!newop ACF
!newop eq
!newop Zar
!newop st
!newop stp
!newop loc
!newop cl
!newop trd
!newop dim
!newop inc
!newop len
!newop locus
!newop Stab
!newop ccl
!newop tp
!newop dcl
!newop acl
!newop Cb
!newop Aut
!newop End
!newop Isog
!newcal U
!newcal G
!newbb R
!newbb C
!newbb Z
!newbb Q
!newbb N

\renewcommand{\P}{\mathcal{P}}

\providecommand{\stR}{{ }^*\R}
\providecommand{\stC}{{ }^*\C}
\providecommand{\dl}{\delta}
\providecommand{\bdl}{\boldsymbol\dl}
\providecommand{\fdl}{\dl_{\"fin}}
\providecommand{\clbdl}{\cl_{\bdl}}
\providecommand{\dind}{\|/^{\bdl}}
\providecommand{\ndind}{\+/^{\bdl}}
\providecommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
\providecommand{\calE}{\mathcal{E}}
\renewcommand{\a}{\overline{a}}
\renewcommand{\b}{\overline{b}}
\renewcommand{\c}{\overline{c}}
\renewcommand{\d}{\overline{d}}
\newcommand{\g}{\overline{g}}
\newcommand{\h}{\overline{h}}
\newcommand{\pp}{\mathfrak{p}}
\newcommand{\qq}{\mathfrak{q}}
\renewcommand{\O}{\mathcal{O}}
\providecommand{\AA}{\mathbb{A}}
\renewcommand{\L}{\mathcal{L}}

\providecommand{\dcleq}{\dcl^{\eq}}
\providecommand{\acleq}{\acl^{\eq}}

\newcommand{\defnstyle}[1]{\textbf{#1}}

\newcommand{\zarcl}[1]{\overline{#1}^{\Zar}}

\newcommand{\interalg}{~~^0}

\newcommand{\modsim}{/\negmedspace\sim}

\providecommand{\concat}{\^}

\newcommand{\Geom}{\mathcal{G}}
\newcommand{\KGeom}{\Geom_K}

\newcommand{\Keq}{K^{\eq}}

\newcommand{\disjointunion}{\mathbin{\dot{\cup}}}
\newcommand{\Disjointunion}{\mathbin{\dot{\bigcup}}}

\newcommand{\sideremark}[1]{{\it \small (#1)}}

\theoremstyle{remark}
\newtheorem{setup}[theorem]{Setup}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{recall}[theorem]{Recall}
\newtheorem{answer}[theorem]{Answer}

\title{Geometry of combinatorially extremal algebraic configurations}
\author{Martin Bays}
\maketitle

Notes for a talk in Paris, 15.12.2017.
Part of a project joint with Emmanuel Breuillard.

## Bounds on finite algebraic configurations

$V (= \C^n$ irreducible algebraic set,
$N (- \N$,
$X_i (= \C$ with $|X_i| <= N$, $i=1...n$.
Then $|V \cap \Pi_i X_i| <= O(N^d)$ where $d=\dim(V)$.

Question{Q}:
  For what $V$ is this exponent $d$ optimal,
  i.e. $|V \cap \Pi_i X_i| </= O(N^{d-\epsilon})$ for any $\epsilon>0$.
  .

\sideremark{also interested in similar questions when co-ordinates $\C$ are 
replaced by higher dimensional varieties}

Theorem[Elekes-Szabó]:
  For $n=3$ and $d=2$,
  the exponent 2 is optimal iff either
  ***) $V$ is in co-ordinatewise correspondence with the graph of the group 
  operation of a 1-dimensional algebraic group $G$,
  i.e. $V$ is a component of the Zariski closure of
  $\{ (\alpha_1(g),\alpha_2(h),\alpha_3(g+h)) : g,h (- G \}$
  where $\alpha_i : G \rightarrow \C$ are finite-to-finite algebraic 
  correspondences,
  ** or $V$ projects to a curve, i.e. $\dim(\pi_{ij}(V))=1$ some $i/=j (- 
  \{1,2,3\}$.
  *e)*
  .
\sideremark{
Hong Wang, Raz-Sharir-deZeeuw: When 2 isn't optimal, 11/6 works.
(Could be that $1+\epsilon$ works for any $\epsilon>0$...)
Raz-Sharir-deZeeuw: case $(n=4, d=3)$.}

##Hrushovski $\bdl$-formalism

Hrushovski ``On Pseudo-Finite Dimensions'' (2013)

***) $K := \Pi_{i-->\U}K_i$,
  $\U (= \P(\omega)$ non-principal ultrafilter,
  $K_i$ expansions of $(\C;+,\cdot)$ in a countable language $\L )= 
  \L_{\"ring}$.

