# A Bollobás-type problem: from root systems to Erdős–Ko–Rado

Patrick J. Browne, Qëndrim R. Gashi and Padraig Ó Catháin  Department of
Mathematics, University of Prishtina
rr. Nena Tereze, p.n., 10000 Pristina, Kosovo
<EMAIL_ADDRESS>Department of Electrical & Electronic Engineering,
Technological University of the Shannon, Limerick, V94 EC5T, Ireland,
<EMAIL_ADDRESS>Fiontar agus Scoil na Gaeilge, Dublin City University,
All Hallows Campus, D09 N920, Ireland<EMAIL_ADDRESS>
###### Abstract.

Motivated by an Erdős–Ko–Rado type problem on sets of strongly orthogonal
roots in the $A_{\ell}$ root system, we estimate bounds for the size of a
family of pairs $(A_{i},B_{i})$ of $k$-subsets in $\\{1,2,\ldots,n\\}$ such
that $A_{i}\cap B_{j}=\emptyset$ and $|A_{i}\cap A_{j}|+|B_{i}\cap B_{j}|=k$
for all $i\neq j$. This is reminiscent of a classic problem of Bollobás. We
provide upper and lower bounds for this problem, relying on classical results
of extremal combinatorics and an explicit construction using the incidence
matrix of a finite projective plane.

MSC 2020: 05D05, 17B22
Keywords: Erdős-Ko-Rado, root system, strongly orthogonal roots

P. J. Browne and P. Ó Catháin acknowledge the National Forum for Enhancement
of Teaching and Learning and Technical University of the Shannon who supported
this publication through Strategic Alignment of Teaching and Learning
Enhancement funding. P. Ó Catháin acknowledges the support of the Conference
Participation Scheme of the Faculty of Humanities and Social Sciences at
Dublin City University.

##  1\. Introduction

Given a vector space $V\leq\mathbb{R}^{\ell+1}$, equipped with the standard
inner product, a subset $R\subseteq V$ is called a root system if the
following conditions are satisfied.

  1. (1)

The roots span $V$, and do not contain the zero vector.

  2. (2)

The only scalar multiple of $\alpha\in R$ also contained in $R$ is $-\alpha$.

  3. (3)

Each root $\alpha\in R$ determines a reflection $x\mapsto x-2\frac{\langle
x,\alpha\rangle}{\langle\alpha,\alpha\rangle}\alpha$. The set $R$ is closed
under these reflections.

  4. (4)

For any $\alpha,\beta\in R$, the value
$2\frac{\langle\alpha,\beta\rangle}{\langle\alpha,\alpha\rangle}$ is an
integer.

It is a remarkable fact that there are precisely four infinite families of
irreducible root systems, together with additional examples in low dimensions.
Root systems play a key role in the classification of semi-simple Lie
algebras, see Chapter 9 of Humphrey’s monograph [7].

######  Definition 1.

Denote by $\varepsilon_{i}$ the $i^{\textrm{th}}$ standard basis vector for
$\mathbb{R}^{\ell+1}$. The set of vectors $\varepsilon_{i}-\varepsilon_{j}$
for $j\neq i$ is the Type A root system, denoted $A_{\ell}$. The vectors
$\alpha_{i}=\varepsilon_{i}-\varepsilon_{i+1}$ with $i=1,2,\ldots,\ell$ form a
basis for $A_{\ell}$; these are called simple roots.

It is a simple calculation to verify that the vectors of Definition 1 indeed
satisfy the axioms of a root system. The inner product of two roots is
negative if and only if their sum is a root; similarly, the inner product of
two roots is positive if and only if their difference is a root. The Dynkin
diagram for the $A_{\ell}$ root system is given below. This shows the $\ell$
simple roots with an edge between them if and only if their sum is a root.

$\alpha_{1}$$\alpha_{2}$$\alpha_{3}$$\alpha_{\ell-1}$$\alpha_{\ell}$ Figure 1.
Dynkin diagram of $A_{\ell}$.

######  Definition 2.

Two roots in $R$ are strongly orthogonal if neither their sum nor their
difference is a root. A subset of $R$ consisting of $k$ roots, each pair of
which is strongly orthogonal is called a strongly orthogonal subset of $R$.
The set of all such subsets is denoted $SOS_{k}(R)$.

For example, a set of simple roots in $A_{\ell}$ is strongly orthogonal if and
only if they correspond to an independent set in the Dynkin diagram (that is,
a subset of vertices in the diagram, no two connected by an edge). Recall that
the support of a vector is the collection of co-ordinates in which the vector
is non-zero. An arbitrary pair of roots in $A_{\ell}$ is strongly orthogonal
if and only if the roots have disjoint supports. While this paper is mostly
concerned with the type $A$ root systems, much is known about $SOS_{k}(R)$ for
an arbitrary root system. The next result characterises the maximal size of a
strongly orthogonal subset in an irreducible root system.

######  Proposition 3 ([1], [2]).

The set $SOS_{k}(R)$ is non-empty in the following cases:

  * (i)

For $k\leq\left\lfloor\frac{\ell+1}{2}\right\rfloor$ when $R=A_{\ell}$.

  * (ii)

For $k\leq\ell$ when $R=B_{\ell}$.

  * (iii)

For $k\leq\ell$ when $R=C_{\ell}$.

  * (iv)

For $k\leq 2\left\lfloor\frac{\ell}{2}\right\rfloor$ when $R=D_{\ell}$.

  * (v)

For $k\leq 3$ when $R=F_{4}$.

  * (vi)

For $k\leq 4$ when $R=E_{6}$.

  * (vii)

For $k\leq 7$ when $R=E_{7}$.

  * (viii)

For $k\leq 8$ when $R=E_{8}$.

  * (ix)

For $k\leq 2$ when $R=G_{2}$.

In all other cases, $SOS_{k}(R)$ is empty.

##  2\. Erdős–Ko–Rado type theorems

Let $n,k\in\mathbb{N}$ and write $[n]:=\\{1,2,\ldots,n\\}$. Denote by
$\mathcal{V}(n,k):=\\{X\subseteq[n]:|X|=k\\}$ the set of $k$-element subsets
of $[n]$. The classical Erdős–Ko–Rado theorem is a cornerstone of extremal
combinatorics.

######  Theorem 4 ([4, 6]).

Let $\mathcal{F}\subseteq\mathcal{V}(n,k)$ and suppose that $X\cap
Y\neq\emptyset$ for all $X,Y\in\mathcal{F}$. Then

$|\mathcal{F}|\leq\binom{n-1}{k-1}\,.$

Moreover, if $n>2k$, equality holds if and only if $\mathcal{F}$ consists of
all elements of $\mathcal{V}(n,k)$ that contain a given element from $[n]$.

Equivalently, let $\Gamma(n,k)$ be the graph which has as vertex set
$\mathcal{V}(n,k)$, with an edge between two $k$-subsets when they have non-
trivial intersection. The first half of Theorem 4 gives an upper bound on the
size of a maximal clique in $\Gamma(n,k)$ and the second half characterises
the maximal cliques precisely. Motivated by Turán’s problem in graph theory,
Bollobás proved a lemma on intersecting set systems; it is notable that the
size of the underlying set does not appear in the bound. This statement of the
result is due to Lovász.

######  Theorem 5 ([3, 8]).

Let $X_{1},\ldots,X_{N}\in\mathcal{V}(n,k_{1})$ and
$Y_{1},\ldots,Y_{N}\in\mathcal{V}(n,k_{2})$ be distinct. If for each
$i,j=1,\ldots,N$,

$X_{i}\cap Y_{i}=\emptyset,\;\;\;X_{i}\cap Y_{j}\neq\emptyset$

then $N\leq\binom{k_{1}+k_{2}}{k_{1}}$.

The obvious configuration of subsets here consists of partitions of a set of
size $k_{1}+k_{2}$ into complementary parts of sizes $k_{1}$ and $k_{2}$. The
content of the theorem is that larger configurations satisfying these
intersection properties cannot exist, even if the size of the underlying set
is increased. Our main result may be stated as a result on the size of a
family of sets satisfying hypotheses similar to those of Bollobás.

######  Theorem 6.

Let $X_{1},\ldots,X_{N}$ and $Y_{1},\ldots,Y_{N}\subseteq\mathcal{V}(n,k)$ be
such that $X_{i}\cap Y_{j}=\emptyset$ for all $1\leq i,j\leq N$. Suppose that
for all $i\neq j$,

$|X_{i}\cap X_{j}|+|Y_{i}\cap Y_{j}|=k\,.$

Then $N\leq n$ for all $k$. If $n>k4^{k}$, then
$N\leq\lfloor\frac{n+2-k}{k}\rfloor$, and the sets attaining this bound may be
constructed explicitly.

