# Maximum number of points on an intersection of a cubic threefold and a non-
degenerate Hermitian threefold

Mrinmoy Datta  Department of Mathematics,
Indian Institute of Technology Hyderabad, Kandi, Sangareddy, Telangana, India
<EMAIL_ADDRESS>and  Subrata Manna  Department of Mathematics,
Indian Institute of Technology Hyderabad, Kandi, Sangareddy, Telangana, India
<EMAIL_ADDRESS>
###### Abstract.

It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian
threefold in ${\mathbb{P}}^{4}(\mathbb{F}_{q^{2}})$ has at most
$d(q^{5}+q^{2})+q^{3}+1$ points in common with a threefold of degree $d$
defined over $\mathbb{F}_{q^{2}}$. He proved the conjecture for $d=2$. In this
paper, we show that the conjecture is true for $d=3$ and $q\geq 7$.

###### Key words and phrases:

Hermitian threefolds, cubic threefolds, rational points, Edoukou’s Conjecture

###### 2010 Mathematics Subject Classification:

Primary 14G05, 14G15, 05B25

The first named author is partially supported by the grant
DST/INT/RUS/RSF/P-41/2021 from the Department of Science and Technology, Govt.
of India and the grant SRG/2021/001177 from Science and Engineering Research
Board, Govt. of India

The second named author is partially supported by a doctoral fellowship from
Council of Scientific and Industrial Research, Govt. of India

##  1\. Introduction

Hermitian varieties over finite fields are fascinating objects in the theory
of algebraic geometry, finite geometry, coding theory, and combinatorics. Bose
and Chakravarti first studied these varieties in [2]. The popularity of
Hermitian varieties grew among coding theorists since the non-degenerate
Hermitian varieties admit a very high number of rational points. Furthermore,
Hermitian varieties come with fascinating point-line incidence structures,
which are attractive to mathematicians working in finite geometry and
combinatorics. From a coding theoretic perspective, the following question has
been studied in [14, 7, 8, 3, 4] among other articles.

######  Question 1.1.

Let $V_{m-1}$ be a non-degenerate Hermitian variety in
${\mathbb{P}}^{m}(\mathbb{F}_{q^{2}})$. Determine the maximum possible number
of $\mathbb{F}_{q^{2}}$-rational points common to a hypersurface of degree $d$
in ${\mathbb{P}}^{m}$ defined over $\mathbb{F}_{q^{2}}$ and $V_{m-1}$.

The case when $d=1$ is completely settled in [2, 5]. When $m=2$, one could
derive from using Bézout’s Theorem that a curve of degree $d$ and a non-
degenerate Hermitian curve intersect at a maximum of $d(q+1)$ many
$\mathbb{F}_{q^{2}}$-rational points. It is fairly easy to produce a plane
curve of degree $d$ defined over $\mathbb{F}_{q^{2}}$ that intersects the
Hermitian curve at $d(q+1)$ points. However, the question becomes much more
difficult as $m$ increases. The case $m=3$, to our knowledge, was first
studied by Sørensen, in his Ph.D. thesis [14]. He conjectured that a non-
degenerate Hermitian surface in ${\mathbb{P}}^{3}(\mathbb{F}_{q^{2}})$ has at
most $d(q^{3}+q^{2}-q)+q+1$ many $\mathbb{F}_{q^{2}}$-rational points in
common with a surface in ${\mathbb{P}}^{3}$ of degree $d$ defined over
$\mathbb{F}_{q^{2}}$. Even though the conjecture was formulated in the early
1990s, the first breakthrough was made by Edoukou in 2007. In [7], he proved
that the conjecture is true for $d=2$. Sørensen’s conjecture was finally
proved in a series of two papers [3, 4]. It is now natural to look at the case
when $m=4$. Edoukou, in [8], determined the answer to Question 1.1 for $d=2$
and $m=4$. Besides, Edoukou proposed the following conjecture:

######  Conjecture 1.

[8, Conjecture 1] Let $F\in\mathbb{F}_{q^{2}}[x_{0},\dots,x_{4}]$ be a
homogeneous polynomial of degree $d$ and $V_{3}$ be a non-degenerate Hermitian
threefold in ${\mathbb{P}}^{4}(\mathbb{F}_{q^{2}})$. If $d\leq q$, then