**
$X (= K^n$ is \defnstyle{internal} if $X = \Pi_{i-->\U}(X_i)$ for some $X_i (= 
K_i^n$.

**
Then $|X| := \Pi_{i-->\U} |X_i| (- \R^\U \cup \{\infty\}$.

**
Let $\xi_0 (- \R_{>0}^\U$ with $\xi_0 > \R$.

**
$\bdl(X) := \st ({ \frac{\log(|X|)}{\log(\xi_0)} }) (- \R_{>=0} \cup 
\{-\infty,\infty\}$
("coarse pseudofinite dimension").

**
Note $\bdl(X\times Y)=\bdl(X)+\bdl(Y)$, and $\bdl(X \cup Y) = 
\max(\bdl(X),\bdl(Y))$.

** $\bdl(\phi) := \bdl(\phi(K))$.
** $\bdl(\Phi) := \inf \{ \bdl(\phi) : \phi (- \Phi \}$.
** $\bdl(a/C) := \bdl(\tp(a/C))$.
*e)*

Assume $\bdl$ is \defnstyle{continuous}:
Given $\phi(x,y)$ and $\alpha (- \R$, for $\epsilon (- \R_{>0}$ exists 
definable $Y$ s.t. \[
\bdl(\phi(x,b)) <= \alpha => b (- Y => \bdl(\phi(x,b)) < \alpha+\epsilon .\]
Can add quantifiers $\exists_{<\xi_0^q}$ for $q (- \Q$ to get continuity.

Fact:--
  **(i) $a==_C b => \bdl(a/C)=\bdl(b/C)$.
  **(ii) $\bdl(ab/C)=\bdl(a/bC)+\bdl(b/C)$.
  **(iii) A partial type $\Phi$ over a countable set $C$ has a realisation
  $K |= \Phi(a)$ with $\bdl(a/C)=\bdl(\Phi)$.
  *ee*
  .

Fix $C_0$ a countable algebraically closed subfield of $K$.
Assume $C_0 (= \dcl({})$.

Definition:
  For $B (= K$,
  ***)$\acl^0(B) := C_0(B)^{\"alg} <= K$;
  **$\dim^0(B) := \trd(C_0(B)/C_0)$.
  *e)*

  Note $a (- \acl^0(B) => \bdl(a/B) = 0$.
  .

## Coherent modularity
#Coherence
Definition:
  $X (= K$ is (coarsely) \defnstyle{coherent} if
  $\dim^0(\a)=\bdl(\a)$ for any $\a (- X^{<\omega}$.
  .
Remark:
  If $\a = a_1...a_n$ and $\bdl(a_i)=\dim^0(a_i)$,
  then $\bdl(\a) <= \dim^0(\a)$,
  and $\{a_1,...,a_n\}$ is coherent iff $\bdl(\a)=\dim^0(\a)$.
  .

Remark:
  For $V$ over $C_0 (= \C$,
  the exponent $d=\dim(V)$ is optimal iff
  for some such structure $K$
  exists coherent $\a (- V(K)$ with $V = \locus(\a/C_0)$.

  Indeed: if $d$ is optimal,
  for $\epsilon>0$, for arbitrarily large $N$,
  have $|X_{i,N}| = N$ s.t. $|V \cap \Pi_i X_{i,N}| > N^{d-\epsilon}$.
  Take $K$ in language with $\Pi_{N --> \U} X_{i,N} =: X_i$ definable.
  Set $\xi_0 := |X_i|$.
  Then $\bdl(V \cap \Pi_i X_i) = d$,
  so say $\a (- V \cap \Pi_i X_i$ with $\bdl(\a)=d$,
  then $\a$ is coherent and generic in $V$.
  Converse is similar.
  .


