V (= \P^3(\C) cubic surface
i.e. V = Z(f), f (- \C[X,Y,Z,W] homogeneous cubic
(Sketch 3 planes, sphere + plane, irred)

For L a generic line, |L \cap V| = 3.

Define: R(a,b,c) iff \{a,b,c\} = L \cap V for some L with |L\cap V| = 3.

For A (= V finite, |R \cap A^3| <= |A|^2.

Orchard problem on V:
    For what large finite A (= V do we have |R \cap A^3| ~ |A|^2?

# Coarse pseudofinite dimension
\U (= \P(\omega) non-principal ufilter
K := \C^\U
X (= K^n is _internal_ if X = \prod_{i --> \U} X_i some X_i (= \C^n,
and _pseudofinite_ if can take X_i finite.
Then |X| = \lim_{i --> \U} |X_i| (- \N^\U.
Let \xi (- \N^\U \\ \N.
d(X) = d_\xi(X) := \st(\log_{\xi}|X|) (- \R \cup \pm\infty

Say pseudofinite A (= V(K) is _R-rich_
if |A| > \N and
and d_A(R \cap A^3) = 2.

Coarse orchard problem on V: What A are R-rich?

A = \prod_{i --> \U} A_i is R-rich iff
\forall m>0. \U i. |R \cap A_i^3| > |A_i|^{2 - 1/m} and |A_i| > m

Aside -- Motivation from ES:
    X irred variety, \Gamma (= X^3 a irred constructible relation
    s.t. projns \Gamma -> X^2 dominant finite fibres

    ES,BB: \Gamma-rich A (= X in general position exist iff
        (up to interalgebraicity) X is a commutative group and \Gamma is +.
    General position: d(A \cap Y) = 0 for all Y (!= X
        Very restrictive!
    Question: What new (X,\Gamma) appear if we relax "general position"?


P (= \P^3(\C) generic plane
C := P \cap V is a cubic curve.
Sps V smooth irreducible, then C is an elliptic curve.
a,b,c collinear <=> a+b+c = 0 for group structure on C.
Pseudofinite subgroups are R-rich.

R-rich A is _planar_ if d_A(R \cap (A \cap \pi)^3) = 2 some plane \pi.
Exists for any V.

Example:
    f = X(X-W)(X+W)
    A = [\{-1,0,1\}:\{-N,...,N\}:\{-N,...,N\}:1] is non-planar R-rich

Theorem[BDZ]:
    Suppose every component of V is smooth.
    Then non-planar R-rich A exist
    iff V is the union of three collinear planes (i.e. with a line in common).

Proof sketch:

Suppose A (= V is R-rich and V is not union of collinear planes.
d := d_A, so d(A) = 1.

Fact:
    Exists countable expansion of (K;+,*,A) by internal sets s.t.
    d(a/C) := inf_{\phi (- \tp(a/C)} d(\phi(K)) is additive:
    d(ab/C) = d(a/bC) + d(b/C).

    a \|/^d_C b <=> d(a/Cb) = d(a/C).
    
    a is ``in weak general position'' (wgp) if for any b
        a \|/^d b => a \|/ b

Get (a_1,a_2,a_3) (- A^3 \cap R with d(a_1a_2a_3) = 2, d(a_i) = 1
so a_i \|/^d a_j.
Adding params, WLOG each a_i is wgp.
WTS: exists plane \pi / {} s.t. a_1,a_2,a_3 (- \pi.

Let's just show: some a_i is not generic in V, i.e. trd(a_i) < 2.
Suppose not.

# Case 1: V is reducible

Ignoring exceptional cases, WLOG f = (X^2+Y^2+Z^2-W^2)W
so V = S \cup P = ``sphere \cup plane at infinity''
Then R(p,s_1,s_2) <=> s_2 = \gamma_p(s_1)
where \gamma_p (- O_3 is the reflection in the plane with normal through p.

Theorem[BDZ]:
    Let X be an irreducible variety,
    let G be a connected algebraic group of automorphisms of X,
    \wirk GX / {},
    and suppose \alpha (- G and x (- X are generic wgp
    with \alpha \|/^d x and \alpha \|/^d \alpha(x).