We will prove this theorem by restating the problem in terms of strongly
orthogonal sets of roots in the $A_{\ell}$ root system.

######  Definition 7.

Let $\Gamma\in SOS_{k}(R),$ we write $|\Gamma|=\sum_{\gamma\in\Gamma}\gamma$
for the sum of the roots in $\Gamma$. A subset $\mathcal{F}\subseteq
SOS_{k}(R)$ is a SOS-clique if and only if for every
$\Gamma_{i},\Gamma_{j}\in\mathcal{F}$ there exists some
$\Gamma_{i,j}\in\mathcal{F}$ such that

(8) $|\Gamma_{i}|-|\Gamma_{j}|=|\Gamma_{i,j}|\,.$

The maximal size of an SOS-clique in $SOS_{k}(R)$ will be denoted
$\mu_{k}(R)$.

A set of strongly orthogonal roots $\Gamma\in SOS_{k}(A_{\ell})$ corresponds
to a pair of $k$-subsets as follows: $X_{i}$ consists of the coordinates where
$|\Gamma|$ is positive, and $Y_{i}$ to the coordinates where $|\Gamma|$ is
negative. The condition on intersections of Theorem 6 is clearly equivalent to
that of Equation (8), hence the upper and lower bounds on $\mu_{k}(A_{\ell})$
given in the next section will yield a proof of Theorem 6.

This is a problem of Erdős–Ko–Rado type: construct a graph in which vertices
are labelled by the vectors $|\Gamma|$ for $\Gamma\in SOS_{k}(R)$; with an
edge between $|\Gamma_{i}|$ and $|\Gamma_{j}|$ if and only if the difference
of their vectors is again the label of a vertex in the graph. The SOS-clique
of Definition 7 is a clique in this graph. As in the Erdős–Ko–Rado theorem, we
give both an upper bound on the size of maximal clique and a characterisation
of all maximal cliques when $\ell$ is sufficiently large in terms of $k$.

##  3\. Main Result

Recall a pair of roots is strongly orthogonal if and only if their supports
are disjoint, and that the set $SOS_{k}(A_{\ell})$ consists of $k$-tuples of
pairwise strongly orthogonal roots. It will be convenient to refer to an
explicit element of $SOS_{k}(A_{\ell})$ on occasion. Define
$\beta_{j}=\sum_{i=1}^{k}\alpha_{i+j}$, then
$\Gamma_{1}=\\{\beta_{j}:j=0,\ldots,k-1\\}$ is such a set. The sum
$|\Gamma_{1}|=\sum_{j=0}^{k-1}\beta_{j}$ is a vector with the first $k$
entries equal to $+1$ followed by $k$ entries equal to $-1$. Let $\mathcal{F}$
be an SOS-clique as in Definition 7. We assume without loss of generality that
$\Gamma_{1}\in\mathcal{F}$. For any $\Gamma\in\mathcal{F}$ distinct from
$\Gamma_{1}$, the supports of $\Gamma$ and $\Gamma_{1}$ intersect in precisely
$k$ co-ordinates. We write $S(\Gamma,\Gamma_{1})$ for this set.

######  Definition 9.

Let $\mathcal{F}\subseteq SOS_{k}(A_{\ell})$ be an SOS-clique. If for all
$\Gamma_{i},\Gamma_{j}\in\mathcal{F}$, the sets $S(\Gamma_{i},\Gamma_{j})$ are
equal, we say that $\mathcal{F}$ is a sunflower.

######  Lemma 10.

Suppose that $\mathcal{F}\subseteq SOS_{k}(A_{\ell})$ is a sunflower. Then
$|\mathcal{F}|\leq\lfloor\frac{\ell+1}{k}\rfloor-1$.

###### Proof.

Let $\mathcal{F}=\\{\Gamma_{1},\Gamma_{2},\ldots,\Gamma_{m}\\}$. Denote by $X$
the set of column indices in which $|\Gamma_{i}|$ and $|\Gamma_{j}|$ agree.
This is a set of size $k$, and it is immediate from Definition 7 that the
remaining non-zero entries of $|\Gamma_{i}|$ are the only non-zero entries in
their respective columns. Hence there exists disjoint $k$-sets
$X_{1},\ldots,X_{m}$ such that the support of $|\Gamma_{i}|=X\cup X_{i}$.
Hence $\ell+1$, the total number of columns, is at least $k|\mathcal{F}|+k$. ∎

Next we show that the number of vectors which do not intersect $\Gamma_{1}$ in
a fixed subset of size $k$ is bounded by a function of $k$.

######  Lemma 11.

Let $\mathcal{F}\subseteq SOS_{k}(A_{\ell})$ be an SOS-clique. Suppose that
$S(\Gamma_{i},\Gamma_{j})=S(\Gamma_{x},\Gamma_{y})$ if and only if
$\\{\Gamma_{i},\Gamma_{j}\\}=\\{\Gamma_{x},\Gamma_{y}\\}$. Then
$|\mathcal{F}|\leq\binom{2k}{k}+1$.

###### Proof.

Consider a fixed $\Gamma_{1}\in\mathcal{F}$. By Definition 7, the support of
every other $\Gamma_{i}\in\mathcal{F}$ intersects the support of $\Gamma_{1}$
in $k$ columns. By hypothesis, these $k$-sets are all distinct. Hence
$|\mathcal{F}|\leq\binom{2k}{k}+1$, by the pigeonhole principle. ∎

We recall a classical result of Ray-Chaudhuri and Wilson, and then we will be
in a position to prove our main theorem.

######  Theorem 12 ([9]).

Let $\mathcal{F}$ be a family of $k$-subsets of $[n]$, and let
$S\subseteq[k-1]$ be of size $s$. Suppose that all pairwise intersections of
elements of $\mathcal{F}$ have size in $S$. Then
$|\mathcal{F}|\leq\binom{n}{s}$.

######  Theorem 13.

For any $k\in\mathbb{N}$, we have $\mu_{k}(A_{\ell})\leq\ell+1$. For
$\ell>k4^{k}$,

$\mu_{k}(A_{\ell})=\left\lfloor\frac{{\ell}+1-k}{k}\right\rfloor\,.$

If $\ell>k4^{k}$ and $|\mathcal{F}|=\mu_{k}(A_{\ell})$ then $\mathcal{F}$ is a
sunflower.

###### Proof.

Let $\Gamma_{1},\Gamma_{2},\ldots,\Gamma_{m}\in\mathcal{F}$ be a maximal
sunflower of $\mathcal{F}$. As in the proof of Lemma 10, there exist disjoint
sets $X,X_{1},\ldots,X_{m}$ of coordinates, each of size $k$, such that the
support of $\Gamma_{i}$ is $X\cup X_{i}$. If there is no vector satisfying the
condition of Lemma 11 then $|\mathcal{F}|=m+1$. Since $\ell+1\geq k(m+1)$, the
conclusion holds.

Now suppose that there exists $\Gamma_{m+1}\in\mathcal{F}$ such that
$S(\Gamma_{1},\Gamma_{m+1})\neq X$. The support of $\Gamma_{m+1}$ intersects
$X\cup X_{i}$ in $k$ points, and so must intersect each $X_{i}$ non-trivially.
Hence $m-1\leq k$. Counting vectors by their relation with $\Gamma_{1}$,
forming a sunflower with $\Gamma_{1}$. There are at most $\binom{2k}{k}-1$
possible intersections with the support of $\Gamma_{1}$ distinct from $X$. For
each such intersection, there are at most $k$ vectors sharing that support.
Hence $|\mathcal{F}|\leq k\binom{2k}{k}+1$. Thus the hypothesis that
$\mathcal{F}$ is not a sunflower leads to an upper bound on $|\mathcal{F}|$
which is independent of $\ell$. For fixed $k$, once
$|\mathcal{F}|>k\binom{2k}{k}+1$, all SOS-cliques are sunflowers.

By Theorem 12, the number of $2k$-sets such that all pairwise intersections
have size $k$ is bounded by $\ell+1$, for any choice of $\ell$ and $k$. This
gives the general upper bound. A standard bound on central binomial
coefficients gives $\binom{2k}{k}\leq 4^{k}-1$. Hence, when $|\mathcal{F}|\geq
k4^{k}$ then $\mathcal{F}$ is necessarily a sunflower. Both the stronger bound
$|\mathcal{F}|\leq\left\lfloor\frac{{\ell}+1-k}{k}\right\rfloor$ and the
characterisation of SOS-cliques meeting the bound follow. ∎

In the interest of presenting an accessible proof, we made no attempt to
optimise the constant $k4^{k}$ in the proof of Theorem 13. It is likely that
this can be improved with a more careful analysis. Trivially,
$\mu_{1}(A_{\ell})=\ell$ for any $\ell\in\mathbb{N}$. The next result
evaluates $\mu_{2}(A_{\ell})$ precisely.

