L}}^{\prime}\supseteq{\mathcal{L}}$ be obtained by adding new
constants for the elements of $C$ and closing off under continuity (adding no
further new constants in the process). Suppose
$\boldsymbol{\delta}^{{\mathcal{L}}^{\prime}}(a)=\boldsymbol{\delta}^{\mathcal{L}}(a/C)$.
Then ${\operatorname{tp}}^{{\mathcal{L}}^{\prime}}(a)$ is wgp.

###### Proof.

If
$\boldsymbol{\delta}^{{\mathcal{L}}^{\prime}}(a/b)=\boldsymbol{\delta}^{{\mathcal{L}}^{\prime}}(a)$,
then
$\boldsymbol{\delta}^{\mathcal{L}}(a/Cb)\geq\boldsymbol{\delta}^{{\mathcal{L}}^{\prime}}(a/b)=\boldsymbol{\delta}^{{\mathcal{L}}^{\prime}}(a)=\boldsymbol{\delta}^{\mathcal{L}}(a/C)$,
so
${\operatorname{trd}}^{0,{\mathcal{L}}}(a/Cb)={\operatorname{trd}}^{0,{\mathcal{L}}}(a/C)$
since ${\operatorname{tp}}^{\mathcal{L}}(a/C)$ is wgp, so

${\operatorname{trd}}^{0,{\mathcal{L}}^{\prime}}(a/b)={\operatorname{trd}}^{0,{\mathcal{L}}}(a/Cb)={\operatorname{trd}}^{0,{\mathcal{L}}}(a/C)={\operatorname{trd}}^{0,{\mathcal{L}}^{\prime}}(a).$

∎

######  Lemma 2.14.

If ${\operatorname{tp}}(a/C)$ is wgp and $d\in{\operatorname{acl}}^{0}(aC)$,
then ${\operatorname{tp}}(d/C)$ is wgp.

###### Proof.

Suppose $d\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{0}_{C}b$. Let $a^{\prime}\equiv_{Cd}a$ with
$a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{Cd}b$. Then
$a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{0}_{C}b$ since $d\in{\operatorname{acl}}^{0}(a^{\prime}C)$, and
${\operatorname{tp}}(a^{\prime}/C)={\operatorname{tp}}(a/C)$ is wgp, so
$a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}b$, and so
$d\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}b$; indeed, if
$d\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}b$, then since also
$a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{Cd}b$, by Transitivity we have
$a^{\prime}d\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}b$, and so in particular
$a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}b$. ∎

######  Lemma 2.15.

If ${\operatorname{tp}}(b/C)$ and ${\operatorname{tp}}(a/Cb)$ are wgp, then so
is ${\operatorname{tp}}(ab/C)$.

###### Proof.

If $ab\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{0}_{C}d$ then $a\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{0}_{Cb}d$ or $b\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{0}_{C}d$, so $a\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{Cb}d$ or
$b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}d$ by the wgp assumptions, so
$ab\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}d$ as required. ∎

######  Lemma 2.16.

If $a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}b$ and ${\operatorname{tp}}(a/C)$ is wgp,
then ${\operatorname{tp}}(a/Cb)$ is wgp.

###### Proof.

If $a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{0}_{Cb}d$, then $a\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{0}_{C}bd$, so $a\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{C}bd$ by wgp, so
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{Cb}d$ as required. ∎

####  2.2.2. Collinearity on planar cubic curves

Let $k$ be a field of characteristic 0 and let $C\subseteq\mathbb{P}^{2}(k)$
be an absolutely irreducible cubic planar curve over $k$. Then collinearity
induces an abelian group structure as follows; see e.g. [19, Theorem 2.1,
Example 2.2] or the references in the proof of Fact 2.18.

######  Fact 2.17.

Let $C^{\prime}$ be the set of non-singular points of $C(k)$. Suppose
$C^{\prime}$ is non-empty, and let $u\in C^{\prime}$. For $a,b,c\in
C^{\prime}$, write $a\circ b=c$ if the divisor of the intersection of some
projective line in $\mathbb{P}^{2}$ with $C(k)$ is the sum of the divisors
corresponding to the points $a,b,c$.

Then $x+y:=u\circ(x\circ y)$ defines an abelian group structure on
$C^{\prime}$, and $x\circ y=(u\circ u)-x-y$ for any $x,y\in C^{\prime}$.

###### Proof.

We give the calculation for the final statement:

$(x\circ y)+y+x=u\circ((x\circ y)\circ y)+x=u\circ x+x=u\circ((u\circ x)\circ
x)=u\circ u.$

∎

######  Fact 2.18.

Let $(C^{\prime};+)$ be as in Fact 2.17. Then for each $n$, the $n$-torsion
subgroup of $C^{\prime}$ is finite.

###### Proof.

We may assume $k$ is algebraically closed. If $C$ is smooth, this is a
standard fact about elliptic curves; see e.g. [22, Example 3.4, p.179].
Otherwise, $C^{\prime}$ is isomorphic to either the multiplicative or the
additive group of the field; this is well-known, we refer to [22, Exercise
1.7.3] for the fact that $C$ is projectively equivalent to either
$y^{2}=x^{3}+x^{2}$ (if the singularity is nodal) or $y^{2}=x^{3}$ (if the
singularity is cuspoidal), and [23, Theorem III.7] which proves (even though
the statement only discusses the rational points) that the group is then
isomorphic to the multiplicative or additive group respectively. Finiteness of
the $n$-torsion is then clear (since $\operatorname{char}(k)=0$). ∎

######  Fact 2.19.

If three points $a,b,c\in C(k)$ are distinct and collinear, then each is non-
singular on $C$.

###### Proof.

Since $C$ is cubic, it has at most one singular point (as in the proof of
Lemma 5.4 below), so say $b$ and $c$ are non-singular. But then by Fact 2.17,
$a=b\circ c$ is also in the group $C^{\prime}$ of non-singular points, so $a$
is also non-singular. ∎

##  3\. Elekes-Szabó for collinearity on a cubic surface

We refer to one-dimensional projective subspaces of $\mathbb{P}^{n}$ as
_lines_ , and two-dimensional projective subspaces as _planes_.

######  Definition 3.1.

Let $K$ be an algebraically closed field. For the purposes of this paper, a
cubic surface $S\subseteq\mathbb{P}^{3}(K)$ is the zero-set
$S=\\{[x:y:z:w]\in\mathbb{P}^{3}(K):f(x,y,z,w)=0\\}$ of a homogeneous degree 3
polynomial $f\in K[X,Y,Z,W]$ with no repeated prime factors; in other words,
$S$ is the set of $K$-rational points of the reduced (but possibly reducible)
projective variety defined by $f$. (The reducedness condition that there are
no repeated prime factors is relevant only in the case that $f$ splits into
linear factors, i.e. that $S$ is a union of planes, where it requires that the
three planes be distinct.)

The zero-sets of the factors $g_{i}$ of $f$ are the irreducible components of
$S$. We will often require these to be smooth, meaning that for each $i$, the
formal derivative $g_{i}^{\prime}$ has no common zero with $g_{i}$ in
$\mathbb{P}^{3}(K)$.

Let $S\subseteq\mathbb{P}^{3}({\mathbb{K}})$ be a cubic surface over
${\mathbb{K}}$. Assume that ${\mathcal{L}}$ contains constants for parameters
defining $S$.

######  Definition 3.2.

Let $R_{S}(x_{1},x_{2},x_{3})$ be the constructible relation:
$(x_{1},x_{2},x_{3})$ is a collinear triple of distinct points on $S$ not
contained in a line contained in $S$.

Note that if $R_{S}(a_{1},a_{2},a_{3})$, then
$a_{i}\in{\operatorname{acl}}^{0}(a_{j},a_{k})$ (for
$\\{i,j,k\\}=\\{1,2,3\\}$).

######  Definition 3.3.

  * •

A wgp Elekes-Szabó triangle (or wES-triangle) for $R_{S}$ is a triple
${\overline{a}}=(a_{1},a_{2},a_{3})\in R_{S}$ such that each
${\operatorname{tp}}(a_{i})$ is wgp and broad, and
$a_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}a_{j}$ for $i\neq j$.

A wES-triangle ${\overline{a}}$ is planar if there is a constructible0 plane
$\pi\subseteq\mathbb{P}^{3}$ with $a_{1},a_{2},a_{3}\in\pi$.

  * •

We say $R_{S}$ has no non-planar wES-triangles if any wES-triangle for $R_{S}$
is planar, for any choice of language ${\mathcal{L}}$ as in Section 2.2
containing constants for parameters defining $S$.