$|V(F)(\mathbb{F}_{q^{2}})\cap V_{3}|\leq d(q^{5}+q^{2})+q^{3}+1.$

Moreover, this bound is attained by a threefold $V(F)$ in ${\mathbb{P}}^{4}$,
if and only if $V(F)$ is a union of $d$ non-tangent hyperplanes
$\Sigma_{1},\dots,\Sigma_{d}$ defined over $\mathbb{F}_{q^{2}}$, such that
$\Sigma_{1},\dots,\Sigma_{d}$ contains a common plane $\Pi$ defined over
$\mathbb{F}_{q^{2}}$ and $\Pi$ intersects $V_{3}$ at a non-degenerate plane
Hermitian curve.

As mentioned above, he proved Conjecture 1 for $d=2$. The conjecture remained
open to date. In this paper, we will prove the conjecture for $d=3$ and $q\geq
7$. It is worth noting that Edoukou’s proof of the conjecture for $d=2$
depends on the well-known classification of quadrics in
${\mathbb{P}}^{4}(\mathbb{F}_{q^{2}})$. However, our approach is more algebro-
geometric. We will in fact provide a proof of Edoukou’s result using our
method in Corollary 4.6.

This paper is organized as follows. In Section 2, we recall various well-known
properties of Hermitian varieties, and in particular non-degenerate Hermitian
threefolds, revisit some preliminary notions from algebraic geometry and some
bounds on the number of rational points on varieties defined over a finite
field. In Section 3 we take a step back to derive an upper bound for the
number of $\mathbb{F}_{q^{2}}$-rational points common to a degenerate
Hermitian surface and a surface of degree $d$ in
${\mathbb{P}}^{3}(\mathbb{F}_{q^{2}})$. In Section 4 we present some results
for general values of $d\leq q$ towards proving Edoukou’s conjecture. Finally,
in Section 5, we prove our main result.

##  2\. Preliminaries

Let us fix $q$, a prime power. As usual, ${\mathbb{F}}_{q}$ and
$\mathbb{F}_{q^{2}}$ denote the finite fields with $q$ and $q^{2}$ elements,
respectively. For $m\geq 0$, we denote by ${\mathbb{P}}^{m}$, the projective
space of dimension $m$ over the algebraic closure
$\overline{{\mathbb{F}}}_{q}$, while ${\mathbb{P}}^{m}(\mathbb{F}_{q^{2}})$
will denote the set of all $\mathbb{F}_{q^{2}}$-rational points on
${\mathbb{P}}^{m}$. Similarly, ${\mathbb{A}}^{m}$ and
${\mathbb{A}}^{m}(\mathbb{F}_{q^{2}})$ will denote the affine space of
dimension $m$ over $\overline{{\mathbb{F}}}_{q}$ and the set of all
$\mathbb{F}_{q^{2}}$-rational points of ${\mathbb{A}}^{m}$ respectively.
Further, for a homogeneous polynomial
$F\in\mathbb{F}_{q^{2}}[x_{0},\dots,x_{m}]$, we denote by $V(F)$, the set of
zeroes of $F$ in ${\mathbb{P}}^{m}$ and by
${\mathsf{V}}(F)=V(F)(\mathbb{F}_{q^{2}})$ the set of all
$\mathbb{F}_{q^{2}}$-rational points of $V(F)$. By an algebraic variety, we
mean a set of zeroes of a certain set of polynomials in the affine space or
projective space, depending on the context. In particular, an algebraic
variety need not be irreducible. We remark that whenever we say that a variety
is nonsingular, we will mean that the variety is nonsingular over
$\overline{{\mathbb{F}}}_{q}$. However, when we say that a hypersurface $V(F)$
is irreducible, we will mean that the polynomial $F$ is irreducible in the
field of its definition. This section is divided into three subsections: in
the first subsection, we recall several known facts about Hermitian varieties
over finite fields; the second subsection is dedicated to revisiting some
useful notions from basic algebraic geometry, while the third subsection
concerns some known bounds on the number of rational points on varieties
defined over a finite field. None of the results mentioned in this section is
new, and their proof can be found in the indicated references.

###  2.1. Geometry of Hermitian threefolds over finite fields

To begin with, let us revisit the definition and several properties of
Hermitian varieties (cf. [2, 5]) that will be used in the latter part of this
paper.