######  Theorem 14.

The sequence $\mu_{2}(A_{\ell})$ is $0,0,1,1,3$ for $1\leq\ell\leq 5$. For
$6\leq\ell\leq 13$, it is equal to $6$ and for $\ell\geq 13$, its value is
$\lfloor(\ell-1)/2\rfloor$.

###### Proof.

Suppose that $\mathcal{F}\subseteq SOS_{2}(A_{\ell})$ is of maximal size. If
$\mathcal{F}$ is a sunflower then $|\mathcal{F}|=\lfloor(l-1)/2\rfloor$. We
will show that otherwise, $|\mathcal{F}|\leq 6$. By the argument of Theorem
13, if $\mathcal{F}$ is not a sunflower, then it contains a sunflower of size
at most $3$.

If $\mathcal{F}$ contains a sunflower of three vectors, then there are
$\ell+1=8$ columns. The following example shows that $6$ vectors are possible
in this case.

$\begin{matrix}+&-&+&-&0&0&0&0\\\ +&-&0&0&+&-&0&0\\\ +&-&0&0&0&0&+&-\\\
+&0&+&0&0&-&0&-\\\ +&0&0&-&+&0&0&-\\\ +&0&0&-&0&-&+&0\end{matrix}$

No seventh vector is possible. The sunflower is forced; if these vectors
intersected on two entries of the same sign a fourth vector would be
impossible without forcing all vectors to satisfy the conditions of Lemma 10.
Each subsequent row is determined by the sequence of signs of its entries.
These are sequence of length $4$ which contain two $1$’s and two $-1$’s and
are mutually orthogonal as vectors in $\mathbb{R}^{4}$. It is easily verified
that there are at most three such, and computation by hand shows that the
configuration above is essentially unique.

Next, suppose that at most two vectors intersect on any pair of columns. The
example of Figure 2 shows that there exist six vectors on seven columns with
this property. By Definition 7, every pair of vectors intersects in some pair
of columns. Thus there can be at most $7$ vectors in total: there are at most
$\binom{4}{2}$ pairs of columns in which a vector can intersect a fixed
vector, and each intersection is unique. In a configuration with seven
vectors, every pair of vectors intersects in two columns; and every pair of
columns in the support of a vector is shared with a unique vector. While it is
possible to find $7$ subsets of an $8$-set intersecting pairwise in a set of
size $2$, as demonstrated below, there is no consistent way to choose columns
of negative entries such that every row has two negative entries.

$\begin{matrix}1&1&1&1&0&0&0&0\\\ 1&1&0&0&1&1&0&0\\\ 0&0&1&1&1&1&0&0\\\
1&0&1&0&1&0&1&0\\\ 1&0&0&1&1&0&0&1\\\ 0&1&0&1&1&0&1&0\\\ 0&1&1&0&1&0&0&1\\\
\end{matrix}$

If the $0$ entries in the above matrix are replaced by $-1$, these are the
non-constant rows of a Hadamard matrix of order $8$. The uniqueness of the
Hadamard matrix of order $8$ up to equivalence shows that $\mu_{2}(A_{7})=6$.

Observe that rows $1,2,3$ of the displayed matrix are forced by the
intersection condition, and that every subsequent vector must have a single
non-zero entry in each of the first three pairs of columns. This leaves only
one additional non-zero entry in each subsequent column, and a hand
computation verifies that the remaining entries must be placed in two columns
to complete the array.

Thus, if $\mathcal{F}\subseteq SOS_{2}(A_{\ell})$ is not a sunflower then
$|\mathcal{F}|\leq 6$. An easy hand computation gives the maximal size of an
SOS-clique for $\ell\leq 5$, while sunflowers are of maximal size when
$\ell\geq 13$; furthermore every maximal clique is a sunflower when $\ell\geq
15$. This completes the proof. ∎

Note that the above two results first appeared in a preprint which also
contains further results on other types of root systems [5]. However,
relations with Erdős–Ko–Rado theory were not discussed in that work.

##  4\. Relation to finite projective planes

In this section we use finite projective planes to construct large SOS-
cliques. Recall that a finite projective plane of order $q$ is an incidence
structure consisting of points and lines which satisfies the following
properties:

  * •

Every line contains $q+1$ points, and each point is incident with $q+1$ lines.

  * •

Any two distinct lines meet at exactly one point, and any two distinct points
lie on a unique line.

  * •

There exist four points, no three lying on a line.

It is an elementary exercise to see that there are $q^{2}+q+1$ points and the
same number of lines in a projective plane of order $q$. The incidence matrix
$N$ for such a plane is a square matrix of size $q^{2}+q+1$, in which columns
are labelled by points, and rows by lines. An entry is $+1$ when a point is
incident with a line and $0$ otherwise. The axioms force $NN^{\top}=qI+J$
where $J$ is the all-ones matrix.

######  Proposition 15.

Let $r_{1},r_{2},\ldots,r_{n}$ be the rows of the incidence matrix of a
projective plane. Then the vectors $r_{1}-r_{i}$ for $2\leq i\leq n$ yield an
SOS-clique of size $q^{2}+q$ in $A_{q^{2}+q}$.

###### Proof.

There is a unique column in which vectors $r_{1}$ and $r_{i}$ are both equal
to $1$, and each vector has $q+1$ entries $1$. Thus $r_{1}-r_{i}$ has $q$
entries $1$ and $q$ entries equal to $-1$.

A second vector $r_{1}-r_{j}$ agrees with $r_{1}-r_{i}$ in precisely $q$
columns. There are two possibilities: if the vectors $r_{1},r_{i},r_{j}$ all
share a $1$ in the same column, then $r_{1}-r_{i}$ and $r_{1}-r_{j}$ do not
agree in any negative entry, but agree in $q$ positive entries. Otherwise,
$r_{i}$ and $r_{j}$ share a non-zero entry in a column disjoint from the
support of $r_{1}$, and each vector intersects $r_{1}$ in a single (distinct)
column. Hence there are $q-1$ columns in which $r_{1}-r_{i}$ and $r_{1}-r_{j}$
share an entry $+1$ and a unique column in which they have an entry $-1$. In
every case, the vectors agree in $q$ columns. These vectors can be trivially
decomposed into sums of $k$ roots of the $A_{q^{2}+q}$ root system, and so
yield an $SOS_{q}(A_{q^{2}+q})$. ∎

As an example consider the projective plane of order $2$ (Fano plane). Figure
2 shows the differences between the first and the $6$ subsequent rows of this
incidence matrix. Note that each pair of vectors intersects in two columns,
but they do not all intersect in the same pair of columns; and so do not
satisfy the conditions of Lemma 10.

$\begin{pmatrix}1&1&1&0&0&0&0\\\ 1&0&0&1&1&0&0\\\ 1&0&0&0&0&1&1\\\
0&1&0&1&0&1&0\\\ 0&1&0&0&1&0&1\\\ 0&0&1&1&0&0&1\\\ 0&0&1&0&1&1&0\end{pmatrix}$

$\begin{matrix}0&+&+&-&-&0&0\\\ 0&+&+&0&0&-&-\\\ +&0&+&-&0&-&0\\\
+&0&+&0&-&0&-\\\ +&+&0&-&0&0&-\\\ +&+&0&0&-&-&0\end{matrix}$

Figure 2. Incidence matrix for the Fano plane, shown next to the first row
minus subsequent rows.

For a prime power $q$, Proposition 15 gives an SOS-clique of size $q^{2}+q$ in
$SOS_{q}(A_{q^{2}+q})$. On the other hand, the bound of Theorem 13 is
$q^{2}+q+1$. Hence our bound is within $1$ of optimality infinitely often. The
following questions are natural.

  * •

Question 1: Is the bound of Theorem 13 ever met with equality?

For fixed $k$, what is maximal size of an SOS-clique, $\mu_{k}(A_{\ell})$,
when $k\leq\ell\leq k4^{k}$?

  * •

Question 2: For fixed $k$, what is the smallest $C$ such that all maximal SOS-
cliques are sunflowers for $\ell>C$?

  * •

Question 3: Do there exist constants $c,C$ such that for every $\ell>C$, an
SOS-clique of size at least $\ell-c$ comes from a projective plane?

The corresponding questions for other root systems also remain open, and will
be the subject of a future investigation by the authors.