######  Lemma 3.4.

Suppose $R_{S}$ has no non-planar wES-triangles.

Let $A_{1},A_{2},A_{3}\subseteq\mathbb{P}^{3}$ be internal sets with
$\boldsymbol{\delta}(A_{i})=1$ and $\boldsymbol{\delta}(\prod_{i}A_{i}\cap
R_{S})=2$.

Then for some plane $\pi\subseteq\mathbb{P}^{3}$, we have
$\boldsymbol{\delta}(\prod_{i}A_{i}\cap R_{S}\cap\pi^{3})=2$ (and hence
$\boldsymbol{\delta}(A_{i}\cap\pi)=1$ for $i=1,2,3$).

###### Proof.

We may assume that the $A_{i}$ are $\emptyset$-definable in our language
${\mathcal{L}}$ (see Section 2.2).

Let ${\overline{a}}=(a_{1},a_{2},a_{3})\in\prod_{i}A_{i}\cap R_{S}$ with
$\boldsymbol{\delta}({\overline{a}})=2$, so
$\boldsymbol{\delta}(a_{i})=1=\boldsymbol{\delta}(a_{i}/a_{j})$ for $i\neq j$.

By Lemma 2.11, let $c\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}{\overline{a}}$ be such that
${\operatorname{tp}}({\overline{a}}/c)$ is wgp. Adding $c$ as constants to the
language and applying Lemma 2.13, we may assume that
${\operatorname{tp}}({\overline{a}})$ is wgp.

By Lemma 2.14, ${\operatorname{tp}}(a_{i})$ is wgp for $i=1,2,3$.

So ${\overline{a}}$ is a wES-triangle, so by assumption we obtain a
constructible0 plane $\pi\ni a_{1},a_{2},a_{3}$.

Then $({\overline{x}}\in\prod A_{i}\cap
R_{S}\cap\pi^{3})\in{\operatorname{tp}}({\overline{a}})$, so

$2=\boldsymbol{\delta}(\prod_{i}A_{i}\cap R_{S})\geq\boldsymbol{\delta}(\prod
A_{i}\cap R_{S}\cap\pi^{3})\geq\boldsymbol{\delta}({\overline{a}})=2.$

∎

The following lemma will allow us to rule out some degenerate cases when
proving planarity of wES-triangles.

######  Lemma 3.5.

Suppose ${\overline{a}}$ is a wES-triangle for $R_{S}$ and some $a_{i}$ is
contained in a constructible0 line. Then ${\overline{a}}$ is planar.

###### Proof.

Say $a_{1}$ is in a constructible0 line $\ell$. By the definition of $R_{S}$,
the projective subspace spanned by $\ell$ and $a_{2},a_{3}$ is a plane $\pi\ni
a_{1},a_{2},a_{3}$. Then the field of definition of $\pi$ is contained in
${\operatorname{acl}}^{0}(a_{2})\cap{\operatorname{acl}}^{0}(a_{3})={\operatorname{acl}}^{0}(\emptyset)$,
where this equality follows from $a_{2}\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{0}a_{3}$, which
follows from $a_{2}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}a_{3}$ and wgp. Since
${\operatorname{acl}}^{0}(\emptyset)$ is named by constants by assumption,
$\pi$ is constructible0. ∎

Recall that planes $P_{1},\ldots,P_{n}$ are said to be collinear if they share
a common line $\ell\subseteq\bigcap P_{i}$. This is the one exceptional case,
where we do get non-planar wES-triangles:

######  Example 3.6.

If $S$ is the union of three distinct collinear planes, then $R_{S}$ has a
non-planar wES-triangle.

Indeed, applying a projective linear transformation (recalling that
${\operatorname{PGL}}_{2}$ acts 3-transitively on $\mathbb{P}^{1}$), we may
assume that $S$ is the projective closure of the union of the three parallel
affine planes $P_{i}:=(i,{\mathbb{K}},{\mathbb{K}})\subseteq{\mathbb{K}}^{3}$
for $i=-1,0,1$.

Then we obtain an Elekes-Szabó configuration by taking grids. Explicitly, for
$n\in{\mathbb{N}}$ let
$A_{n}:=([-1,1]\times(-n,n)\times(-n,n))\cap{\mathbb{Z}}^{3}$, so
$|A_{n}|=3(2n-1)^{2}$. Then the number $T$ of collinear triples
$((-1,a,b),(0,a^{\prime},b^{\prime}),(1,2a^{\prime}-a,2b^{\prime}-b))$ in
${A_{n}}^{3}$ satisfies $(cn)^{4}\leq T\leq(2n-1)^{4}$ for some $c>0$
independent of $n$. So setting
$A:=\prod_{n\rightarrow{\mathcal{U}}}A_{n}\subseteq S$ and
$\xi:=\prod_{n\rightarrow{\mathcal{U}}}n\in{\mathbb{R}}_{S}^{\mathcal{U}}$, we
have $\boldsymbol{\delta}(A)=2$ and $\boldsymbol{\delta}(A^{3}\cap R_{S})=4$.

Let $(a_{1},a_{2},a_{3})\in A^{3}\cap R_{S}$ with $a_{i}\in P_{i}$ and
$\boldsymbol{\delta}(a_{1},a_{2},a_{3})=4$. Then
$\boldsymbol{\delta}(a_{i})=2$ and
$a_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}a_{j}$. Also
${\operatorname{tp}}(a_{i})({\mathbb{K}})$ is wgp in $P_{i}$ for $i=1,2,3$;
indeed, if $C\subseteq P_{i}$ is a curve, then $|A\cap C|\leq c^{\prime}\xi$
for some finite $c^{\prime}$ by considering the co-ordinate projections, so
$\boldsymbol{\delta}({\operatorname{tp}}(a_{i})({\mathbb{K}})\cap
C)\leq\boldsymbol{\delta}(A\cap C)\leq 1<2=\boldsymbol{\delta}(a_{i})$. It
follows that $P_{i}={\operatorname{locus}}^{0}_{S}a_{i}$ and, by Lemma 2.10,
that $a_{i}$ is wgp. Then ${\overline{a}}$ is a wES-triangle, and it is not
planar, since if $\pi\ni a_{1},a_{2},a_{3}$ is a constructible0 plane then
$a_{i}\in\mathbb{P}_{i}\cap\pi$ contradicts
$P_{i}={\operatorname{locus}}^{0}_{S}a_{i}$.

######  Theorem 3.7.

Suppose $S\subseteq\mathbb{P}^{3}({\mathbb{K}})$ is a cubic surface with
smooth irreducible components. Then $R_{S}$ has no non-planar wES-triangles if
and only if $S$ is not the union of three collinear planes.

Hence the conclusion of Lemma 3.4 holds for $R_{S}$.

###### Proof.

There are four cases.

  * •

$S$ is irreducible. We prove in Theorem 5.42 that $R_{S}$ then has no non-
planar wES-triangle.

  * •

$S$ has two irreducible components. Then $S$ is the union of a plane and a
smooth quadric surface, and we prove in Corollary 4.11 that $R_{S}$ has no
non-planar wES-triangle.

  * •

$S$ has three irreducible components. Then $S$ is the union of three distinct
planes, and we have two subcases.

    * –

The planes are collinear. Then $R_{S}$ has non-planar wES-triangles by Example
3.6.

    * –

The planes are not collinear. Proposition 4.6 shows that $R_{S}$ then has no
non-planar wES-triangle.

∎

We deduce a finitary statement. (The version stated in the introduction,
Theorem 1.1, is precisely the symmetric case $A_{1}=A_{2}=A_{3}=A$.)

######  Corollary 3.8.

For any $\epsilon>0$ there exists $\eta>0$ and $N_{0}\in{\mathbb{N}}$ such
that if $S\subseteq\mathbb{P}^{3}({\mathbb{C}})$ is a cubic surface with
smooth irreducible components which is not the union of three collinear
planes, and $A_{1},A_{2},A_{3}\subseteq S$ are finite subsets with
$|A_{1}|=|A_{2}|=|A_{3}|=:N\geq N_{0}$ such that
$|R_{S}\cap\prod_{i}A_{i}|>N^{2-\eta}$, then for some plane
$\pi\subseteq\mathbb{P}^{3}({\mathbb{C}})$ we have
$|R_{S}\cap\pi^{3}\cap\prod_{i}A_{i}|>N^{2-\epsilon}$.

###### Proof.

Otherwise, there is $\epsilon>0$ and for each $n\in{\mathbb{N}}$ such a cubic
surface $S_{n}\subseteq\mathbb{P}^{3}({\mathbb{C}})$ and finite subsets giving
a counterexample with $|(A_{i})_{n}|=N>n$ and $\eta=\frac{1}{n}$.