######  Definition 2.1 ([2]).

Let $A=(a_{ij})$ ($0\leq i,j\leq m$) be an $(m+1)\times(m+1)$ matrix with
entries in $\mathbb{F}_{q^{2}}$. We denote by $A^{(q)}$ the matrix whose
$(i,j)$-th entry is given by $a_{ij}^{q}$. The matrix $A$ is said to be a
Hermitian matrix if $A\neq 0$ and $A^{T}=A^{(q)}$. A Hermitian variety of
dimension $m-1$, denoted by $V_{m-1}$, is the set of zeroes of the polynomial
$x^{T}Ax^{(q)}$ inside ${\mathbb{P}}^{m}$, where $A$ is an $(m+1)\times(m+1)$
Hermitian matrix and $x=(x_{0},\dots,x_{m})^{T}$. From a finite-geometric
perspective, the authors of [2] denoted by $V_{m-1}$ the
$\mathbb{F}_{q^{2}}$-rational points on the variety given by the equation
$x^{T}Ax^{(q)}=0$. Strictly speaking, from an algebro-geometric point of view,
one should denote this set by $V_{m-1}(\mathbb{F}_{q^{2}})$. But to maintain
the classical notation, we will stick to the symbol $V_{m-1}$ to denote the
set of all $\mathbb{F}_{q^{2}}$-rational points on $V_{m-1}$. The Hermitian
variety is said to be non-degenerate if $\mathrm{rank}\ A=m+1$ and degenerate
otherwise.

It was established in [2, Equation 5.6] that if the rank of a Hermitian matrix
is $r$, then by a suitable change of coordinates over $\mathbb{F}_{q^{2}}$,
one can describe the corresponding Hermitian variety by the zero set of the
polynomial

(1) $x_{0}^{q+1}+x_{1}^{q+1}+\dots+x_{r-1}^{q+1}=0.$

We note that the polynomial $x_{0}^{q+1}+\dots+x_{r-1}^{q+1}$ is irreducible
over the algebraic closure of ${\mathbb{F}}_{q}$ whenever $r\geq 3$. Thus, a
Hermitian variety of rank at least three is always absolutely irreducible,
i.e. the variety is irreducible over $\overline{{\mathbb{F}}}_{q}$. Moreover,
if $V_{m-1}$ is non-degenerate, then it is non-singular. As mentioned in the
Introduction, the linear sections of Hermitian varieties are rather well-
understood thanks to [2, 5]. The following result is a straightforward
consequence of [2, Theorem 7.4] and [5, Theorem 3.1].

######  Theorem 2.2.

Let $V_{m-1}$ be a non-degenerate Hermitian variety in ${\mathbb{P}}^{m}$ and
$\Sigma$ be a hyperplane of ${\mathbb{P}}^{m}$ defined over
$\mathbb{F}_{q^{2}}$.

  1. (a)

[2, Theorem 7.4] if $\Sigma$ is tangent to $V_{m-1}$ at a point $P\in
V_{m-1}$, then $\Sigma\cap V_{m-1}$ is a cone over a non-degenerate Hermitian
variety $V_{m-3}$ contained in a hyperplane of $\Sigma$ with center at $P$.

  2. (b)

[5, Theorem 3.1] 111Strictly speaking, the result by Chakravarti is about the
section of non-degenerate Hermitian varieties with hyperplanes that are polar
hyperplanes at external points. However, it is not difficult to show that any
non-tangent hyperplane can be realized as a polar hyperplane to the non-
degenerate Hermitian variety at an external point. if $\Sigma$ is not a
tangent to $V_{m-1}$, then $\Sigma\cap V_{m-1}$ is a non-degenerate Hermitian
variety $V_{m-2}$ in $\Sigma$.

We now describe the possibilities arising out of a line intersecting a
Hermitian variety.

######  Lemma 2.3.

[2, Section 7] Any line in ${\mathbb{P}}^{m}(\mathbb{F}_{q^{2}})$ satisfies
precisely one of the following.

  1. (i)

The line intersects $V_{m-1}$ at precisely $1$ point.

  2. (ii)

The line intersects $V_{m-1}$ at precisely $q+1$ points.

  3. (iii)

The line is contained in $V_{m-1}$.

