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\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.

\section{Bounds on finite algebraic configurations}

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

\begin{question} \label{Q}
  For what $V$ is this exponent $d$ optimal,
\\  i.e.\ $|V \cap \Pi_i X_i| \not\leq  O(N^{d-\epsilon})$ for any $\epsilon>0$.
\end{question}

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

\begin{theorem}[Elekes-Szab\'o]
  For $n=3$ and $d=2$,
\\  the exponent 2 is optimal iff either
  \begin{itemize}\item $V$ 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
  $\{ (\alpha_1(g),\alpha_2(h),\alpha_3(g+h)) : g,h \in G \}$
  where $\alpha_i : G \rightarrow \C$ are finite-to-finite algebraic 
  correspondences,
  \item or $V$ projects to a curve, i.e.\ $\dim(\pi_{ij}(V))=1$ some $i\neq j \in 
  \{1,2,3\}$.
  \end{itemize}
\end{theorem}
\sideremark{
Hong Wang, Raz-Sharir-deZeeuw: When 2 isn't optimal, 11/6 works.
\\(Could be that $1+\epsilon$ works for any $\epsilon>0$\ldots )
\\Raz-Sharir-deZeeuw: case $(n=4, d=3)$.}

\section{Hrushovski $\bdl$-formalism}

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

\begin{itemize}\item $K := \Pi_{i\rightarrow \U}K_i$,
\\  $\U \subseteq  \P(\omega)$ non-principal ultrafilter,
\\  $K_i$ expansions of $(\C;+,\cdot)$ in a countable language $\L \supseteq 
  \L_{\operatorname{ring}}$.

\item
$X \subseteq  K^n$ is \defnstyle{internal} if $X = \Pi_{i\rightarrow \U}(X_i)$ for some $X_i \subseteq  
K_i^n$.

\item
Then $|X| := \Pi_{i\rightarrow \U} |X_i| \in \R^\U \cup \{\infty\}$.

\item
Let $\xi_0 \in \R_{>0}^\U$ with $\xi_0 > \R$.

\item
$\bdl(X) := \st \left({\frac{\log(|X|)}{\log(\xi_0)}}\right) \in \R_{\geq 0} \cup 
\{-\infty,\infty\}$
("coarse pseudofinite dimension").

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

\item $\bdl(\phi) := \bdl(\phi(K))$.
\item $\bdl(\Phi) := \inf \{ \bdl(\phi) : \phi \in \Phi \}$.
\item $\bdl(a/C) := \bdl(\tp(a/C))$.
\end{itemize}

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

\begin{fact} \mbox{}
  \begin{enumerate}[(i)]\item $a\equiv _C b \Rightarrow  \bdl(a/C)=\bdl(b/C)$.
  \item $\bdl(ab/C)=\bdl(a/bC)+\bdl(b/C)$.
  \item A partial type $\Phi$ over a countable set $C$ has a realisation
  $K \vDash  \Phi(a)$ with $\bdl(a/C)=\bdl(\Phi)$.
  \end{enumerate}
\end{fact}

Fix $C_0$ a countable algebraically closed subfield of $K$.
\\Assume $C_0 \subseteq  \dcl(\emptyset )$.

\begin{definition}
  For $B \subseteq  K$,
  \begin{itemize}\item$\acl^0(B) := C_0(B)^{\operatorname{alg}} \leq  K$;
  \item$\dim^0(B) := \trd(C_0(B)/C_0)$.
  \end{itemize}

  Note $a \in \acl^0(B) \Rightarrow  \bdl(a/B) = 0$.
\end{definition}

\section{Coherent modularity}
\subsection{Coherence}
\begin{definition}
  $X \subseteq  K$ is (coarsely) \defnstyle{coherent} if
  $\dim^0(\a)=\bdl(\a)$ for any $\a \in X^{<\omega}$.
\end{definition}
\begin{remark}
  If $\a = a_1\ldots a_n$ and $\bdl(a_i)=\dim^0(a_i)$,
\\  then $\bdl(\a) \leq  \dim^0(\a)$,
\\  and $\{a_1,\ldots ,a_n\}$ is coherent iff $\bdl(\a)=\dim^0(\a)$.
\end{remark}

\begin{remark}
  For $V$ over $C_0 \subseteq  \C$,
\\  the exponent $d=\dim(V)$ is optimal iff
  for some such structure $K$
  exists coherent $\a \in V(K)$ 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 $\Pi_{N \rightarrow  \U} X_{i,N} =: X_i$ definable.
\\  Set $\xi_0 := |X_i|$.
\\  Then $\bdl(V \cap \Pi_i X_i) = d$,
\\  so say $\a \in V \cap \Pi_i X_i$ with $\bdl(\a)=d$,
\\  then $\a$ is coherent and generic in $V$.
\\  Converse is similar.
\end{remark}


\subsection{Geometries}
Recall a \defnstyle{pregeometry} is a closure operator $\cl$ on a set $S$ 
satisfying exchange
\[a \in \cl(Cb) \setminus \cl(C) \Rightarrow  b \in \cl(Ca)\]
and finite character ($\cl(A) = \cup_{A_0 \subseteq _{\operatorname{fin}} A} \cl(A_0)$).