Let
$S:=\prod_{n\rightarrow{\mathcal{U}}}S_{n}\subseteq\mathbb{P}^{3}({\mathbb{K}})$.
We can write each $S_{n}$ as the zero set of an irreducible cubic polynomial,
or the product of irreducible linear or quadratic polynomials. It is a
standard and easy fact that smoothness and irreducibility are polynomially
definable properties of the coefficients of such a polynomial, so $S$ is also
a cubic surface with smooth irreducible components. In the case of three
planes, collinearity is also definable, so $S$ is not the union of collinear
planes. We have $\prod_{n\rightarrow{\mathcal{U}}}R_{S_{n}}=R_{S}$.

Setting $A_{i}:=\prod_{n\rightarrow{\mathcal{U}}}(A_{i})_{n}$ and
$\boldsymbol{\delta}:=\boldsymbol{\delta}_{\xi}$ where $\xi:=|A_{i}|$, we
contradict Theorem 3.7. ∎

######  Question 3.9.

How does $\eta$ depend on $\epsilon$? Can we take $\eta=\frac{1}{2}\epsilon$?

Also, does the following incomparable version of Corollary 3.8 hold
(corresponding to a version of Theorem 3.7 for the “fine” pseudofinite
dimension)? For any $c>0$ there exists $C>0$ and $N_{0}\in{\mathbb{N}}$ such
that if $S\subseteq\mathbb{P}^{3}({\mathbb{C}})$ is a smooth irreducible cubic
surface which is not the union of three collinear planes, and
$A_{1},A_{2},A_{3}\subseteq S$ are finite subsets with
$|A_{1}|=|A_{2}|=|A_{3}|=:N\geq N_{0}$ such that
$|R_{S}\cap\prod_{i}A_{i}|>CN^{2}$, then for some plane
$\pi\subseteq\mathbb{P}^{3}({\mathbb{C}})$ we have
$|R_{S}\cap\pi^{3}\cap\prod_{i}A_{i}|>cN^{2}$.

##  4\. The reducible case

In this section, we denote group operations and actions by concatenation.

###  4.1. Coherent group actions

In this subsection, we apply the results of [4] on nilpotence of
characteristic 0 algebraic groups admitting Zariski dense approximate
subgroups, or rather their reformulation in terms of _wgp-coherence_ given in
[2], to obtain an analogous criterion (Theorem 4.4 below) for an algebraic
group action.

######  Definition 4.1.

We say the graph $\Gamma_{G}=\\{(x,y,z):xy=z\\}$ of the group operation of a
connected algebraic group $G$ over ${\mathbb{K}}$ is wgp-coherent if there are
wgp in $G$ internal subsets $A_{1},A_{2},A_{3}\subseteq G$ such that
$\boldsymbol{\delta}(\prod_{j}A_{j}\cap\Gamma_{G})=2\boldsymbol{\delta}(A_{1})=2\boldsymbol{\delta}(A_{2})=2\boldsymbol{\delta}(A_{3})$.

The following fact is [2, Theorem 2.15$(i)\Rightarrow(iv)$]; it is stated
there with an assumption that $G$ is over ${\mathbb{C}}$, but this assumption
is not used for this part (only for $(iv)\Rightarrow(v)$).

######  Fact 4.2.

If the graph of the group operation of a connected algebraic group $G$ over
${\mathbb{K}}$ is wgp-coherent, then $G$ is nilpotent.

######  Lemma 4.3.

Let $G$ be a connected algebraic constructible0 group. Let $(V_{b})_{b}$ be a
constructible0 left-invariant family of proper subvarieties of $G$, where
left-invariance means that for all $b$ and $g\in G$ there is $b^{\prime}$ such
that $gV_{b}=V_{b^{\prime}}$.

Suppose $g\in G$ is generic in $G$ with ${\operatorname{tp}}(g)$ wgp, and let
(by Lemma 2.10 and Remark 2.9) $\epsilon>0$ be such that
$\boldsymbol{\delta}({\operatorname{tp}}(g)\cup\\{x\in
V_{b}\\})\leq\boldsymbol{\delta}(g)-\epsilon$ for all $b$.

Let $h\in G$ with $h\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}g$. Then also
$\boldsymbol{\delta}({\operatorname{tp}}(hg)\cup\\{x\in
V_{b}\\})\leq\boldsymbol{\delta}(hg)-\epsilon$ for all $b$.

###### Proof.

Fix $b$. Let $\alpha:=\boldsymbol{\delta}({\operatorname{tp}}(hg)\cup\\{x\in
V_{b}\\})$. We may assume that $\boldsymbol{\delta}(hg/b)=\alpha$ and
$h,g\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{hg}b$; indeed,
let (by Extension) $c$ realise an extension of the partial type
${\operatorname{tp}}(hg)\cup\\{x\in V_{b}\\}$ to a complete type over $b$ such
that also $\boldsymbol{\delta}(c/b)=\alpha$, then say
$h^{\prime},g^{\prime},c\equiv h,g,hg$ with
$h^{\prime},g^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{c}b$ (we can find such
$h^{\prime},g^{\prime}$ by applying Extension to the type over $c$
corresponding to ${\operatorname{tp}}(h,g/hg)$). Since
${\operatorname{tp}}(h^{\prime},g^{\prime},c)={\operatorname{tp}}(h,g,hg)$ and
our statement only concerns this type, we may assume
$(h,g,hg)=(h^{\prime},g^{\prime},c)$.

Then

$\displaystyle\alpha$ $\displaystyle=\boldsymbol{\delta}(hg/b)$
$\displaystyle=\boldsymbol{\delta}(g,h/b)-\boldsymbol{\delta}(g,h/hg,b)$
$\displaystyle=\boldsymbol{\delta}(g,h/b)-\boldsymbol{\delta}(g,h/hg),$

and

$\boldsymbol{\delta}(g,h/b)=\boldsymbol{\delta}(h/b)+\boldsymbol{\delta}(g/h,b)\leq\boldsymbol{\delta}(h)+\boldsymbol{\delta}(g)-\epsilon$

(since $g\in h^{-1}V_{b}$). But
$\boldsymbol{\delta}(h)+\boldsymbol{\delta}(g)-\boldsymbol{\delta}(g,h/hg)=\boldsymbol{\delta}(hg)$
(by independence of $h$ from $g$), so we conclude
$\alpha\leq\boldsymbol{\delta}(hg)-\epsilon$, as required. ∎

######  Theorem 4.4.

Let $G$ be an algebraic group acting on an irreducible variety $X$ by regular
maps, and suppose $G$, $X$, and the action are all constructible0.

Let $g\in G$ and $x\in X$, and suppose ${\operatorname{tp}}(g)$ and
${\operatorname{tp}}(x)$ are wgp and $g\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}x$ and
$g\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}gx$.

Let $Y:={\operatorname{locus}}^{0}_{G}(g)$, and let
$H:=\left<{Y^{-1}Y}\right>\leq G$.

Suppose further that no non-trivial element of $H$ fixes
$Z:={\operatorname{locus}}^{0}_{X}(x)$ pointwise, i.e. $\\{h\in H:\forall z\in
Z.\;hz=z\\}=\\{1\\}$.

Then $H$ is nilpotent.

###### Proof.

First we argue that by changing $x$ and the action, we may assume that $x$ has
trivial stabiliser under $H$.

######  Claim 4.4.1.

For some $n\in{\mathbb{N}}$, generics of $Z^{n}$ have trivial stabiliser under
the diagonal action of $H$ on $Z^{n}$.

###### Proof.

Let $(z_{i})_{i\in\omega}$ be a sequence of independent generics of $Z$, i.e.
each $z_{i}$ is generic in $Z$ over $z_{<i}:=\\{z_{j}:j<i\\}$. Then the
stabilisers $(H_{z_{<m}})_{m}$ form a descending chain, which stabilises by
Noetherianity of algebraic subgroups (which follows e.g. from
$\omega$-stability of $\operatorname{ACF}_{0}$). So for some $n$,
$H_{z_{<n}}=H_{z_{<n}z_{n}}$. Then $z_{n}$ lies in the set

$\\{z\in Z:\forall h\in H.\;(hz_{<n}=z_{<n}\Rightarrow hz=z)\\}=\bigcap_{h\in
H_{z_{<n}}}{\operatorname{Fix}}(h),$

which is closed (since the action is by regular maps) and is
$z_{<n}$-constructible0, so is equal to $Z$ by genericity of $z_{n}$. Hence
any element of $H_{z_{<n}}$ is in the pointwise stabiliser of $Z$, which is
trivial by assumption. ∎${}_{\mathit{\ref{c:trivStab}}}$

Now let $n$ be as in the claim, and let $\bar{x}=(x_{i})_{i<n}$ be a
$\boldsymbol{\delta}$-independent $n$-tuple of copies of $x$ over $g$, i.e.
$x_{i}\equiv_{g}x$ and $x_{i}\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{g}x_{<i}$. By wgp, $\bar{x}$ is generic in
$Z^{n}$. So $\bar{x}$ has trivial stabiliser with respect to the diagonal
action of $H$ on $X^{n}$, and $g\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}\bar{x}$ and
$g\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}g\bar{x}$. So by redefining $x$ as $\bar{x}$,
we may assume that $x$ has trivial stabiliser with respect to $H$.