Let us summarize the known facts regarding the linear sections of non-
degenerate Hermitian threefolds in ${\mathbb{P}}^{4}$. From now onwards, we
will denote by $V_{3}$ a non-degenerate Hermitian threefold in
${\mathbb{P}}^{4}$. It is well-known that a non-degenerate Hermitian threefold
does not contain a plane. One can derive this from [2, Theorem 9.1] and also
from a much stronger result [10, Lemma 2.1]. We have the following definitions
thanks to Lemma 2.3.

######  Definition 2.4.

Let $\ell$ be a line in ${\mathbb{P}}^{4}(\mathbb{F}_{q^{2}})$. The line
$\ell$ is called

  1. (a)

a tangent line if it intersects $V_{3}$ at exactly $1$ point.

  2. (b)

a secant line if it intersects $V_{3}$ at exactly $q+1$ points.

  3. (c)

a generator if it is contained in $V_{3}$.

######  Remark 2.5.

In particular, if $\Sigma$ is a hyperplane in ${\mathbb{P}}^{4}$ defined over
$\mathbb{F}_{q^{2}}$, and $V_{3}$ denotes the non-degenerate Hermitian
threefold in ${\mathbb{P}}^{4}$, then the following two possibilities occur:

  1. (a)

if $\Sigma$ is not a tangent to $V_{3}$, then $\Sigma\cap V_{3}$ is a non-
degenerate Hermitian surface in $\Sigma$.

  2. (b)

if $\Sigma$ is a tangent to $V_{3}$ at $P\in V_{3}$, then $\Sigma\cap V_{3}$
is a cone over a nonsingular plane Hermitian curve $V_{1}$ with center at $P$.
If $V_{1}$ is contained in a plane $\Pi\subset\Sigma$, then it is understood
that $P\not\in\Pi$.

Since any line $\ell$ with $P\in\ell\subseteq V_{3}$, is contained in
$T_{P}(V_{3})$, the tangent hyperplane to $V_{3}$ at $P$, it follows from part
(b) that the non-degenerate Hermitian threefold contains $q^{3}+1$ lines
defined over $\mathbb{F}_{q^{2}}$ passing through $P$. Also as a consequence
of [2, Theorem 9.2], the non-degenerate Hermitian threefold $V_{3}$ in
${\mathbb{P}}^{4}$ contains exactly $(q^{5}+1)(q^{3}+1)$ lines defined over
$\mathbb{F}_{q^{2}}$.

From [2, Section 7], we know that the intersection of a Hermitian variety with
a linear subspace is a Hermitian variety. In particular, if $\Pi$ is a plane
in ${\mathbb{P}}^{4}$ defined over $\mathbb{F}_{q^{2}}$, then $V_{3}\cap\Pi$
is also a Hermitian variety. There are three possibilities :

  1. (a)

$\Pi\cap V_{3}$ is a non-degenerate Hermitian curve: in this case, $|\Pi\cap
V_{3}|=q^{3}+1$.

  2. (b)

$\Pi\cap V_{3}$ is a union of $q+1$ lines through a common point: in this case
$|\Pi\cap V_{3}|=q^{3}+q^{2}+1$.

  3. (c)

$\Pi\cap V_{3}$ is a line: in this case $|\Pi\cap V_{3}|=q^{2}+1$.

In particular, if $\Pi$ is a plane defined over $\mathbb{F}_{q^{2}}$ that is
contained in the tangent hyperplane $T_{P}(V)$ to $V_{3}$ at a point $P\in
V_{3}$ and $P\not\in\Pi$, then $\Pi\cap V_{3}$ is a non-degenerate Hermitian
curve.

For a fixed line $\ell$ in ${\mathbb{P}}^{4}$ defined over
$\mathbb{F}_{q^{2}}$, We will denote by $\mathcal{B}(\ell)$, the book of
planes containing $\ell$, that is,

$\mathcal{B}(\ell):=\\{\Pi:\ell\subset\Pi,\Pi\ \text{is a plane in}\
{\mathbb{P}}^{4}\ \text{defined over }\ \mathbb{F}_{q^{2}}\\}.$

It is easy to see that $|\mathcal{B}(\ell)|=q^{4}+q^{2}+1$. Moreover, if
$\Sigma$ is a hyperplane in ${\mathbb{P}}^{4}$ defined over
$\mathbb{F}_{q^{2}}$ containing $\Pi$, we denote by
$\mathcal{B}_{\Sigma}(\ell)$ the book of planes in $\Sigma$ containing $\ell$,
that is,

$\mathcal{B}_{\Sigma}(\ell):=\\{\Pi\in\mathcal{B}(\ell):\Pi\subset\Sigma\\}.$

It is fairly easy to show that $|\mathcal{B}_{\Sigma}(\ell)|=q^{2}+1$.