## References

  * [1] Y. Agaoka and E. Kaneda.  Strongly orthogonal subsets in root systems.  Hokkaido Math. J., 31:107–136, 2002.
  * [2] L. Bedulli, A. Gori, and F. Podestà.  Homogeneous hyper-complex structures and the Joyce’s construction.  Differential Geom. Appl., 29(4):547–554, 2011.
  * [3] B. Bollobás.  On generalized graphs.  Acta Mathematica Academiae Scientiarum Hungaricae, 16:447–452, 1965\.
  * [4] P. Erdős, C. Ko, and R. Rado.  Intersection theorems for systems of finite sets.  Quart. J. Math. Oxford Ser. (2), 12:313–320, 1961.
  * [5] Q. R. Gashi.  On a problem involving strongly orthogonal roots.  Max-Planck-Institut für Mathematik preprint series, 17, 2016.
  * [6] Chris Godsil and Karen Meagher.  Erdős-Ko-Rado theorems: algebraic approaches, volume 149 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2016.
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# A note on the definability of genus for Zariski geometries

Darío García<EMAIL_ADDRESS>Universidad de Los Andes
Facultad de Ciencias
Departamento de Matemáticas, Edificio H.  ,  Pedro Rizzo
<EMAIL_ADDRESS>Universidad de Antioquia
Facultad de Ciencias Exactas y Naturales
Instituto de Matemáticas  and  Joel Torres del Valle<EMAIL_ADDRESS>Universidad de Antioquia
Facultad de Ciencias Exactas y Naturales
Instituto de Matemáticas

(Date: January 2021)

###### Abstract.

In this work we propose a notion of genus in the context of Zariski geometries
and we obtain natural generalizations of the Riemann–Hurwitz Theorem and the
Hurwitz Theorem in the context of very ample Zariski geometries. As a
corollary, we show that such notion of genus cannot be first-order definable
in the full language of a Zariski geometry.

###### Key words and phrases:

Zariski geometry, Riemann–Hurwitz Theorem, Hurwitz Theorem.

###### 2010 Mathematics Subject Classification:

03C99-14H99

##  1\. Introduction

Classification theorems play an important role in the development of several
areas in mathematics. In Model Theory, one of the main classification results
appeared in [6], where Boris Zilber proved that for any strongly minimal set
$M$, its geometry is either trivial, locally projective, or there is a pseudo-
plane definable in $M$. As pointed out by himself in [7], this result and the
faithful belief that “mathematical perfect structures cannot be overlook in
the natural progression of mathematics”, motivated Zilber to conjecture in [6]
the following stronger classification:

Trichotomy conjecture. For any given strongly minimal set $M$, the pregeometry
$(M,\mathrm{acl})$ corresponding to $M$ and its algebraic closure operator
lies in one of the following classes:

  * •

Trivial pregeometries. For any $A\subseteq M$,
$\mathrm{acl}(A)=\displaystyle\bigcup_{a\in A}\mathrm{acl}(\\{a\\})$.

  * •

Locally modular pregeometries. For any closed sets $A,B\subseteq M$,

$\dim(A\cup B)+\dim(A\cap B)=\dim(A)+\dim(B).$

  * •

Non-locally modular pregeometries.

In particular, if the first two cases do not occur, then there is an
algebraically closed field $K$ definable in $M$ and the only structure induced
on $K$ from $M$ is definable in the field structure itself. (This property is
called purity of $K$.)

The veracity of the Trichotomy conjecture would have resulted in a very strong
classification theorem for strongly minimal structures. However, it was shown
in [3] by Ehud Hrushovski that the Trichotomy conjecture is false in general.
Indeed, Hrushovski proved that there are strongly minimal theories which are
not locally modular but do not even interpret an infinite group. Nevertheless,
in a joint work [2] Hrushovski and Zilber defined an important class of
strongly minimal structures where the Trichotomy conjecture holds. These
structures are called Zariski geometries. Roughly speaking, a Zariski geometry
is a strongly minimal structure with some topological properties that resemble
properties of the Zariski topology over a smooth algebraic curve.

An important goal in the early development of the subject was to provide finer
classification theorems for Zariski geometries, beyond Zilber’s trichotomy. In
[2] Zariski geometries are classified into the following classes: trivial
Zariski geometries, locally modular Zariski geometries, non-locally modular
Zariski geometries, ample Zariski geometries and, very ample Zariski
geometries.

It can be shown that a Zariski geometry is non-locally modular if and only if
it is ample (see, for example, [4, Lemma 3.1] for a proof). Moreover, very
ample Zariski geometries are characterized as smooth curves over algebraically
closed fields:

######  Theorem 1.1 (Theorem A and Proposition 1.1 in [2]).

Any smooth curve $C$ over an algebraically closed field may be seen as a very
ample Zariski geometry, where the topology on each $C^{n}$ is the restriction
of the Zariski topology $\tau_{\mathrm{Zar}}^{n}$. Conversely, if $D$ is a
very ample Zariski geometry, there exists an algebraically closed field $K$
and a smooth curve $C$ over $K$ such that the Zariski geometry corresponding
to $C$ is isomorphic to $D$. Moreover, the curve $C$ and the field $K$ are
unique up to a curve isomorphism and up to a field isomorphism. ∎

The results of this paper are related to the notion of genus for Zariski
geometries. In Section 2 we present all the necessary preliminaries and
provide a recollection of the main results used to show the classification
theorems for Zariski geometries. In particular, Proposition 2.11 is used to
guarantee the uniqueness of the curve $C$ and the field $K$ given in Theorem
1.1, and we also recall the definition of $D$-manifolds for a Zariski geometry
$D$. In Section 3 we prove a slight generalization of Proposition 2.11, and
use Theorem 1.1 to define a notion of genus for Zariski geometries that agrees
with the genus for smooth curves over algebraically closed fields. This notion
gives rise to corresponding generalizations of the Riemann–Hurwitz and Hurwitz
theorems for algebraic curves (cf. Proposition 3.5 and Theorem 3.7).
Furthermore, these results are used to prove in Theorem 3.8 that the notion of
genus in very ample Zariski geometries cannot be first-order definable in the
so-called _full language_ , where there is a predicate for every closed subset
of the Zariski geometry.

###  1.1. Acknowledgements

This work is mostly contained in the third author’s Master thesis [5], under
the supervision of the first and the second authors.

The second and third authors were supported by Fundación para la Promoción de
la Investigación y la Tecnología del Banco de la República de Colombia
(Project 3.963: Estudio del Espacio de Configuraciones de S.L.L. de
nivel-$\delta$).

The third author is grateful for the partial financial support from Dirección
de Posgrados, Universidad de Antioquia through a grant from Fondo General de
Becas de Maestría.

##  2\. Preliminaries

In this section we will present all the necessary preliminaries in Zariski
geometries that we will require in the later sections. We begin by introducing
Zariski geometries and the classification results for Zariski geometries
provided by Hrushovski and Zilber in [2].

Notation 1. Let $M$ be a topological space and let $X,Y\subseteq M$.
Throughout this work $X\subseteq_{\mathrm{cl}}Y$ means that $X$ is a closed
subset of $Y$; $\mathrm{cl}(X)$ is the topological closure of $X$ and
dimension means Krull dimension, which is the supremum of $n$ such that there
exists an ascending chain of closed sets $X_{0}\subseteq
X_{1}\subseteq\cdots\subseteq X_{n}\subseteq X$, and it is well defined
because the topological spaces we will consider are Noetherian. We write
$\dim(X)$ for the dimension of $X$. For any set $C\subseteq X\times Y$ and
$a\in X$ we write $C(a,Y)$ for the set $\\{y\in Y:(a,y)\in C\\}$ and $|X|$ is
the cardinal of $X$.

######  Definition 2.1.

A Zariski geometry $\mathfrak{D}$ consists of an infinite set $D$ together
with a collection of Noetherian topologies
$\mathfrak{D}=\langle(D^{n},\tau_{n}):n\in\mathbb{N}\rangle$ such that the
following axioms hold:

  * (Z0)

Let $f_{i}:D^{m}\to D$ be a projection $f_{i}(\vec{x})=x_{\sigma(i)}$ or a
constant map $f_{i}(\vec{x})=c$ (where $\sigma$ is a bijective function
$\sigma:\\{1,\ldots,m\\}\to\\{1,\ldots,m\\}$), then the map $f:D^{m}\to D^{n}$
sending $\vec{x}$ to $(f_{1}(\vec{x}),\ldots,f_{n}(\vec{x}))$ is a continuous
function. Also, the diagonal sets $\Delta_{ij}^{n}:=\\{\vec{x}\in
D^{n}:x_{i}=x_{j}\\}$ are closed for all $1\leq i<j\leq n$ for
$n\in\mathbb{N}$.