Now let $g_{0}:=g$, and iteratively define $g_{i}$ for $i\in\omega$ as
follows.

Suppose $g_{i}$ satisfies:

(*) $g_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}x\text{ and
}g_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}g_{i}x\text{ and
}{\operatorname{tp}}(g_{i})\text{ is wgp, and if }i>0\text{ then }g_{i}\in H.$

Let $g_{i}^{\prime},x^{\prime}\equiv_{g_{i}x}g_{i},x$ with
$g_{i}^{\prime},x^{\prime}\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{g_{i}x}g_{i},x$. Then
$g_{i},g_{i}^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}x$; indeed,
$g_{i}^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}g_{i},x$, since
$g_{i}^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}g_{i},g_{i}x$, since
$g_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}g_{i}x$. By symmetry, also
$g_{i},g_{i}^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}x^{\prime}$.

Let $g_{i+1}:=g_{i}^{\prime-1}g_{i}$, so $x^{\prime}=g_{i+1}x$, and so
$g_{i+1}$ satisfies (*): $g_{i+1}\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}x$ and
$g_{i+1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}g_{i+1}x$ and (by Lemma 2.16, Lemma 2.14, and
Lemma 2.15) ${\operatorname{tp}}(g_{i+1})$ is wgp.

Now $g_{i+1}\in H$; for $i>0$ this follows from $g_{i}\in H$, and for $i=0$ it
follows from $g_{0}\in Y$. But $x$ has trivial stabiliser with respect to
$H\ni g_{i+1}$, so $g_{i+1}$ is interdefinable with $g_{i+1}x$ over $x$, so
$\boldsymbol{\delta}(g_{i+1})=\boldsymbol{\delta}(g_{i+1}/x)=\boldsymbol{\delta}(g_{i+1}x/x)\leq\boldsymbol{\delta}(g_{i+1}x)=\boldsymbol{\delta}(g_{i+1}x/g_{i+1})=\boldsymbol{\delta}(x/g_{i+1})=\boldsymbol{\delta}(x)$.
Meanwhile,
$\boldsymbol{\delta}(g_{i+1})\geq\boldsymbol{\delta}(g_{i+1}/g_{i}^{\prime})=\boldsymbol{\delta}(g_{i}/g_{i}^{\prime})=\boldsymbol{\delta}(g_{i})$.

So this iterative process yields a sequence $(g_{i})_{i<\omega}$ with non-
strictly increasing $\boldsymbol{\delta}$ bounded above by
$\boldsymbol{\delta}(x)$. For $i>0$, it follows from the independence that
$g_{i}$ is generic in $(Y^{-1}Y)^{i}$, which by Zilber indecomposability is
equal to $H$ for all large enough $i$, say for all $i\geq i_{0}$.

Let $\eta:=\sup_{i}\boldsymbol{\delta}(g_{i})$.

######  Claim 4.4.2.

There exist internal sets $A_{1},A_{2},A_{3}\subseteq H$ such that:

  1. (i)

$\boldsymbol{\delta}(A_{j})\leq\eta$;

  2. (ii)

$\boldsymbol{\delta}(\prod_{j}A_{j}\cap\Gamma_{H})\geq 2\eta$;

  3. (iii)

each $A_{j}$ is wgp in $H$.

###### Proof.

We prove this by $\aleph_{1}$-compactness.

Let $((X^{k}_{b})_{b})_{k\in\omega}$ enumerate the constructible0 left-
invariant families of proper subvarieties of $H$. Now for $i>i_{0}$, each
$g_{i}$ and each $g_{i}^{-1}$ can be written as $hg$ with $g\equiv g_{i_{0}}$
and $h\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}g$. Then
${\operatorname{tp}}(g)\equiv{\operatorname{tp}}(g_{i_{0}})$ is wgp and
generic in $H$, so by Lemma 4.3, for each $k$ we find $\epsilon_{k}>0$ such
that

$\forall b.\;\boldsymbol{\delta}({\operatorname{tp}}(g_{i})({\mathbb{K}})\cap
X^{k}_{b})\leq\boldsymbol{\delta}(g_{i})-\epsilon_{k}\leq\eta-\epsilon_{k},$

for all $i>i_{0}$, and the same inequality holds for
${\operatorname{tp}}(g_{i}^{-1})$.

Now given $m>0$, by definition of $\eta$ and the independence
$g_{i}^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}g_{i}$, we can find $i>i_{0}$ such that
$\boldsymbol{\delta}(g_{i},g_{i}^{\prime})\geq 2\eta-\frac{1}{m}$. Then a
compactness argument yields $\emptyset$-definable sets
$A_{1}\ni(g_{i}^{\prime})^{-1}$, $A_{2}\ni g_{i}$, and $A_{3}\ni g_{i+1}$ such
that $\boldsymbol{\delta}(A_{j})\leq\eta+\frac{1}{m}$, and
$\bigwedge_{k<m}\forall b.\;\boldsymbol{\delta}(A_{j}\cap
X^{k}_{b})\leq\eta-\epsilon_{k}+\frac{1}{m}$; the compactness argument for
this last condition may require some explanation, one way to see it is as
follows: ${\operatorname{tp}}(g_{i})$ is equivalent to the conjunction of a
chain of $\emptyset$-definable sets $\bigwedge_{l\in\omega}B_{l}$, and if
there were $\beta>0$ such that for all $l$ there is $b$ with
$\boldsymbol{\delta}(B_{l}\cap X^{k}_{b})\geq\eta-\epsilon_{k}+\beta$, then by
$\aleph_{1}$-compactness we would find $b_{0}$ such that
$\boldsymbol{\delta}(B_{l}\cap X^{k}_{b_{0}})\geq\eta-\epsilon_{k}+\beta$ for
all $l$, contradicting $\boldsymbol{\delta}({\operatorname{tp}}(g_{i})\cap
X^{k}_{b_{0}})\leq\eta-\epsilon_{k}$.

Now if $V_{b_{0}}\subsetneq H$ is a proper subvariety (equivalently, since $H$
is connected, a lower dimension subvariety), it is part of a constructible0
family $(V_{b})_{b}$ of such, and then $\\{hV_{b}:h\in H,\;b\\}$ is a left-
invariant constructible0 family of such. So every proper subvariety of $H$ is
of the form $X^{k}_{b}$ for some $k$ and $b$.

Meanwhile, $((g_{i}^{\prime})^{-1},g_{i},g_{i+1})\in\Gamma_{H}$, so

$\boldsymbol{\delta}(\prod_{j}A_{j}\cap\Gamma_{H})\geq\boldsymbol{\delta}((g_{i}^{\prime})^{-1},g_{i},g_{i+1})=\boldsymbol{\delta}(g_{i},g_{i}^{\prime})\geq
2\eta-\frac{1}{m}.$

The claim now follows by $\aleph_{1}$-compactness.
∎${}_{\mathit{\ref{c:wgpCoh}}}$

Hence $\Gamma_{H}$ is wgp-coherent, and nilpotence of $H$ follows from Fact
4.2. ∎

######  Remark 4.5.

Although the characteristic 0 assumption is necessary for Theorem 4.4 as
stated, it may be interesting to consider a positive characteristic version.
In particular, the main result of [6] can be seen to be of this form.

###  4.2. Non-collinear planes

Let $X$ be the union of three non-collinear planes, $P_{1},P_{2},P_{3}$. We
will show that if $P_{1},P_{2}$ and $P_{3}$ are not collinear, then $R_{X}$
has no non-planar wES-triangles.

This will serve as a warm-up exercise for the case considered in the next
subsection where $X$ is the union of a smooth quadric and a plane; there the
argument will be similar in outline, but the geometry will be somewhat more
complicated.

######  Proposition 4.6.

Suppose $P_{1},P_{2}$ and $P_{3}$ are not collinear, then $R_{X}$ has no non-
planar wES-triangles.