If $P,Q\in V_{3}$ are distinct points and $\ell$ is a generator of $V_{3}$
through $P$ and $Q$, then the tangent spaces $T_{P}(V_{3})$ and $T_{Q}(V_{3})$
intersect at a plane $\Pi$ such that $\Pi\cap V_{3}=\ell$. Also, if a plane
$\Pi$ intersects $V_{3}$ at $q+1$ lines through $P$, then $\Pi\subset
T_{P}(V_{3})$.

As mentioned in the Introduction, Edoukou [8] dealt with the sections of
quadric threefolds and non-degenerate Hermitian threefolds. We record his
results here for future references.

######  Theorem 2.6.

[8, Theorem 4.1] Let $\mathcal{Q}$ be a quadric threefold in
${\mathbb{P}}^{4}$ defined over $\mathbb{F}_{q^{2}}$. Then
$|Q(\mathbb{F}_{q^{2}})\cap V_{3}|\leq 2q^{5}+q^{3}+2q^{2}+1$.

Besides proving the above-mentioned theorem, Edoukou also provided the five
highest possible number of points of intersection between a quadric and a non-
degenerate Hermitian threefold. We present an excerpt of this result [8,
Theorem 4.4] below. For future reference, we first introduce the following
definition.

######  Definition 2.7.

Let $V_{3}$ be a non-degenerate Hermitian threefold in ${\mathbb{P}}^{4}$. A
quadric threefold $Q$ in ${\mathbb{P}}^{4}$ defined over $\mathbb{F}_{q^{2}}$
is said to be of

  1. (a)

Type I with respect to $V_{3}$, if $Q$ is a union of two distinct non-tangent
hyperplanes $\Sigma_{1}$ and $\Sigma_{2}$ such that $\Sigma_{1}\cap\Sigma_{2}$
intersects $V_{3}$ at a nonsingular Hermitian plane curve.

  2. (b)

Type II with respect to $V_{3}$, if $Q$ is a union of two distinct non-tangent
hyperplanes $\Sigma_{1}$ and $\Sigma_{2}$ such that $\Sigma_{1}\cap\Sigma_{2}$
intersects $V_{3}$ at a union of $q+1$ lines passing through a common point.

  3. (c)

Type III with respect to $V_{3}$, if $Q$ is a union of two hyperplanes
$\Sigma_{1}$ and $\Sigma_{2}$, where $\Sigma_{1}$ is a tangent to $V_{3}$ and
$\Sigma_{2}$ is a non-tangent and $\Sigma_{1}\cap\Sigma_{2}$ intersects
$V_{3}$ at a nonsingular plane Hermitian curve.

In addition to Theorem 2.6, Edoukou proved the following result.

######  Proposition 2.8.

[8, Theorem 4.4] Let $Q$ be a quadric threefold in ${\mathbb{P}}^{4}$ defined
over $\mathbb{F}_{q^{2}}$. Then

$|Q(\mathbb{F}_{q^{2}})\cap V_{3}|\begin{cases}=2(q^{5}+q^{2})+q^{3}+1,\
&\mathrm{if}\ $Q$\ \mathrm{is\ of\ Type\ I}\\\ =2q^{5}+q^{3}+q^{2}+1,\
&\mathrm{if}\ $Q$\ \mathrm{is\ of\ Type\ II}\\\ =2q^{5}+2q^{2}+1,\
&\mathrm{if}\ $Q$\ \mathrm{is\ of\ Type\ III}\\\ \ \leq 2q^{5}+q^{2}+1,\
&\mathrm{otherwise}.\end{cases}$

We conclude the current subsection by recalling the Sørensen’s bound from [4].

######  Theorem 2.9.

[4] Let $F\in\mathbb{F}_{q^{2}}[x_{0},x_{1},x_{2},x_{3}]$ be a homogeneous
polynomial of degree $d$ and let $V_{2}$ be a non-degenerate Hermitian surface
in ${\mathbb{P}}^{3}(\mathbb{F}_{q^{2}})$. If $d\leq q$, then

$|V(F)(\mathbb{F}_{q^{2}})\cap V_{2}|\leq d(q^{3}+q^{2}-q)+q+1.$

Moreover, this bound is attained by $V(F)$ if and only if $V(F)$ is a union of
$d$ planes $\Pi_{1},\dots,\Pi_{d}$ such that $\Pi_{1},\dots,\Pi_{d}$ intersect
at a line $\ell$ that is a secant to $V_{2}$.