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

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

\begin{definition}
  A geometry $(S,\cl)$ is \defnstyle{modular} if for $a,b \in S$ and $C \subseteq  S$,
\\  if $a \in \cl(bC) \setminus \cl(C)$
  then exists $c \in \cl(C)$ such that $a \in \cl(bc)$.
\end{definition}

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

\begin{definition}[SKIP]
  \begin{itemize}\item
  $(S_1,\cl_1),(S_2,\cl_2)$ 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 $X_i \subseteq  
  S_i$.
  \item A \underline{subgeometry} of a geometry $(S;\cl)$ is $(X;\cl\negmedspace\restriction_X)$ where $X\subseteq S$ and 
  $\cl\negmedspace\restriction_X(A) = \cl(A) \cap X$.
  \end{itemize}
\end{definition}

\begin{fact}
  $(S,\cl)$ modular geometry.

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

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

\subsection{Coherent modularity}
Hrushovski observes that incidence bounds yield modularity.

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

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

\begin{lemma}
  For $X \subseteq  K$ coherent,
\\  $\ccl(X) := \{ c \in \acl^0(X) : c \textrm{ is coherent} \}$
  is coherent.
\end{lemma}

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

\begin{proposition}
  If $X$ is coherent and $X=\ccl(X)$,
\\  then $\Geom_X := \Geom(X;\acl^0) \subseteq  \KGeom$ is a modular geometry.
\end{proposition}

\begin{remark}
  If $Ba$ is coherent and $a \notin \acl^0(B)$,
\\  then exists $a' \equiv _B a$ with $\bdl(a'/Ba) = \bdl(a'/B)$,
\\  and then $a' \notin \acl^0(Ba)$,
\\  and so $Baa'$ is coherent.
\end{remark}

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


\subsection{Projective subgeometries of $\KGeom$}

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

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 $M \in \operatorname{Mat}(\End_{C_0}(G))$.
\\Same holds for $\dim(\a)=1$, with $G := \G_a$ and $g_i = g_j$.
\\So -

\begin{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.
\end{theorem}


\section{Higher dimension}

\begin{theorem}[Elekes-Szabo]
  $V \subseteq  W_1\times W_2\times W_3$, $\dim(W_i)=k$, $\dim(V)=2k$, all irreducible 
  complex.
\\  (Nice example: $W_i$ surfaces in $\C^3$, $V$ collinearity.)
\\  $X_i \subseteq  W_i$, $|X_i| = N$, in \defnstyle{general position}:
\\  for $W_i' \not\subseteq  W_i$ proper subvariety,
\\  $|X_i \cap W_i'| \in O_{\deg(W_i')}(1)$.
\\  Then either $|V \cap \Pi_i X_i| \leq  O(N^{2-\eta})$ or $V$ is in 
  correspondence with a group operation (or trivial case).
\end{theorem}

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

\subsection{Coarse general position}

\begin{definition}
  $\a \in K^{<\omega}$ is in \defnstyle{coarse general position} if
  \[ \dim^0(a/B) < \dim^0(a) \Rightarrow  \bdl(a) = 0 \]
  for any $B \subseteq  K$.
\end{definition}

\begin{definition} \mbox{}
  \begin{itemize}\item $\Keq := \bigcup_n K^n$.
  \item $X \subseteq  \Keq$ is (coarsely) \defnstyle{coherent} if every $a \in X$ is in 
    coarse general position and
    $\dim^0(\a)=\bdl(\a)$ for any $\a \in X^{<\omega}$.
  \item $\ccl(X) := \{ x \in \acleq(X) : \{x\} \;\textrm{is coherent}\}$.
  \end{itemize}
\end{definition}
\begin{theorem}
  Suppose $X = \ccl(X)$.
\\  Then $(X,\acl^{\eq^0})$ is modular.
\end{theorem}

As in Evans-Hrushovski, using abelian group configuration,
\\$\G$ projective geometry of $\dim \geq 4$ in $\mathbb{P}(\Keq,\acl^{eq})$
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 $F \leq  
\End^0_{C_0}(G)$.

So obtain corresponding analogue of 1-dimensional theorem, generalising 
Elekes-Szabo.
\end{document}