###### Proof.

Applying a projective linear transformation, we may assume
$P_{1}=\\{[0:y:z:w]:y,z,w\\}$ and $P_{2}=\\{[x:0:z:w]:x,z,w\\}$; since such a
transformation preserves collinearity and coplanarity, this does not affect
the existence of non-planar wES-triangles.

For $q\in\mathbb{P}^{3}\setminus(P_{1}\cup P_{2})$, denote by
$\gamma_{q}:P_{1}\to P_{2}$ the function which sends $p\in P_{1}$ to the
unique point $p^{\prime}\in P_{2}$ such that $p,p^{\prime}$ and $q$ are
collinear.

Let $q,q^{\prime}\in\mathbb{P}^{3}\setminus(P_{1}\cup P_{2})$. We can write
$q=[a:1:c:d]$ and $q^{\prime}=[1:b^{\prime}:c^{\prime}:d^{\prime}]$. Then we
calculate $\gamma_{q^{\prime}}^{-1}\circ\gamma_{q}:P_{1}\rightarrow P_{1}$ as

$\displaystyle\gamma^{-1}_{q^{\prime}}(\gamma_{q}([0:y:z:w]))$
$\displaystyle=\gamma^{-1}_{q^{\prime}}([-ay:0:z-cy:w-dy])$
$\displaystyle=[0:b^{\prime}ay:z+(c^{\prime}a-c)y:w+(d^{\prime}a-d)y].$

So $\gamma_{q^{\prime}}^{-1}\circ\gamma_{q}$ corresponds to the element of
${\operatorname{PGL}}_{3}\cong{\operatorname{Aut}}(P_{1})$ with matrix

$\begin{bmatrix}b^{\prime}a&0&0\\\ c^{\prime}a-c&1&0\\\
d^{\prime}a-d&0&1\end{bmatrix};$

in other words, $\gamma_{q^{\prime}}^{-1}\circ\gamma_{q}$ corresponds to

$((c^{\prime}a-c,d^{\prime}a-d),b^{\prime}a)\in{\mathbb{G}}_{a}^{2}\rtimes{\mathbb{G}}_{m}$

for the action of ${\mathbb{G}}_{a}^{2}\rtimes{\mathbb{G}}_{m}$ on $P_{1}$
defined by

$((a,b),c)[0:y:z:w]:=[0:cy:z+ay:w+by].$

We identify $\gamma^{-1}_{q^{\prime}}\circ\gamma_{q}$ with this element of
${\mathbb{G}}_{a}^{2}\rtimes{\mathbb{G}}_{m}$.

Let $(p_{1},p_{2},q)\in X^{3}$ be a wES-triangle for the collinearity relation
$R_{X}\subseteq P_{1}\times P_{2}\times P_{3}$. By the definition of $R_{X}$,
$(p_{1},p_{2},q)$ span a line which is not contained in $X$, so no two
elements of the triple are in the same plane $P_{i}$, so we may assume
$p_{1}\in P_{1}$, $p_{2}\in P_{2}$, and $q\in P_{3}\setminus\\{P_{1}\cup
P_{2}\\}$.

Take $(p^{\prime}_{1},q^{\prime})\equiv_{p_{2}}(p_{1},q)$ with
$p^{\prime}_{1},q^{\prime}\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{p_{2}}p_{1},q$. Let
$g:=\gamma^{-1}_{q^{\prime}}\circ\gamma_{q}\in{\mathbb{G}}_{a}^{2}\rtimes{\mathbb{G}}_{m}$,
so $g(p_{1})=p_{1}^{\prime}$ (where the action of
${\mathbb{G}}_{a}^{2}\rtimes{\mathbb{G}}_{m}$ on $P_{1}$ is that defined
above). We will now verify the conditions of Theorem 4.4 for
$g\in{\mathbb{G}}_{a}^{2}\rtimes{\mathbb{G}}_{m}$ and $p_{1}\in P_{1}$.

Since $q^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}p_{2}$, we get
$q^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}p_{1},q$ by transitivity, hence
$q^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}_{q}p_{1}$ by monotonicity, and then since
$q\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}p_{1}$ we obtain
$q,q^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}p_{1}$ and hence
$g\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}p_{1}$. Similarly,
$g\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}p_{1}^{\prime}$. Note that
${\operatorname{tp}}(g)$ is wgp: indeed, since
$q^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}q$ and ${\operatorname{tp}}(q^{\prime})$ is
wgp, ${\operatorname{tp}}(q^{\prime}/q)$ is wgp by Lemma 2.16, and then by
Lemma 2.15 ${\operatorname{tp}}(q^{\prime},q)$ is wgp, and as
$g\in{\operatorname{acl}}(q,q^{\prime})$ we conclude ${\operatorname{tp}}(g)$
is wgp by Lemma 2.14.

Let
$Y:={\operatorname{locus}}^{0}_{G}(g)\subseteq{\mathbb{G}}_{a}^{2}\rtimes{\mathbb{G}}_{m}$
and $Z:={\operatorname{locus}}^{0}_{S}(p_{1})\subseteq P_{1}$. Let $H:=\langle
YY^{-1}\rangle\leq{\mathbb{G}}_{a}^{2}\rtimes{\mathbb{G}}_{m}$. Note that for
any $p=[0:y:z:w]\in P_{1}$ and any
$h=((a,b),c)\in{\mathbb{G}}_{a}^{2}\rtimes{\mathbb{G}}_{m}$, we have
$h(p)=[0:cy:z+ay:w+by]=p$ if and only if $ay=(c-1)z$ and $by=(c-1)w$. If
$c\neq 1$, then $h$ only fixes the projective point $[0:c-1:a:b]$; then since
$Z$ is infinite, $h$ cannot fix $Z$ pointwise. If $c=1$, then $h$ fixes the
line $L_{0}=\\{[0:0:z^{\prime}:w^{\prime}]:z^{\prime},w^{\prime}\\}=P_{1}\cap
P_{2}$, which is constructible0. Suppose a non-trivial element $h\in H$ fixes
$Z$ pointwise. Then by the argument above, we must have $h=((a,b),1)$ for some
$a,b$ and $Z=L_{0}$. Then $p_{1}$ is in a constructible0 line, hence
$(p_{1},p_{2},q)$ is planar by Lemma 3.5, and we are done. Therefore, we may
assume no non-trivial element $h\in H$ fixes $Z$ pointwise. We may now apply
Theorem 4.4 and conclude that $H\leq G_{a}^{2}\rtimes{\mathbb{G}}_{m}$ is a
nilpotent algebraic subgroup.

######  Claim.

$H$ is of one of the following forms:

  1. (a)

$(V,1)$ for a vector subgroup $V\leq{\mathbb{G}}_{a}^{2}$;

  2. (b)

$\\{((\lambda-1)v,\lambda):\lambda\in{\mathbb{G}}_{m}\\}$ for some
$v\in{\mathbb{G}}_{a}^{2}$.

###### Proof.

We use only that $H$ is a connected nilpotent algebraic subgroup of
${\mathbb{G}}_{a}^{2}\rtimes{\mathbb{G}}_{m}$. Since
${\mathbb{G}}_{a}^{2}\rtimes{\mathbb{G}}_{m}$ is solvable non-nilpotent and
connected, $\dim(H)\leq 2$, so $H$ is abelian.

Suppose $H$ is not of the form in (a). Then the projection
$\pi_{2}(H)\leq{\mathbb{G}}_{m}$ must be the whole of ${\mathbb{G}}_{m}$ since
$H$ is connected, so say $(v,2)\in H$. Then if $(w,\lambda)\in H$, then
$(v,2)\cdot(w,\lambda)=(\lambda v+w,2\lambda)=(2w+v,\lambda
2)=(w,\lambda)\cdot(v,2)$, so $w=(\lambda-1)v$. ∎

Let $C:={\operatorname{locus}}^{0}_{S}(q)\subseteq P_{3}$. Say $q=[a:1:c:d]$
and $q^{\prime}=[1:b^{\prime}:c^{\prime}:d^{\prime}]$.

If $H\leq({\mathbb{G}}_{a}^{2},1)$, then $g\in({\mathbb{G}}_{a}^{2},1)$, so
$b^{\prime}a=1$. Then $q=[1:b^{\prime}:cb^{\prime}:db^{\prime}]$, so $q$ is
contained in the plane $\pi$ spanned by $q^{\prime}$ and $P_{1}\cap
P_{2}=\\{[0:0:z:w]:z,w\\}$. Note that $q\in\pi$,
$q^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{0}q$, and $\pi$
is $q^{\prime}$-constructible0. Therefore, $C\subseteq\pi$, which is collinear
with $P_{1}$ and $P_{2}$. By assumption, $P_{3}$ is not collinear with $P_{1}$
and $P_{2}$. Thus, $C\subseteq\pi\cap P_{3}$ is a line. By Lemma 3.5,
$(p_{1},p_{2},q)$ is planar.