#Geometries
Recall a \defnstyle{pregeometry} is a closure operator $\cl$ on a set $S$ 
satisfying exchange
\[a (- \cl(Cb) \\ \cl(C) => b (- \cl(Ca)\]
and finite character ($\cl(A) = \cup_{A_0 (=_{\"fin} A} \cl(A_0)$).

The associated \defnstyle{geometry} is
\[ \|P(S) := (S \\ \cl({}) ) / \{ \cl(x)=\cl(y) \} .\]

For $A(=S$, $\dim(A) = \min \{ |A_0| : A_0 (= A (= \cl(A_0) \}$.

Definition:
  A geometry $(S,\cl)$ is \defnstyle{modular} if for $a,b (- S$ and $C (= S$,
  if $a (- \cl(bC) \\ \cl(C)$
  then exists $c (- \cl(C)$ such that $a (- \cl(bc)$.
  .

If $V$ is a vector space over a division ring $R$,
$\|P(V) := \|P(V;\left<\cdot\right>_R)$, is modular.
$\KGeom:=\|P(K;\acl^0)$ is not modular: $a=c_1\cdot b+c_2$.

Definition[SKIP]:
  ***)
  $(S_1,\cl_1),(S_2,\cl_2)$ geometries.
  The \defnstyle{coproduct} is the geometry $(S_1\disjointunion 
  S_2,\cl_1\disjointunion \cl_2)$ where $(\cl_1\disjointunion \cl_2)(X_1 
  \disjointunion X_2) = \cl_1(X_1) \disjointunion \cl_2(X_2)$ for $X_i (= 
  S_i$.
  ** A _subgeometry_ of a geometry $(S;\cl)$ is $(X;\cl|_X)$ where $X(=S$ and 
  $\cl|_X(A) = \cl(A) \cap X$.
  *e)*
  .

Fact:
  $(S,\cl)$ modular geometry.

  Say $a,b (- S$ are \defnstyle{collinear} \sideremark{equiv: non-orthogonal} 
  if $a (- \cl(bc)$ for some $c /= a$.

  Then $(S,\cl)$ is the coproduct of the subgeometries on the collinearity 
  equivalence classes,
  and each class of $\dim>3$ is a projective geometry $\|P(V)$ over a division 
  ring (coordinatisation theorem of projective geometry).
  .

# Coherent modularity
Hrushovski observes that incidence bounds yield modularity.

If $y=a\cdot x+b$, $x,y,a,b (- K \\ C_0$,
then $\{x,y,a,b\}$ is not coherent:

Theorem[Szemeredi-Trotter for $\C$, due to Zahl]:
  For $P,L \subseteq  \C^2$ with $|P|,|L| \leq  N^2$,
  \[ |\{ ((x,y),(a,b)) \in P \times L : y = a\cdot x + b \}| <= O(N^{\frac 
  83}) .\]
  .

Lemma:
  For $X (= K$ coherent,
  $\ccl(X) := \{ c (- \acl^0(X) : c \textrm{ is coherent} \}$
  is coherent.
  .

Using generalisations of Szemeredi-Trotter proved in various levels of 
generality by various authors (one by Elekes-Szabó suffices for our purposes),

Proposition:
  If $X$ is coherent and $X=\ccl(X)$,
  then $\Geom_X := \Geom(X;\acl^0) (= \KGeom$ is a modular geometry.
  .

Remark:
  If $Ba$ is coherent and $a (/- \acl^0(B)$,
  then exists $a' ==_B a$ with $\bdl(a'/Ba) = \bdl(a'/B)$,
  and then $a' (/- \acl^0(Ba)$,
  and so $Baa'$ is coherent.
  .

Hence: $\Geom_X = \Disjointunion_i \Geom_i$ where for each $i$,
$\Geom_i = \{*\}$ or $\Geom_i (= \P(V_i) (= \KGeom$,
some $V_i$ vector space over a division ring $F_i$.