  * (Z1)

Let $C\subseteq_{\mathrm{cl}}D^{m}$ and $\pi_{mn}:D^{m}\to D^{n}$ a projection
map. Then there is a proper closed set
$F\subseteq_{\mathrm{cl}}\mathrm{cl}(\pi_{mn}(C))$ such that
$\pi_{mn}(C)\supseteq\mathrm{cl}(\pi_{mn}(C))-F$.

  * (Z2)

$D$ is irreducible and uniformly one-dimensional, that is, if
$C\subseteq_{\mathrm{cl}}D^{n}\times D$, then there exists $N$ such that for
all $\vec{a}\in D^{n}$ either, $|C(\vec{a},D)|\leq N$ or $C(\vec{a},D)=D$.

  * (Z3)

(Dimension Theorem) If $C$ is an irreducible closed set in $D^{n}$ and
$\Delta_{ij}^{n}$ the $ij$-diagonal, then any component of
$C\cap\Delta_{ij}^{n}$ has dimension $\geq\dim(C)-1$.

The above definition is based on properties that have topological nature.
Nevertheless, we are interested in the first-order properties of these
structures, so it is necessary to introduce some languages in order to express
such properties.

######  Definition 2.2.

Let $\mathfrak{D}$ be a Zariski geometry.

  * •

The full language of $\mathfrak{D}$ is denoted as
$\mathcal{L}_{\mathrm{full}}(\mathfrak{D})$ and contains a $n$-ary relation
symbol for each closed subset of $D^{n}$, for all $n\in\mathbb{N}$.

  * •

The natural language of $\mathfrak{D}$ is the sub-language
$\mathcal{L}_{\mathrm{nat}}(\mathfrak{D})$ of
$\mathcal{L}_{\mathrm{full}}(\mathfrak{D})$ consisting of all automorphism
invariant subsets of $D^{n}$ for all $n$.

######  Example 2.3 (Smooth curves).

Let $C$ be an irreducible smooth algebraic curve over an algebraically closed
field $K$. Consider on each power of $C$ the restriction of the Zariski
topology $\tau^{n}_{\mathrm{Zar}}$. Then
$\mathfrak{C}:=\langle(C^{n},\tau_{\mathrm{Zar}}^{n}):n\in\mathbb{N}\rangle$
is a Zariski geometry. Indeed, (Z0) follows from the fact that the underlying
functions are regular and regular functions are continuous ([1, Remark
3.1.1]). (Z1) follows from quantifier elimination for algebraically closed
fields ([4, pp. 109-110]) and (Z2) follows from strong minimality of
algebraically closed fields. Finally, (Z3) follows from Dimension Theorem for
smooth curves (see e.g., [1, Theorem I.7.2]).

The following lemma summarizes the basic model-theoretical properties of
Zariski geometries. The reader is refereed to Sections 2 and 4 of the work [2]
for a proof.

######  Lemma 2.4.

Let $\mathfrak{D}$ be a Zariski geometry regarded as a
$\mathcal{L}$-structure, where $\mathcal{L}$ is any of the languages
$\mathcal{L}_{\mathrm{nat}}(\mathfrak{D})$ or
$\mathcal{L}_{\mathrm{full}}(\mathfrak{D})$. Then the following is true about
$\mathfrak{D}$:

  1. (1)

$\mathfrak{D}$ eliminates quantifiers and is strongly minimal.

  2. (2)

Any $\mathcal{L}$-elementary extension of $\mathfrak{D}$ can be endowed with
the structure of a Zariski geometry.

  3. (3)

Any universal domain $\mathfrak{D}\preceq\mathfrak{D}^{*}$ is a Zariski
geometry.∎

Remember that in a Noetherian space any closed set $X$ may be written
(uniquely) as the union of irreducible sets, called the (irreducible)
components of $X$.

If $X$ is irreducible, we say that a property $P$ holds generically on $X$, if
$P$ holds for all elements of $X$ outside a proper closed subset or,
equivalently, a generic point of $X$ satisfies the property $P$.

The following two definitions are the key in the classification theorems for
Zariski geometries provided in [2]. These definitions allow the existence of
families of curves enriched with some geometrical structure.

For the rest of this work
$\mathfrak{D}=\langle(D^{n},\tau_{n}):n\in\mathbb{N}\rangle$ is an arbitrary
Zariski geometry.

######  Definition 2.5.

A plane curve in $D^{2}$ is an irreducible one-dimensional subset of $D^{2}$.
A family of plane curves is a pair $(C,E)$ where
$E\subseteq_{\mathrm{cl}}D^{n}$ is irreducible closed;
$C\subseteq_{\mathrm{cl}}E\times D^{2}$ is such that $C(\vec{e})$ is a plane
curve for generic $\vec{e}\in E$.

######  Definition 2.6.

We say that $\mathfrak{D}$ is ample if there exists a family of plane curves
$(C,E)$ such that for generic $\vec{a},\vec{b}\in D^{2}$ there is $\vec{e}\in
E$ such that the curve $C(\vec{e})$ passes through both $\vec{a}$ and
$\vec{b}$.

We say that $\mathfrak{D}$ is very ample if there exists a family of plane
curves $(C,E)$ making $\mathfrak{D}$ an ample Zariski geometry and such that
for any $\vec{a},\vec{b}\in D^{2}$ there is a $C(\vec{e})$ passing through
just one of $\vec{a}$, $\vec{b}$.

######  Example 2.7 ([4], pp. 115).

Let $C$ be a quasiprojective smooth algebraic curve in $\mathbf{P}_{K}^{n}$,
where $K$ is an algebraically closed field and consider on each power of $C$
the restriction of the Zariski topology, $\tau_{\mathrm{Zar}}^{n}$. As it was
mentioned before
$\mathfrak{C}=\langle(C^{n},\tau_{\mathrm{Zar}}^{n}):n\in\mathbb{N}\rangle$ is
a Zariski geometry. We claim that $\mathfrak{C}$ is very ample. Indeed, the
product $C\times C$ may be embedded in some projective space and let us choose
$N$ to be the smallest positive integer such that there is an embedding
$f:C\times C\to\mathbf{P}^{N}_{K}$. Let us define the projective variety
$\mathrm{Grass}(1,N)=\left\\{H\subseteq\mathbf{P}_{K}^{N}:H\text{ is a
hyperplane}\right\\}$ and $D=\\{(H,\vec{x}):\vec{x}\in f^{-1}(H)\\}$. For each
$H\in\mathrm{Grass}(1,N)$ we have $\dim(H\cap f(C\times C))\leq 1$. Thus
$(D,\mathrm{Grass}(1,N))$ defines a very ample family of curves. Indeed, by
Bertini’s Theorem (see e.g., [1, pp. 179]) it follows that for generic
$H\in\mathrm{Grass}(1,N)$, $D(H)=\\{\vec{x}\in C\times C:H(\vec{x})\in H\\}$
is irreducible closed of dimension 1 in $C\times C$ because $f$ is one-to-one.
Clearly, if $H$ contains $f(\vec{x})$ it follows that $\vec{x}\in D(H)$.
Finally, if $\vec{x},\vec{y}$ are two distinct points in $C\times C$, then
$f(\vec{y})\neq f(\vec{x})$ so there exists a generic hyperplane containing
$f(\vec{x})$ and not $f(\vec{y})$. Thus $\mathfrak{C}$ is very ample.

###  2.1. Classical classification theorems

Zilber’s trichotomy conjecture was introduced as a way of classifying (via
geometric stability theory) the models of strongly minimal theories and when
it was refuted by E. Hrushovski the question remaining was: for which
structures does the Trichotomy Conjecture holds? Zariski geometries turned out
to be a large class of mathematical structures where the conjecture holds. In
this sense the Trichotomy Conjecture gives a classification result for the
class of Zariski geometries. Results of this kind are the classical ones, and
we will now state them. (Later we will find finer classification results for
Zariski geometries in a geometrical sense by the introduction of a notion of
“genus” for very ample Zariski geometries.)

######  Lemma 2.8 (Lemma 3.1 in [4]).

A Zariski geometry is ample if and only if it is non-locally modular. ∎

######  Lemma 2.9 (Trichotomy Theorem, Lemma 6.11 and Proposition 7.10 in
[2]).

If $\mathfrak{D}$ is an ample Zariski geometry, then there exists an
algebraically closed field $K$ interpretable in $\mathfrak{D}$. Moreover, the
only structure induced from $\mathfrak{D}$ on $K$ is the field structure
itself.∎

We have already mentioned that Zariski geometries can be classified via
Zilber’s trichotomy (Lemma 2.9). The following theorem states that in essence,
the very ample Zariski geometries are just the smooth curves over
algebraically closed fields.