It remains to consider case (b) of the Claim. So say
$H=\\{((\lambda-1)v,\lambda):\lambda\in{\mathbb{G}}_{m}\\}$ where
$v=(v_{1},v_{2})\in{\mathbb{G}}_{a}^{2}$. Now
$g=((c^{\prime}a-c,d^{\prime}a-d),b^{\prime}a)\in H$, so
$(c^{\prime}a-c,d^{\prime}a-d)=(b^{\prime}a-1)v$, so
$(a,1,c,d)=a(1,0,c^{\prime}-b^{\prime}v_{1},d^{\prime}-b^{\prime}v_{2})+(0,1,v_{1},v_{2})$.
So $q$ is on the line $L$ spanned by
$[1:0:c^{\prime}-b^{\prime}v_{1}:d^{\prime}-b^{\prime}v_{2}]$ and
$[0:1:v_{1}:v_{2}]$. But $\\{v\\}$ is constructible0: indeed, $Y$ and hence
$H$ is constructible0, and $\\{(v,2)\\}$ is the fibre over
$2\in{\mathbb{G}}_{m}$ of the projection. Hence $L$ is
$q^{\prime}$-constructible0. Since $q\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{0}q^{\prime}$
(since $q\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}q^{\prime}$ and ${\operatorname{tp}}(q)$ is
wgp), we get $C\subseteq L$. As $\dim^{0}(C)\geq 1$, we must have $C=L$, and
so $C$ is a line. By Lemma 3.5 again, $(p_{1},p_{2},q)$ is planar. ∎

###  4.3. Quadric surface

Let $S\subseteq\mathbb{P}^{3}$ be a cubic surface over ${\mathbb{K}}$ with two
smooth irreducible components. Since $S$ is cubic, one component is a plane
$P$, and the other is a smooth quadric surface $Q$.

We will prove (in Corollary 4.11 below) that our collinearity relation $R_{S}$
for $S$ has no non-planar wES-triangles. For this purpose, we are free to
transform $S$ by projective linear transformations of $\mathbb{P}^{3}$, since
they preserve collinearity and planarity.

By the principal axis theorem, see e.g. [21, IV.1.4, Theorem 1]), after
applying a projective linear transformation we may assume $Q$ has projective
equation $\sum_{i<4}\lambda_{i}x_{i}^{2}=0$ for some
$\lambda_{i}\in{\mathbb{K}}$. Since $Q$ is smooth we must have
$\lambda_{i}\neq 0$ for all $i$, so by a further projective linear
transformation we may assume

$Q=\\{[x_{0}:x_{1}:x_{2}:x_{3}]:\sum_{i<4}x_{i}^{2}=0\\}=\\{[{\overline{x}}]:{\overline{x}}\cdot{\overline{x}}=0\\},$

where we denote by $\cdot$ the standard scalar product on ${\mathbb{K}}^{4}$.

(It would be tempting at this point to apply a further projective linear
transformation to send $P$ to the plane $x_{3}=0$ “at infinity”, and work in
affine co-ordinates $(x_{0},x_{1},x_{2})$ with $Q$ as the “sphere”
$\sum_{i<3}x_{i}^{2}=1$. Some of the constructions below become more
transparent with this setup, and the reader may want to consider them in these
terms. However, this is possible only in the case that, writing
$P=\\{[{\overline{x}}]:{\overline{d}}\cdot{\overline{x}}=0\\}$, we have
${\overline{d}}\cdot{\overline{d}}\neq 0$, so to handle both cases uniformly
we will continue to work projectively.)

Now for $x\in\mathbb{P}^{3}\setminus Q$, denote by $\gamma_{x}:Q\rightarrow Q$
the involution defined by collinearity with $x$; more precisely, the map such
that for any $y\in Q$, the divisor of the intersection of $X=Q\cup P$ and the
line through $x$ and $y$ is precisely $x+y+\gamma_{x}(y)$.

For $x\in\mathbb{P}^{3}$, let $\widetilde{x}$ be its lift to a one-dimensional
subspace of ${\mathbb{K}}^{4}$, namely
$\widetilde{x}=\\{{\overline{x}}:[{\overline{x}}]=x\\}\cup\\{0\\}$, and for
$A\subseteq\mathbb{P}^{3}$ let $\widetilde{A}:=\bigcup_{a\in
A}\widetilde{a}\subseteq{\mathbb{K}}^{4}$. Denote the orthogonal space to a
subspace $V\leq{\mathbb{K}}^{4}$ by $V^{\perp}=\\{w:\forall v\in V.\;w\cdot
v=0\\}$. We use the same notation for a projective subspace
$P\subseteq\mathbb{P}^{3}$, i.e. $P^{\perp}$ is the projective subspace with
$\widetilde{P^{\perp}}=(\widetilde{P})^{\perp}$.

######  Lemma 4.7.

Let $x\in\mathbb{P}^{3}\setminus Q$. Let $\widetilde{\gamma}_{x}$ be the
element of
$O_{4}\setminus{\operatorname{SO}}_{4}\subseteq{\operatorname{GL}}_{4}$ which
is the identity on $\widetilde{x}^{\perp}$ and acts as $-1$ on
$\widetilde{x}$. Then $\gamma_{x}$ is the restriction to $Q$ of the image in
${\operatorname{PO}}_{4}\leq{\operatorname{PGL}}_{4}\widetilde{=}{\operatorname{Aut}}(\mathbb{P}^{3})$
of $\widetilde{\gamma}_{x}$ under projectivisation.

###### Proof.

These conditions do uniquely define an element
$\widetilde{\gamma}_{x}\in{\operatorname{GL}}_{4}$, since
$\widetilde{x}^{\perp}\cup\widetilde{x}$ span ${\mathbb{K}}^{4}$, since
$x\notin Q$. The matrix with respect to a corresponding orthonormal basis is
then $\operatorname{diag}(-1,1,1,1)$, so $\widetilde{\gamma}_{x}\in
O_{4}\setminus{\operatorname{SO}}_{4}$. In particular,
$\widetilde{\gamma}_{x}$ preserves $\cdot$, so its image $\gamma^{\prime}_{x}$
in ${\operatorname{PGL}}_{4}$ does restrict to a map $Q\rightarrow Q$.

Let $y=[{\overline{y}}]\in Q$. It remains to check that
$y,x,\gamma^{\prime}_{x}(y)$ are collinear. But indeed,
$\widetilde{\gamma}_{x}({\overline{y}})\in{\overline{y}}+\widetilde{x}$, so
$\widetilde{y},\widetilde{x},\widetilde{\gamma^{\prime}_{x}(y)}$ are linearly
dependent. ∎

Compositions of pairs of these involutions therefore generate a subgroup of
${\operatorname{PSO}}_{4}$, and since we aim to apply Theorem 4.4, we are
interested in which such subgroups are nilpotent, as well as which
subvarieties of $Q$ have non-trivial pointwise stabilisers. The following fact
clarifies these matters.

######  Fact 4.8.

  1. (i)

$Q\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$, and
${\operatorname{PSO}}_{4}\cong{\operatorname{PGL}}_{2}\times{\operatorname{PGL}}_{2}$.
We fix such isomorphisms, and treat them as identities.

Then the action of ${\operatorname{PSO}}_{4}$ on $Q$ respects these product
decompositions, i.e. $(\alpha,\beta)(x,y)=(\alpha(x),\beta(y))$.

Each subvariety $\\{x\\}\times\mathbb{P}^{1}\subseteq\mathbb{P}^{3}$ and
$\mathbb{P}^{1}\times\\{y\\}\subseteq\mathbb{P}^{3}$ is a line.

  2. (ii)

Any connected nilpotent algebraic subgroup of ${\operatorname{PSO}}_{4}$ is
abelian.

  3. (iii)

For any $\sigma\in{\operatorname{PSO}}_{4}\setminus\\{1\\}$, the fixed set
${\operatorname{Fix}}(\sigma)\subseteq Q$ is a finite union of points and
lines.

###### Proof.

  1. (i)

This is explained in e.g. [14, §18.2]. In fact, $Q\subseteq\mathbb{P}^{3}$ is
the image of the Segre embedding of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ into
$\mathbb{P}^{3}$; see [17, Exercise I.2.15].