In particular, if $V(F)$ is not a union of planes, then the inequality in the
above theorem is strict. We will use this observation in Proposition 4.3.

###  2.2. Preliminaries from algebraic geometry

Here we recall various basic results from algebraic geometry that will be
instrumental for proving our main theorem. We will make use of the notions of
dimension, degree and singularity of a variety, as can be found in standard
textbooks of Algebraic Geometry, for example, the book of Harris [9]. We begin
with the following definitions. A variety is said to be of pure dimension or
equidimensional if all the irreducible components of the variety have equal
dimension. Further, two equidimensional varieties $X,Y\subset{\mathbb{P}}^{m}$
are said to intersect properly if $\mathrm{codim}(X\cap Y)=\mathrm{codim}\
X+\mathrm{codim}\ Y.$ It turns out that good upper bounds for the number of
rational points on varieties defined over a finite field, depend on the degree
and dimension of the variety. Because of this reason, the following
proposition from [9] will be indispensable for us.

######  Proposition 2.10.

[9, Cor. 18.5] Let $X$ and $Y$ be equidimensional varieties that intersect
properly in ${\mathbb{P}}^{m}$. Then $\deg(X\cap Y)\leq\deg X\deg Y$.

######  Proposition 2.11.

Let $F,G\in\mathbb{F}_{q^{2}}[x_{0},x_{1},\dots,x_{m}]$ be nonconstant
homogeneous polynomials having no common factors. Then

  1. (a)

$V(F)$ and $V(G)$ intersect properly.

  2. (b)

$V(F)\cap V(G)$ is equidimensional of dimension $m-2$.

  3. (c)

$\deg(V(F)\cap V(G))\leq\deg F\deg G$.

###### Proof.

Part (a) is well-known in Algebraic Geometry and is generally proved using
Krull’s principal ideal theorem. Part (b) is proved using Macaulay’s
unmixedness theorem (See [15, Chapter 7, Theorem 26]). Part (c) follows
straightaway from Proposition 2.10. ∎

We conclude this subsection by recording a very important observation that
follows from the two propositions above.

######  Remark 2.12.

Suppose $V$ is a Hermitian variety in ${\mathbb{P}}^{m}(\mathbb{F}_{q^{2}})$
of rank at least $3$. Since a Hermitian variety is absolutely irreducible, if
$X$ is any hypersurface in ${\mathbb{P}}^{m}$ defined over
$\mathbb{F}_{q^{2}}$ with $\deg X\leq q$, then $X$ and $V$ intersect properly.
As a consequence, $\deg(X\cap V)\leq(q+1)\deg X.$

###  2.3. Basic upper bounds

In this subsection, we recall some well-known upper bounds on the number of
rational points on varieties defined over a finite field ${\mathbb{F}}_{q}$
with given degrees and dimensions. We begin with a result by J. -P. Serre [12]
that is undoubtedly one of the fundamental results. This was independently
proved by Sørensen [13].

######  Theorem 2.13 (Serre’s inequality).

Let $F\in{\mathbb{F}}_{q}[x_{0},x_{1},\dots,x_{m}]$ be a nonconstant
homogeneous polynomial of degree $d\leq q$. Then

$|{\mathsf{V}}(F)|\leq dq^{m-1}+p_{m-2},$

where $p_{m-2}=1+q+\dots+q^{m-2}$. Moreover, equality holds if and only if
$V(F)$ is a union of $d$ hyperplanes defined over ${\mathbb{F}}_{q}$ all
containing a common linear subspace of codimension $2$.

In particular, any plane curve defined over ${\mathbb{F}}_{q}$ of degree
$d\leq q$ has at most $dq+1$ many ${\mathbb{F}}_{q}$-rational points.
Furthermore, a plane curve of degree $d$ that admits exactly $dq+1$ many
${\mathbb{F}}_{q}$-rational points is a union of $d$ lines passing through a
common point. Among various other known bounds for plane curves defined over
finite fields, we would make use of the following result [1, Corollary 2.5] by
Aubry and Perret.