    Then G is nilpotent.
Proof idea:
    Obtain \Gamma_\cdot-rich wgp subset of G by approximating with arbitrary 
    long compositions;
    conclude by Balog-Szemeredi-Gowers + Breuillard-Green-Tao arguments.
    .

\alpha := \gamma_p\o\gamma_{p'} (- SO_3 =: G and X=S,
this contradicts non-nilpotency of SO_3.

Other reducible cases are similar, with other groups (PSO_4, G_a^2 ><| G_m).

# Case 2: V irreducible
Geiser involutions: \gamma_a(b) := c if R(a,b,c).
\gamma_a (- Bir(V), but (\gamma_a)_a *don't* generate an algebraic subgroup,
so can't use above theorem.

Use incidence bounds instead.

E_X := \{ ((a,a'),(b,b')) : \exists x. R(a,b,x) /\ R(a',b',x) \} <= X^2 \times 
X^2

Take a_1',a_2' \|/^d==_{a_3} a_1,a_2,
e := ((a_1,a_1'),(a_2,a_2'))
then |= E_X(e)

Theorem[Tóth]:
    Let P be a pseudofinite set of points in \P^2(\C),
    let L be a pseudofinite set of lines  in \P^2(\C).
    Then d(\{ (p,l) (- P \times L : p (- l \}) <=
        max( \frac23(d(P) + d(L)), d(P), d(L) )

Theorem[ES,Chernikov-Galvin-Starchenko]:
    \Gamma (= X \times Y constructible binary relation,
    A (= X, B (= Y pseudofinite, d(B) < 2d(A).
    Let E := \Gamma \cap (A\times B).

    d(E) < d(A) + d(B)/2
    unless E contains some A'\times B' with |A'|=2, d(B')>0.

    Moreover,
    d(E \\ \bigcup \{ A'\times B' (= E : |A'|=2, d(B')>0 \})
        < d(A) + d(B)/2

d(e) = 3 = d(a_1,a_1') + d(a_2,a_2')/2

~~> WMA e (- A' \times B' where
    A'=\{(a_1,a_1'),(p,p')\}, \bdl(B')>0.

WLOG (p,p') (/- \acl(a_1,a_1')  (chromatic argument)

[draw diamond]
B' ~~> curve C of fixed points of composition
    \gamma_{a_1} o \gamma_{a_1'} o \gamma_{p'} o \gamma_{p}

Key Lemma:
    C and a_1,a_1',p,p' all lie in a plane.

WLOG (a_2,a_2') (- C  (not obv)

\pi := plane spanned by e
d(e / \pi) > 0
d(e / \pi) < 3 (by wgp)
Tóth: d(a_2,\pi/a_1) <= ... contradiction

Proof of Key Lemma:
    Fact:
        V ~= blow up of \P^2 at 6 points
        \pi : V --> \P^2
        Pic(V) ~= \Zl (+)_{i<6} \Ze_i
        l = \pi^*(line), e_i = exceptional divisor of ith blowup
        
        C (= V irred curve
        [C] = al - \sum_{i<6} b_ie_i
        deg(C) = 3a - \sum_i b_i

        p (- V
        m := \mu_p(C)
        d := deg(C)
        C' := \gamma_p(C)
        then
        m' = d - 2m
        d' = 2d - 3m

        P_i (- C singular points
        \sum_i \mu_{P_i}(C)(\mu_{P_i}(C)-1) <=
            2g_a(C) = (a-1)(a-2) - \sum_{i<6} b(b-1)
    
    Reduce to case all multiplicities and degrees equal to m,d
    so m = d-2m, d=2d-3m
    so d=3m

    Obtain 5 points on C of mult m.
    Say [C] = al - \sum_{i<6} b_ie_i
    Then 5m(m-1) <= (a-1)(a-2) - \sum b_i(b_i-1)
    so 5(d/3)(d/3 - 1) <= a^2 + 2 - d - \sum b_i^2
        <= a^2 + 2 - d - (1/6)(3a - d)^2  (Cauchy-Schwarz)
    ==> (2/9)d^2 - (2/3)d - 2 <= -(1/2)(a-d)^2 <= 0
    ==> d < 6
    ==> m < 2

    So m=1 and d=3.

    More argument yields planarity.