# Projective subgeometries of $\KGeom$

Example:
  Let $G$ be a 1-dimensional algebraic group over $C_0$,
  let $F <= \End^0_{C_0}(G) := \Q (x)_\Z \End_{C_0}(G)$
  be a division subring.
  Then $G(K)/G(C_0)$ is an $F$-vector space.
  Let $g_1,...,g_n (- G$ be independent generics.
  Then $\|P_F( <{ g_1/G(C_0),...,g_n/G(C_0) }>_F )$
  is a subgeometry of $\KGeom$.
  .
Theorem[Evans-Hrushovski '91]:
  Any projective subgeometry $\G (= \KGeom$ of dim $>=3$ of the above form.
\[ \xymatrix{
::: \G \ar@{^{(}->}[dr] \ar@{^{(}->}[d]^{\cl} \\
::: \mathbb{P}_F(G(K)/G(C_0)) \ar[r] & \KGeom } \]
  .

So suppose $\a$ coherent and each $(a_i,a_j)$ is collinear in $\ccl(\a)$,
and $\dim(\a)>1$.
Then there is a 1-dimensional group $G$
and $g_i$ $\acl^0$-interalgebraic with $a_i$,
s.t. $\locus(\g) = \ker(M)^0$ for some $M (- \"Mat(\End_{C_0}(G))$.
Same holds for $\dim(\a)=1$, with $G := \G_a$ and $g_i = g_j$.
So -

Theorem:
  The exponent $d=\dim(V)$ is optimal for $V$ iff,
  up to finite-to-finite correspondences on the co-ordinates,
  $V$ is a product of algebraic subgroups of powers of 1-dimensional algebraic 
  groups.
  .


## Higher dimension

Theorem[Elekes-Szabo]:
  $V (= W_1\times W_2\times W_3$, $\dim(W_i)=k$, $\dim(V)=2k$, all irreducible 
  complex.
  (Nice example: $W_i$ surfaces in $\C^3$, $V$ collinearity.)
  $X_i (= W_i$, $|X_i| = N$, in \defnstyle{general position}:
  for $W_i' (/= W_i$ proper subvariety,
  $|X_i \cap W_i'| (- O_{\deg(W_i')}(1)$.
  Then either $|V \cap \Pi_i X_i| <= O(N^{2-\eta})$ or $V$ is in 
  correspondence with a group operation (or trivial case).
  .

Remark:
  Example showing necessity of general position:
  $V :=$ graph of $(a_1,b_1)*(a_2,b_2) = (a_1+a_2+b_1^2b_2^2,b_1+b_2)$,
  $X_{i,N} := \{-N^4,...,N^4\}\times\{-N,...,N\} (= \C^2 =: W_i$.
  .

# Coarse general position

Definition:
  $\a (- K^{<\omega}$ is in \defnstyle{coarse general position} if
  \[ \dim^0(a/B) < \dim^0(a) => \bdl(a) = 0 \]
  for any $B (= K$.
  .

Definition:--
  ***) $\Keq := \bigcup_n K^n$.
  ** $X (= \Keq$ is (coarsely) \defnstyle{coherent} if every $a (- X$ is in 
    coarse general position and
    $\dim^0(\a)=\bdl(\a)$ for any $\a (- X^{<\omega}$.
  ** $\ccl(X) := \{ x (- \acleq(X) : \{x\} \;\textrm{is coherent}\}$.
  *e)*
  .
Theorem:
  Suppose $X = \ccl(X)$.
  Then $(X,\acl^{\eq^0})$ is modular.
  .

As in Evans-Hrushovski, using abelian group configuration,
$\G$ projective geometry of $\dim >=4$ in $\|P(\Keq,\acl^{eq})$
factors
\[ \xymatrix{
::: \G \ar@{^{(}->}[dr] \ar@{^{(}->}[d]^{\cl} \\
::: \mathbb{P}_F(G(K)/G(C_0)) \ar[r] & \mathbb{P}(\Keq,\acleq) } \]
for some abelian algebraic group $G$ and a division ring $F <= 
\End^0_{C_0}(G)$.

So obtain corresponding analogue of 1-dimensional theorem, generalising 
Elekes-Szabo.