######  Theorem 2.10 (Theorem A in [2]).

Let $\mathfrak{D}$ be a very ample Zariski geometry. Then there is a smooth
algebraic curve $C$ over an algebraically closed field $K$ such that the
Zariski geometries $\mathfrak{D}$ and
$\mathfrak{C}=\langle(C^{n},\tau_{\mathrm{Zar}}^{n}):n\in\mathbb{N}\rangle$
are isomorphic. That is, there exists a homeomorphism $h:D\to C$ such that the
induced map $h^{[n]}:D^{n}\to C^{n}$ defines a homeomorphism for each
$n\in\mathbb{N}$. ∎

The curve and the field mentioned in Theorem 2.10 are unique up to a curve
isomorphism and a field isomorphism. The precise statement is contained in the
following result:

######  Proposition 2.11 (Proposition 1.1 in [2]).

Let $C$ be a smooth curve over an algebraically closed field $F$ and
$C^{\prime}$ a smooth curve over a field $F^{\prime}$. Suppose
$h:\mathfrak{C}\to\mathfrak{C}^{\prime}$ is an isomorphism of Zariski
geometries. Then there is a field isomorphism $h_{F}:F\to F^{\prime}$ such
that $h$ is an isomorphism of algebraic varieties with respect to the
identification of fields $h_{F}$. (In other words, $h_{F}$ induces a field
isomorphism $h_{F}^{\\#}:F(C)\to F^{\prime}(C^{\prime})$ of the fields of
rational functions over $C$ and over $C^{\prime}$.) ∎

###  2.2. Manifolds

In general it is not true that Axiom (Z3) hols for arbitrary closed subsets of
$D^{n}$ for the various $n\in\mathbb{N}$. In order to find a large collection
of sets where (Z3) remains true it is necessary to define special imaginary
sorts, and to identify the regular subsets (were (Z3) is preserved). It can be
proved that in Zariski geometries imaginaries can be eliminated down to
special sorts and that an analogue of Weil’s group chunk is valid in the
context (see [2, Lema 5.11]).

Let $\mathfrak{D}$ be a Zariski geometry, $H\leq S_{n}$ and let $H$ act on
$D^{n}$ by permuting coordinates. The quotient $S_{H}:=D^{n}/H$ is called the
special imaginary sort corresponding to $H$.

######  Definition 2.12.

Let $\Delta_{C}$ be the diagonal of $C\times C$ for $C\subseteq D^{n}$ closed
irreducible with $\dim(C)=k$.

  1. (1)

A point $\vec{p}\in C$ is regular on $C$ if there exists a closed irreducible
set $G\subseteq D^{n}\times D^{n}$ with $\dim(D^{n}\times
D^{n})-\dim(G)=\dim(C)$ such that $\Delta_{C}$ is the unique irreducible
component of $G\cap(C\times C)$ passing through $(\vec{p},\vec{p})$.

  2. (2)

If $C$ is any closed set in $D^{n}$, $\vec{p}$ is regular if $\vec{p}$ lies in
a unique component of $C$, whose dimension is equal to $\dim(C)$ and is
regular on that component.

  3. (3)

Let $S_{H}$ be a special imaginary sort, $\pi:D^{n}\to S_{H}$ the quotient
projection, and let $C$ be a closed subset of $S_{H}$. A point $\vec{p}\in C$
is regular on $C$ if $\vec{p}=\pi(\vec{p*})$ for some regular $\vec{p*}$ on
$\pi^{-1}(C)$.

######  Example 2.13.

Let $C$ be a smooth plane curve over an algebraically closed field $K$, and
let $\vec{p}$ be a non-singular point of $C$. Then $\vec{p}$ is a regular
point of the Zariski geometry
$\mathfrak{C}=\langle(C^{n},\tau_{\mathrm{Zar}}^{n}):n\in\mathbb{N}\rangle$
corresponding to $C$. (See [4, pp. 113-114] for the details.)

######  Definition 2.14.

Let $\mathfrak{D}$ be a Zariski geometry. A $D$-manifold consist of a set $X$
together with a finite number of injective maps $\langle\varphi_{i}:U_{i}\to
X:1\leq i\leq n\rangle$ such that

  * •

The set $X$ is covered by the $\varphi(U_{i})$’s, that is,
$X=\varphi_{1}(U_{1})\cup\cdots\cup\varphi_{n}(U_{n})$.

  * •

Each $U_{i}$ is an irreducible regular subset of some special sort.

  * •

For each $1\leq i,j\leq n$, the set $\\{(x,y)\in U_{i}\times
U_{j}:\varphi_{i}(x)=\varphi_{j}(y)\\}$ is a closed irreducible subset of
$U_{i}\times U_{j}$ and projects onto nonempty open subsets of both
$U_{i},U_{j}$.

The most important relation between Zariski geometries and manifolds is given
by the following lemma, asserting that any field interpretable in a Zariski
geometry has a manifold structure where the field operations are morphisms
(i.e., the graph of the map is irreducible closed), and such field is a pure
field.

######  Lemma 2.15 (Lemma 5.12 and Proposition 7.10 in [2]).

If a field $K$ is definably interpretable in $\mathfrak{D}$, then $K$ may be
endowed with a manifold structure such that the field operations are
morphisms. Moreover, $V$ is a $K$-manifold, i.e., an algebraic variety over
$K$, if and only if it is a $\mathfrak{D}$-manifold. ∎

##  3\. Results

The main lemma of this paper consists of a strengthening of Proposition 2.11
in the case where $F=F^{\prime}$. Our proof is a straightforward adaptation of
the proof of Proposition 1.1 in [2], but we included it for sake of
completeness.

###  3.1. Main Lemma

######  Lemma 3.1.

Let $C$, $C^{\prime}$ be a smooth quasi-projective algebraic curves over an
algebraically closed field $F$. Let
$h_{C}:\mathfrak{C}\to\mathfrak{C}^{\prime}$ be an injective morphism of the
corresponding Zariski geometries. Then there is a field homomorphism
$h_{F}:F\to F$ such that $h_{C}$ induces a rational morphism $h^{\\#}:C\to
C^{\prime}$ of algebraic varieties with respect to $h_{F}$. In other words,
$h_{F}$ induces a field homomorphism $h:F(C^{\prime})\to F(C)$.

Proof. Since the Zariski geometries $\mathfrak{C}$ and $\mathfrak{C}^{\prime}$
are ample, by Lemma 2.9 there are algebraically closed fields $K$ and
$K^{\prime}$ such that $K$ is interpretable in $\mathfrak{C}$ and $K^{\prime}$
is interpretable in $\mathfrak{C}^{\prime}$. Due to the fact that $C$ and
$C^{\prime}$ are algebraic curves over $F$, there exist $F$-definable
isomorphisms $g:F\to K$ and $g^{\prime}:F\to K^{\prime}$. By Lemma 2.15, $K$
has a $C$-manifold structure making the field operations morphisms. In
particular, $K$ is also an $F$-manifold and $g$ defines an isomorphism of
manifolds. Since $h_{C}$ is injective, there is a partial morphism
$h^{-1}_{C}:\mathfrak{C}^{\prime}\to\mathfrak{C}$ that extends to a morphism
$(\mathfrak{C}^{\prime},K^{\prime})\to(\mathfrak{C},K)$, by taking $h_{K}$ as
the map $g\circ g^{\prime-1}:K^{\prime}\to K$.

Let $p$ be the characteristic of the fields involved and consider the map
given by $\mathrm{Frob}(x)=x^{p}$. Let $F(C)$ and $F(C^{\prime})$ be the
fields of rational functions over $C$ and $C^{\prime}$, respectively,
considered as functions over the projective line $\mathbf{P}_{F}^{1}$, and let
$F_{p}(C)$ and $F_{p}(C^{\prime})$ be their corresponding perfect closures.
For any given $m\in\mathbb{Z}^{+}$, we can consider the map $h_{F}^{[m]}:F\to
F$ defined by $h_{F}^{[m]}:=g^{\prime-1}\circ\mathrm{Frob}^{m}\circ
h_{K}^{-1}\circ g$. During this proof we will view
$h_{m}=(h_{F}^{[m]},h_{C}^{-1},h_{K})$ as a map from the triple
$(F,C^{\prime},K^{\prime})$ to the triple $(F,C,K)$.

_Claim 1:_ For every $m\in\mathbb{N}$, the morphism $h_{m}$ maps
$F_{p}(C^{\prime})$ into $F_{p}(C)$.