  2. (ii)

If $N\leq{\operatorname{PSO}}_{4}$ is connected nilpotent, then so are its
images in ${\operatorname{PGL}}_{2}$ under the projection maps, so it suffices
to verify that any nilpotent connected algebraic subgroup of
${\operatorname{PGL}}_{2}$ is abelian. This follows for example from the
easily verified fact that any nilpotent Lie algebra of dimension $\leq 2$ is
abelian, and the simplicity of ${\operatorname{PGL}}_{2}$.

  3. (iii)

Say $\sigma=(\alpha,\beta)$. Then
${\operatorname{Fix}}(\sigma)={\operatorname{Fix}}(\alpha)\times{\operatorname{Fix}}(\beta)$.
But ${\operatorname{Fix}}(\alpha)\subseteq\mathbb{P}^{1}$ is either finite or
the whole of $\mathbb{P}^{1}$, so the result follows from (i).

∎

######  Lemma 4.9.

Let $a,b,c\in\mathbb{P}^{3}\setminus Q$, suppose $a\notin\\{b\\}\cup
b^{\perp}$, and suppose $\gamma_{c}\gamma_{a}$ commutes with
$\gamma_{b}\gamma_{c}$. Then $c\in\ell\cup\ell^{\perp}$, where $\ell$ is the
line spanned by $a,b$.

###### Proof.

Let $\widetilde{\gamma}_{a},\widetilde{\gamma}_{b},\widetilde{\gamma}_{c}$ be
as in Lemma 4.7. We use the notation $E_{\lambda}(T)$ for the
$\lambda$-eigenspace of $T$.

######  Claim 4.9.1.

$E_{1}(\widetilde{\gamma}_{a}\widetilde{\gamma}_{b})=\widetilde{\ell^{\perp}}=\left<{\widetilde{a},\widetilde{b}}\right>^{\perp}\leq{\mathbb{K}}^{4}$.

###### Proof.

By Lemma 4.7, $\left<{\widetilde{a},\widetilde{b}}\right>^{\perp}$ is
contained in the fixed sets of both $\widetilde{\gamma}_{a}$ and
$\widetilde{\gamma}_{b}$, and hence of the composition; for the converse,
suppose $\widetilde{\gamma}_{a}\widetilde{\gamma}_{b}(\beta)=\beta$. Then
$\widetilde{\gamma}_{a}(\beta)=\widetilde{\gamma}_{b}(\beta)$, so
$\beta-\widetilde{\gamma}_{a}(\beta)=\beta-\widetilde{\gamma}_{b}(\beta)$ is
in the intersection of the $(-1)$-eigenspaces of $\widetilde{\gamma}_{a}$ and
$\widetilde{\gamma}_{b}$, which is trivial since $a\neq b$, so
$\widetilde{\gamma}_{a}(\beta)=\beta=\widetilde{\gamma}_{b}(\beta)$, so
$\beta\in\widetilde{a}^{\perp}\cap\widetilde{b}^{\perp}=\left<{\widetilde{a},\widetilde{b}}\right>^{\perp}$.
∎${}_{\mathit{\ref{c:fixedLine}}}$

Now
$\gamma_{c}\gamma_{a}\gamma_{b}\gamma_{c}=\gamma_{b}\gamma_{c}\gamma_{c}\gamma_{a}=\gamma_{b}\gamma_{a}$,
so $\gamma_{b}\gamma_{a}\gamma_{c}=\gamma_{c}\gamma_{a}\gamma_{b}$, and so
$\widetilde{\gamma}_{b}\widetilde{\gamma}_{a}\widetilde{\gamma}_{c}=\lambda\widetilde{\gamma}_{c}\widetilde{\gamma}_{a}\widetilde{\gamma}_{b}$
for some scalar $\lambda\in{\mathbb{K}}$.

Now if $x\in\widetilde{\ell^{\perp}}$, then
$\widetilde{\gamma}_{b}\widetilde{\gamma}_{a}\widetilde{\gamma}_{c}(x)=\lambda\widetilde{\gamma}_{c}\widetilde{\gamma}_{a}\widetilde{\gamma}_{b}(x)=\lambda\widetilde{\gamma}_{c}(x)$,
so $\widetilde{\gamma}_{c}(x)\in
E_{\lambda}(\widetilde{\gamma}_{b}\widetilde{\gamma}_{a})$. So
$\dim(E_{\lambda}(\widetilde{\gamma}_{b}\widetilde{\gamma}_{a}))\geq 2$.

Suppose $\lambda\neq 1$. Now
$E_{1}(\widetilde{\gamma}_{b}\widetilde{\gamma}_{a})$ is the 2-dimensional
space $\widetilde{\ell^{\perp}}$ by (a symmetric version of) the claim, so
$\widetilde{\gamma}_{b}\widetilde{\gamma}_{a}\in{\operatorname{SO}}_{4}$ has a
basis of eigenvectors with eigenvalues $(\lambda,\lambda,1,1)$, so
$\lambda^{2}=1$ and hence $\lambda=-1$.

Then $E_{-1}(\widetilde{\gamma}_{b}\widetilde{\gamma}_{a})=\widetilde{\ell}$,
since it is orthogonal to
$E_{1}(\widetilde{\gamma}_{b}\widetilde{\gamma}_{a})$. So in particular, if
$a^{\prime}\in\widetilde{a}\setminus\\{0\\}$, then
$-a^{\prime}=\widetilde{\gamma}_{b}\widetilde{\gamma}_{a}(a^{\prime})=\widetilde{\gamma}_{b}(-a^{\prime})$,
so $-a^{\prime}\in(\widetilde{b})^{\perp}$, contradicting the assumption
$a\notin b^{\perp}$.

So $\lambda=1$ and we conclude that $\widetilde{\gamma}_{c}$ preserves
$(\widetilde{\ell})^{\perp}$, hence $\gamma_{c}$ preserves $\ell^{\perp}$.

Since $\widetilde{\gamma}_{c}\in O_{4}$, also $\gamma_{c}$ preserves
$\ell:=\ell=(\ell^{\perp})^{\perp}$.

Now suppose $c\notin\ell^{\perp}$, so say
$x\in\widetilde{\ell}\setminus\widetilde{c}^{\perp}$. Then
$\widetilde{\gamma}_{c}(x)-x\in\widetilde{c}\setminus\\{0\\}$ by Lemma 4.7,
but $\widetilde{\gamma}_{c}(x)\in\widetilde{\ell}$ as $\gamma_{c}$ preserves
$\ell$, so $\widetilde{c}\cap\widetilde{\ell}\neq\\{0\\}$, so $c\in\ell$, as
required. ∎

######  Theorem 4.10.

Suppose $a\in Q$, $c\in\mathbb{P}^{3}\setminus Q$, ${\operatorname{tp}}(a)$
and ${\operatorname{tp}}(c)$ are wgp, and
$c\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}a$ and
$c\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\boldsymbol{\delta}}\gamma_{c}(a)$.

Then either ${\operatorname{locus}}^{0}_{Q}(a)$ or
${\operatorname{locus}}^{0}_{\mathbb{P}^{3}}(c)$ is contained in a line.

###### Proof.

Suppose not.

Let $Z:={\operatorname{locus}}^{0}_{Q}(a)\subseteq Q$. Let
$\theta:\mathbb{P}^{3}\setminus Q\rightarrow{\operatorname{PO}}_{4}$ be the
(constructible0) embedding given by Lemma 4.7, so
$(\theta(c))a=\gamma_{c}(a)$, and let
$Y:={\operatorname{locus}}^{0}_{{\operatorname{PO}}(4)}(\theta(c))\subseteq{\operatorname{PO}}_{4}$.
Note that (by considering determinants)
$Y^{-1}Y=Y^{2}\subseteq{\operatorname{PSO}}_{4}$.

By Fact 4.8(iii), since $Z$ is not contained in a line, no non-trivial element
of ${\operatorname{PSO}}_{4}$ fixes $Z$ pointwise.

By Theorem 4.4, the subgroup $H$ of ${\operatorname{PSO}}_{4}$ generated by
$Y^{-1}Y=Y^{2}$ is nilpotent, and hence abelian by Fact 4.8(ii).

Let $(c_{1},c_{2})\in{\operatorname{locus}}^{0}_{\mathbb{P}^{3}}(c)^{2}$ be
generic. If $c_{1}\in(c_{2})^{\perp}$, then this closed condition holds for
all $(x,y)\in{\operatorname{locus}}^{0}_{\mathbb{P}^{3}}(c)^{2}$, which
implies that ${\operatorname{locus}}^{0}_{\mathbb{P}^{3}}(c)\subseteq Q$,
contradicting $c\notin Q$.