_Proof of Claim 1:_ The map $h_{F}^{[m]}$ may be seen as a map from
$\mathbf{P}_{F}^{1}$ to $\mathbf{P}_{F}^{1}$ by sending $\infty\mapsto\infty$.
Let us take $f:C^{\prime}\to\mathbf{P}_{F}^{1}$ in $F_{p}(C^{\prime})$. Then,
the set $\Gamma_{f}$ (the graph of $f$) is Zariski-closed and irreducible in
$C^{\prime}\times\mathbf{P}_{F}^{1}$. We now show that $h_{m}(f)\in F_{p}(C)$,
and for this we will prove that $h_{m}(\Gamma_{f})$ is irreducible and
Zariski-closed in $C\times\mathbf{P}_{F}^{1}$ and the defining function
$h_{m}(f):C\to\mathbf{P}_{F}^{1}$ is not constantly infinite.

The map $g^{\prime}$ is a morphism. Since the composition of morphisms is a
morphism, it follows that the map $g^{\prime}\circ f:C^{\prime}\to K$ is a
morphism. Thus $\Gamma_{g^{\prime}\circ f}$ (the graph of $g^{\prime}\circ f$)
is irreducible and Zariski-closed in $C^{\prime}\times K$. By assumption,
$h_{m}$ is a morphism so $h_{m}(\Gamma_{g^{\prime}\circ f})$ is irreducible
and Zariski-closed in $C\times K$. Furthermore, $h_{m}(\Gamma_{g^{\prime}\circ
f})$ is the graph of the map $\mathrm{Frob}^{-m}\circ h_{K}\circ
g^{\prime}\circ h_{m}(f)$. Since Frob and $g^{\prime}$ are morphisms of
Zariski geometries, $\Gamma_{\mathrm{Frob}}$ and $\Gamma_{g^{\prime}}$ are
closed irreducible sets. Thus $\Gamma_{h(f)}$ must be also irreducible and
closed. Now, if $h_{m}(f)$ is constantly $\infty$, then its graph is a
component of $C\times F\times K$ which is impossible since this product is
irreducible. This shows that $h_{m}(\Gamma_{f})$ is an irreducible Zariski-
closed subset of $C\times\mathbf{P}_{F}^{1}$ and that $h(f)$ is not constantly
$\infty$, so $h_{m}(f)\in F_{p}(C)$. Therefore, $h_{m}$ maps
$F_{p}(C^{\prime})$ to $F_{p}(C)$, and we have finished the proof of Claim 1.

Note that if $p=0$, then $F(C)$ and $F(C^{\prime})$ are perfect fields and the
morphism $h_{0}$ maps $F(C^{\prime})$ to $F(C)$. From now on, let us assume
that $p>0$ and consider the collection $\mathcal{F}$ of all subfields of
$F_{p}(C)$ containing $F(C)$ and finitely generated over $F$ whose perfect
closure is $F_{p}(C)$, together with the collection $\mathcal{F}^{\prime}$ of
all subfields of $F_{p}(C^{\prime})$ containing $F(C^{\prime})$ and finitely
generated over $F$ whose perfect closure is $F_{p}(C^{\prime})$. Notice that
$h_{m}$ maps fields in $\mathcal{F}^{\prime}$ to fields in $\mathcal{F}$, and
in particular $h_{m}$ maps $F(C^{\prime})$ to some field
$L_{0}\in\mathcal{F}$.

Claim 2: If $L,L^{\prime}\in\mathcal{F}$, there is some $m_{0}\in\mathbb{Z}$
such that $L^{\prime p^{m_{0}}}=L$, that is,
$\mathrm{Frob}^{m_{0}}(L)=L^{\prime}$.

Proof of Claim 2: Assume for a moment that $p\neq 0$ and notice that
$F_{p}(C)=\bigcup_{m\in\mathbb{N}}F(C)^{1/p^{m}}$, that is, $F_{p}(C)$ is the
smallest field containing all $p^{m}$-th roots of elements in $F(C)$. Now, by
the choice of $\mathcal{F}$, for any $L,L^{\prime}\in\mathcal{F}$ it follows
that their respective perfect closures $L_{p}$ and $L^{\prime}_{p}$ satisfy

$L_{p}=\bigcup_{m\in\mathbb{N}}L^{1/p^{m}}=F_{p}(C)=\bigcup_{m\in\mathbb{N}}(L^{\prime})^{1/p^{m}}=L^{\prime}_{p}.$

Notice that $F(C)\subseteq L\subseteq F_{p}(C)$. So, for each one of the
finitely many generators $\xi_{i}$ of $L$ over $F$, there is some
$n_{i}\in\mathbb{Z}$ such that $\xi_{i}\in(L^{\prime})^{1/p^{n_{i}}}$. By
picking $m_{1}$ to be the product of the integers $n_{i}$, we obtain
$L\subseteq(L^{\prime})^{1/p^{m_{1}}}$. In other words,
$\mathrm{Frob}^{m_{1}}(L)\subseteq L^{\prime}.$ A similar procedure shows that
for some $m_{2}\in\mathbb{N}$ we have
$\mathrm{Frob}^{m_{2}}(L^{\prime})\subseteq L$, or equivalently,
$L^{\prime}\subseteq\mathrm{Frob}^{-m_{2}}(L)$.

Hence, we have $\operatorname{Frob}^{m_{1}}(L)\subseteq
L^{\prime}\subseteq\operatorname{Frob}^{-m_{2}}(L)$, and since
$[\operatorname{Frob}^{-m_{2}}(L):\operatorname{Frob}^{m_{1}}(L)]=p^{m_{1}+m_{2}}$,
there must exists some $n\in\mathbb{N}$ such that
$[L^{\prime}:\operatorname{Frob}^{m_{1}}(L)]=p^{n}$. Therefore, by taking
$m=m_{1}-n\in\mathbb{Z}$, we have that
$\operatorname{Frob}^{m}(L)=\operatorname{Frob}^{-n}(\operatorname{Frob}^{m_{1}}(L))=L^{\prime}.$
This concludes the proof of Claim 2.

Since $h_{0}$ is a map over $F$ sending $F_{p}(C^{\prime})$ to $F_{p}(C)$,
$L=h_{0}(F(C^{\prime}))$ is a field in $\mathcal{F}$, as well as
$L^{\prime}=F(C)$. By Claim 2, there is $m_{0}\in\mathbb{Z}$ such that
$\operatorname{Frob}^{m_{0}}(L)=L^{\prime}$, and we have two cases:

  1. (i)

If $m_{0}\leq 0$, then
$h_{0}(F(C^{\prime}))=\operatorname{Frob}^{-m_{0}}(F(C))\subseteq F(C)$, so
$h_{0}$ maps $F(C^{\prime})$ into $F(C)$.

  2. (ii)

If $m_{0}>0$ we have

$\displaystyle F(C)$
$\displaystyle=\operatorname{Frob}^{m_{0}}(h_{0}(F(C^{\prime})))=(\operatorname{Frob}^{m_{0}}\,\circ(g^{\prime})^{-1}\circ
h_{K}^{-1}\circ g)(F(C^{\prime}))$
$\displaystyle=((g^{\prime})^{-1}\circ\operatorname{Frob}^{m_{0}}\circ
h_{K}^{-1}\circ g)(F(C^{\prime}))$ $\displaystyle=h_{F}^{[m]}(F(C^{\prime})).$

In any case, there is $m\in\mathbb{N}$ such that
$h_{F}^{[m]}(F(C^{\prime}))\subseteq F(C)$, and by Claim 1 we have obtained a
morphism $h=h_{m}$ that maps $F(C^{\prime})$ to $F(C)$, satisfying the
conclusion of the lemma. ∎

###  3.2. The genus of a very ample Zariski geometry

In Algebraic Geometry, the notion of genus is presented as a numerical
invariant for one-dimensional algebraic varieties over an algebraically closed
field $K$. In this subsection we will show that using Theorem 2.10 a similar
notion could be derived for very ample Zariski geometries.

######  Definition 3.2.

We say that a very ample Zariski geometry $\mathfrak{D}$ has genus $g$,
denoted $\mathrm{gen}(\mathfrak{D})=g$, if $\mathfrak{D}$ is isomorphic to the
Zariski geometry of a smooth curve $C$ of genus $g$. (We will also write
$\mathrm{gen}(C)$ for the genus of the curve $C$.)

According to Theorem 2.10 the genus of a Zariski geometry is always defined
and it follows from Proposition 2.11 that it is invariant under isomorphisms
of Zariski geometries.

######  Lemma 3.3 (Riemann–Hurwitz Theorem).

Let $C_{1}$ and $C_{2}$ be smooth quasiprojective algebraic curves over an
algebraically closed field $K$. If there exists a non-constant morphism of
curves $f:C_{1}\to C_{2}$, then $g_{2}\leq g_{1}$, where $g_{i}$ is the genus
of $C_{i}$, for $i=1,2$.