Let $\ell$ be the line spanned by $c_{1}$ and $c_{2}$. Then by Lemma 4.9,
${\operatorname{locus}}^{0}_{\mathbb{P}^{3}}(c)\setminus
Q\subseteq\ell\cup\ell^{\perp}$; but
${\operatorname{locus}}^{0}_{\mathbb{P}^{3}}(c)$ is irreducible, so it must be
contained in a line, contrary to assumption. ∎

######  Corollary 4.11.

$R_{S}$ has no non-planar wES-triangles.

###### Proof.

Let $(a_{1},a_{2},a_{3})\in R_{S}$ be a wES-triangle.

Permuting, we may assume that $a_{3}\in P\setminus Q$ and $a_{1},a_{2}\in Q$,
so $a_{2}=\gamma_{a_{3}}(a_{1})$. So by Theorem 4.10,
${\operatorname{locus}}^{0}_{S}(a_{1})$ or
${\operatorname{locus}}^{0}_{S}(a_{3})$ is contained in a line, and we
conclude by Lemma 3.5. ∎

##  5\. The irreducible case

Let $S\subseteq\mathbb{P}^{3}({\mathbb{K}})$ be (the $K$-points of) an
irreducible smooth cubic surface over ${\mathbb{K}}$. Assume that
${\mathcal{L}}$ contains constants for parameters defining $S$.

###  5.1. Collinearity on an irreducible smooth cubic surface

######  Definition 5.1.

$x\in S$ is good if $x$ is not contained in any line contained in $S$.

######  Fact 5.2.

$S$ contains only finitely many (namely 27) lines. In particular, the set of
good points is Zariski open, and so a generic point of $S$ is good.

###### Proof.

[17, Theorem V.4.9] ∎

Let $\Pi$ be the (Grassmannian) variety of projective planes in
$\mathbb{P}^{3}$.

####  5.1.1. Algebraic geometry preliminaries

Our primary reference for the algebraic geometry we will use is [17], and
especially [17, Chapter V] on surfaces and [17, Section V.4] on cubic surfaces
in $\mathbb{P}^{3}$.

We adopt the convention that varieties are reduced but not necessarily
irreducible. A curve is a 1-dimensional variety, and a surface is a
2-dimensional variety. A curve _in_ a variety is a 1-dimensional closed
subvariety of the surface. Except where otherwise mentioned, all varieties are
over the algebraically closed field ${\mathbb{K}}$, and we often use the same
notation for a variety and for its set of ${\mathbb{K}}$-points.

We work with Weil divisors [17, II.6]. Recall that on a smooth irreducible
variety, Weil divisors coincide with Cartier divisors [17, Proposition
II.6.11, Remark II.6.11.1A]. A morphism $f:X\rightarrow Y$ of smooth
irreducible varieties induces homomorphisms
$f^{*}:{\operatorname{Div}}(Y)\rightarrow{\operatorname{Div}}(X)$ of divisor
groups and $f^{*}:{\operatorname{Pic}}(Y)\rightarrow{\operatorname{Pic}}(X)$
of divisor class groups [17, Corollary II.6.16,Exercise II.6.8(a)]; context
will disambiguate which is meant by $f^{*}$.

We work with the multiplicity of a point $p$ on a curve $C$ in a surface,
denoted $\mu_{p}(C)$, as in [17, Remark V.3.5.2].

We also work with blow-ups at points; see [17, Section V.3] (where such blow-
ups are termed “monoidal transformations”).

####  5.1.2. Some geometry on the surface $S$

######  Fact 5.3.

  1. (a)

$S$ is the result of blowing up $\mathbb{P}^{2}$ at six distinct points
$P_{1},\ldots,P_{6}$, no three of which are collinear, and which do not lie on
a common conic.

  2. (b)

The divisor class group ${\operatorname{Pic}}(S)$ is freely generated as an
abelian group by $l,e_{1},\ldots,e_{6}$, where $e_{i}$ is the class of the
exceptional divisor at $P_{i}$ and $l$ is the lift of the divisor class of a
line in $\mathbb{P}^{2}$.

  3. (c)

The divisor class of a hyperplane section, $S\cap\pi$ for a plane $\pi\in\Pi$,
is $3l-\sum_{i=1}^{6}e_{i}$; conversely, any divisor with this class is a
hyperplane section.

  4. (d)

A curve $C$ in $S$ with divisor class $al-\sum_{i=1}^{6}b_{i}e_{i}$ has degree
$d(C)=3a-\sum_{i=1}^{6}b_{i}$ as a curve in $\mathbb{P}^{3}$.

###### Proof.

  1. (a)

[17, Remark V.4.7.1]]

  2. (b)

[17, Proposition V.4.8(a)]

  3. (c)

[17, Proposition V.4.8(c)], and the converse statement follows from the fact
[17, Corollary V.4.7] that the projective embedding is that of this (very
ample) linear system.

  4. (d)

[17, Proposition V.4.8(e)]

∎

######  Lemma 5.4.

If $C\subseteq S$ is an irreducible curve which is not contained in any plane
$\pi\subseteq\mathbb{P}^{3}$, and $a_{1},a_{2},a_{3}\in C$ are three distinct
points on $C$, then $\sum_{i=1}^{3}\mu_{a_{i}}(C)\leq d(C)$.

If $C$ is contained in a plane and is not a line, we still have
$\sum_{i=1}^{2}\mu_{a_{i}}(C)\leq d(C)$ for any two distinct points
$a_{1},a_{2}\in C$.

###### Proof.

First suppose $C$ is not contained in a plane, let $\pi\in\Pi$ be the plane
containing $a_{1},a_{2},a_{3}$, and let $D$ be $\pi\cap S$.

Then $d(C)$ is precisely the intersection product $C.D$; this can be seen from
first principles or by calculating with divisor classes (namely: $D$ has class
$3l-\sum_{i}e_{i}$, so if $C$ has class $al-\sum_{i}b_{i}e_{i}$ then, by the
formulae for the intersection pairing in [17, Proposition V.4.8(b)], we have
$C.D=3a-\sum_{i}b_{i}=d(C)$).

But by [17, Exercise V.3.2], $C.D=\sum_{a\in
A}\mu_{a}(C)\mu_{a}(D)\geq\sum_{i}\mu_{a_{i}}(C)$, where $A$ is a set of
generalised intersection points which includes $a_{1},a_{2},a_{3}$.

For the case that $C$ is contained in a plane $\pi_{0}$ and $a_{1},a_{2}\in C$
are given, the same argument applies by taking $\pi$ to be any plane through
$a_{1},a_{2}$ distinct from $\pi_{0}$. ∎

For $a\in S$, let $C_{a}$ be the intersection of $S$ with the tangent plane at
$a$,

$C_{a}:=T_{a}(S)\cap S.$

######  Fact 5.5.

If $a\in S$ is good, then $C_{a}$ is irreducible, $d(C_{a})=3$, and $a$ is its
unique singular point: $\mu_{a}(C_{a})=2$, and $\mu_{b}(C_{a})=1$ for $b\in
C\setminus\\{a\\}$.

###### Proof.

$d(C_{a})=3$ by Fact 5.3(c,d).

Let $C^{\prime}$ be the irreducible component of $C_{a}$ containing $a$; this
is unique since $d(C_{a})=3$ and $a$ is good. Then $\mu_{a}(C^{\prime})>1$;
this follows from [17, Remark V.3.5.2] since $a$ is singular on $C^{\prime}$
(or, more directly, from the proof of [17, Theorem I.5.1]).

So by Lemma 5.4 applied with $a$ and any other point $b$ on $C^{\prime}$, we
must have $\mu_{a}(C^{\prime})=2$, $\mu_{b}(C^{\prime})=1$, and
$d(C^{\prime})=3$, and hence $C_{a}=C^{\prime}$ is irreducible. ∎

####  5.1.3. Geiser involutions

Given a good point $a\in S$, the Geiser involution $\gamma_{a}$ through $a$ is
the birational map $\gamma_{a}\in\operatorname{Bir}(S)$ defined generically on
$S$ by setting $\gamma_{a}(x)$ to be the third point of intersection (counting
multiplicity) with $S$ of the line in $\mathbb{P}^{3}$ through $a$ and $x$.

This $\gamma_{a}$ lifts to an automorphism of the blowup $\widetilde{S}_{a}$
of $S$ at $a$,
$\widetilde{\gamma}_{a}\in\operatorname{Aut}(\widetilde{S}_{a})$. Namely (as
described in e.g. [8, 2.6.1] and [5, 4.1,Involutions.3]), the morphism
$\widetilde{S}_{a}\rightarrow\mathbb{P}^{2}$ of linear projection through $a$
is a double covering, and $\widetilde{\gamma}_{a}$ is the corresponding
involution which exchanges the sheets of this covering; in particular, it