For a further discussion about Lemma 3.3 we recommend [1] from pp. 299 on.
Together Lemma 3.1 and Lemma 3.3 show the following:

######  Proposition 3.4.

Let $C_{1}$ and $C_{2}$ be smooth quasi-projective algebraic curves over an
algebraically closed field $F$. Suppose that there exists an injective
morphism $f:\mathfrak{C}_{1}\to\mathfrak{C}_{2}$ of very ample Zariski
geometries. Then there exists a non-constant morphism of curves
$f_{*}:C_{1}\to C_{2}$. In particular,
$\mathrm{gen}(C_{2})\leq\mathrm{gen}(C_{1})$.

Proposition 3.4 is relating the two structures considered over the curves
$C_{1}$, $C_{2}$, namely, the structure of curve and that of Zariski geometry.

######  Theorem 3.5 (Riemann–Hurwitz for very ample Zariski geometries).

Let $\mathfrak{D}_{1}$ and $\mathfrak{D}_{2}$ be very ample Zariski geometries
and assume that there is an algebraically closed field $K$ interpretable in
both geometries. Also assume that there exists an injective morphism
$f:\mathfrak{D}_{1}\to\mathfrak{D}_{2}$ of Zariski geometries, then
$\mathrm{gen}(\mathfrak{D}_{2})\leq\mathrm{gen}(\mathfrak{D}_{1}).$

Proof. By Theorem 2.10 we can find smooth curves over $K$, say $C_{1}$ and
$C_{2}$, such that considered as Zariski geometries $\mathfrak{C}_{1}$ is
isomorphic to $\mathfrak{D}_{1}$ and $\mathfrak{C}_{2}$ is isomorphic to
$\mathfrak{D}_{2}$. Let $i:\mathfrak{D}_{1}\to\mathfrak{C}_{1}$ and
$j:\mathfrak{D}_{2}\to\mathfrak{C}_{2}$ be isomorphisms of Zariski geometries.
Then we have an injective morphism of Zariski geometries $j\circ f\circ
i^{-1}:\mathfrak{C}_{1}\to\mathfrak{C}_{2}$. By Lemma 3.1 we may find a non-
constant morphism of curves $\iota:C_{1}\to C_{2}$. By Lemma 3.3 we have that
$\mathrm{gen}(C_{2})\leq\mathrm{gen}(C_{1})$ so
$\mathrm{gen}(\mathfrak{D}_{2})\leq\mathrm{gen}(\mathfrak{D}_{1})$. ∎

######  Corollary 3.6.

Let $\mathfrak{D}$ be a very ample Zariski geometry and let $K$ be an
algebraically closed field interpretable in $\mathfrak{D}$. Suppose that there
exists an injective morphism of Zariski geometries
$f:\mathbf{P}_{K}^{1}\to\mathfrak{D}$. Then $\mathrm{gen}(\mathfrak{D})=0$.

Proof. According to Theorem 3.5,
$\mathrm{gen}(\mathfrak{D})\leq\mathrm{gen}(\mathbf{P}_{K}^{1})=0$. ∎

We now see one more application of Proposition 2.11.

######  Theorem 3.7 (Hurwitz Theorem).

Let $\mathfrak{D}$ be a very ample Zariski geometry interpreting an
algebraically closed field $K$ of characteristic $0$. Assume that
$g=\mathrm{gen}(\mathfrak{D})\geq 2$. Then there exists at most $84(g-1)$
automorphisms of $\mathfrak{D}$.

Proof. There exists a smooth quasi-projective algebraic curve over $K$ which
considered as Zariski geometry is isomorphic to $\mathfrak{D}$ and
$\mathrm{gen}(\mathfrak{D}):=\mathrm{gen}(C)$. By Proposition 2.11 any
automorphism of $\mathfrak{D}$ induces an automorphism of $C$ (as algebraic
curve), but by Hurwitz Theorem for algebraic curves (see [1, pp. 305]) $C$ has
at most $84(g-1)$ automorphisms, so $\mathfrak{D}$ has at most the same number
of automorphisms. ∎

###  3.3. Non-definability of genus

We wonder if there exists a first order sentence $\Phi^{g}$ such that a very
ample Zariski geometry $\mathfrak{D}$ satisfies $\Phi^{g}$ if and only if
$\mathfrak{D}$ has genus $g$. The following theorem answers the question in
the negative for several cases.

######  Theorem 3.8 (Non-definability of genus).

Let $\mathfrak{D}$ be a very ample Zariski geometry of genus $g\geq 2$
interpreting an algebraically closed field of characteristic 0. Thus if
$\mathcal{L}$ is one of the languages
$\mathcal{L}_{\mathrm{nat}}(\mathfrak{D})$ or
$\mathcal{L}_{\mathrm{full}}(\mathfrak{D})$, there is no
$\mathcal{L}$-sentence $\Phi^{g}$ ensuring that $\mathfrak{D}$ has genus $g$,
thus the genus of $\mathfrak{D}$ is not definable.

We need the following lemma (see [2, §6] for a proof):

######  Lemma 3.9.

Let $\mathfrak{D}$ be a very ample Zariski geometry. Then for some elementary
extension $\mathfrak{D}^{*}$ of $\mathfrak{D}$, we may find an array
$f=\langle f_{ij}:i,j<\omega\rangle$ such that $f$ is indiscernible. That is,
for any $n,m<\omega$ the first order type of any $(n,m)$-rectangle from $f$ is
constant.

Proof of Theorem 3.8 (The proof running is valid in both, the natural and the
full languages of $\mathfrak{D}$.) Let $\mathfrak{D}$ be a very ample Zariski
geometry interpreting an algebraically closed field of characteristic zero and
with genus $g\geq 2$. Assume that there is a first order formula $\Phi^{g}$
asserting that $\mathfrak{D}$ has genus $g$. By Hurwitz Theorem 3.7
$\mathfrak{D}$ has at most a finite number of automorphisms and this number is
bounded above by $84(g-1)$. According to the previous lemma, if we extend to
some sufficiently large elementary extension of $\mathfrak{D}$, say
$\mathfrak{D}^{*}$, there we can find an indiscernible array of elements
$f=\langle f_{ij}:i,j\in\mathbb{N}\rangle$. By definition, for any $(i,j)$
fixed we have $\mathrm{tp}(f_{ij})=\mathrm{tp}(f_{nm})$ for any
$n,m\in\mathbb{N}$. If we assume $\mathfrak{D}^{*}$ to be a universal domain
of $\mathfrak{D}$, it follows from homogeneity that may find an automorphism
of $\mathfrak{D}^{*}$, say $\sigma_{nm}^{ij}$ sending $f_{ij}$ to $f_{nm}$.
This argument shows that there exists at least $\aleph_{0}$ distinct
automorphisms of $\mathfrak{D}$. However, since
$\mathfrak{D}\preceq\mathfrak{D}^{*}$, $\mathfrak{D}^{*}\models\Phi^{g}$ so
$\mathfrak{D}^{*}$ also has genus $g$. Thus the number of automorphisms of
$\mathfrak{D}^{*}$ is bounded by above $84(g-1)$, a contradiction. ∎

In the proof of Theorem 3.8 it is implicit that for a Zariski geometry of
genus $\geq 2$ interpreting an algebraically closed field of characteristic
zero, there exists an elementary extension of $\mathfrak{D}$, say
$\mathfrak{D}^{*}$ such that
$\mathrm{gen}(\mathfrak{D})\neq\mathrm{gen}(\mathfrak{D}^{*})$. Therefore, the
genus of $\mathfrak{D}$ cannot be definable even by a set of
$\mathcal{L}$-sentences, where $\mathcal{L}$ is either
$\mathcal{L}_{\mathrm{nat}}(\mathfrak{D})$ or
$\mathcal{L}_{\mathrm{full}}(\mathfrak{D})$.

##  4\. Final remarks

We are not aware of a definition of genus that is intrinsic to the definition
of Zariski geometries, without appealing to Theorem 2.10. This question
remains open so far. Nevertheless, we have shown that for any given definition
of the genus of a very large Zariski geometry, which agrees with the notion of
genus of smooth curves, the Riemann–Hurwitz and Hurwitz theorems must be
valid, and Theorem 3.8 must hold.

So, even though the genus can be presented as a natural geometric invariant
for Zariski geometries, we have shown that it is not definable by any
collection of sentences in the language
$\mathcal{L}_{\mathrm{full}}(\mathfrak{C})$ or
$\mathcal{L}_{\mathrm{nat}}(\mathfrak{C})$. This kind of phenomenon occurs
even for more elementary theories such as the theory of $\mathbb{Q}$-vector
spaces in the language $\mathcal{L}=\\{0,+,(\lambda_{q}:q\in\mathbb{Q})\\}$
where the dimension is not first-order definable.

## References

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