######  Theorem 2.14.

[1, Corollary 2.5] Let $\mathcal{C}$ be a plane absolutely irreducible
algebraic curve defined over ${\mathbb{F}}_{q}$. Then
$|\\#C({\mathbb{F}}_{q})-(q+1)|\leq(d-1)(d-2)\sqrt{q}$, where
$d=\deg\mathcal{C}$.

For nonsingular curves, the above bound is known as Hasse-Weil bound. However,
the result by Aubry and Perret applies to a larger family of curves. To solve
our problem, it is imperative that we consider bounds on varieties that are
not necessarily hypersurfaces. To this end, we have the following result by
Lachaud and Rolland.

######  Proposition 2.15.

[11, Prop. 2.3] Let $X$ be an equidimensional projective (resp. affine)
variety defined over a finite field ${\mathbb{F}}_{q}$. Further assume that
$\dim X=\delta$ and $\deg X=d$. Then

$|X({\mathbb{F}}_{q})|\leq dp_{\delta}\ \ \ \ \ \ (\mathrm{resp.}\ \
|X({\mathbb{F}}_{q})|\leq dq^{\delta}),$

where $p_{\delta}=1+q+\dots+q^{\delta}$.

The following result from [11] on varieties that are irreducible over
${\mathbb{F}}_{q}$, but not absolutely irreducible, will be very important.

######  Proposition 2.16.

[11, Prop. 3.8] Let $X$ be a projective variety defined over
${\mathbb{F}}_{q}$ that is irreducible over ${\mathbb{F}}_{q}$, but not
absolutely irreducible. Then
$X({\mathbb{F}}_{q})=\mathrm{Sing}(X)({\mathbb{F}}_{q})$, where
$\mathrm{Sing}(X)$ is the subset of $X$ consisting of all the singular points
of $X$ in $\overline{{\mathbb{F}}}_{q}$.

The above result shows, in particular, that if $X$ is a projective variety
defined over ${\mathbb{F}}_{q}$, that is irreducible over ${\mathbb{F}}_{q}$
but not absolutely irreducible, then any ${\mathbb{F}}_{q}$-rational point on
$X$ is a singular point of $X$. We conclude this section by stating a recent
result that will be useful later.

######  Theorem 2.17.

[6, Theorem 3.1] Let $Y\subset{\mathbb{P}}^{3}$ be a surface of degree $d$
defined over ${\mathbb{F}}_{q}$ and $P\in Y({\mathbb{F}}_{q})$. Then one of
the following holds:

  1. (a)

$Y$ contains a plane defined over ${\mathbb{F}}_{q}$,

  2. (b)

$Y$ contains a cone over a plane curve defined over ${\mathbb{F}}_{q}$ with
center at $P$,

  3. (c)

$\\#\\{\ell\subset{\mathbb{P}}^{3}\mid\ell\ \mathrm{is\ a\ line\ such\ that}\
P\in\ell\subset Y\\}\leq d(d-1)$.

######  Remark 2.18.

We explain the application of Theorem 2.17 in case of cubic surfaces. Suppose
$Y$ is a cubic surface in ${\mathbb{P}}^{3}$ that does not contain any plane
defined over ${\mathbb{F}}_{q}$ and $P\in Y({\mathbb{F}}_{q})$. If $Y$
contains a cone over a plane curve $Y^{\prime}$ defined over
${\mathbb{F}}_{q}$ with center at $P$, then $\deg Y^{\prime}=3$. Indeed, if
$\deg Y^{\prime}=1$ or $2$, then $Y$ contains a plane defined over
${\mathbb{F}}_{q}$, which contradicts our assumption. Moreover, it is clear
that $Y^{\prime}$ must be irreducible over ${\mathbb{F}}_{q}$. The Theorem
above thus implies that such a cubic surface $Y$ is either a cone over a plane
irreducible cubic curve $Y^{\prime}$ with center at $P$, or $Y$ may contain at
most $6$ lines passing through $P$.

##  3\. Intersection of a singular Hermitian surface and a surface of degree
$d$

Let $V_{2}^{\prime}$ denote a singular Hermitian surface of rank $3$ in
${\mathbb{P}}^{3}$. Geometrically, the Hermitian variety $V_{2}^{\prime}$ is a
cone over a plane nonsingular Hermitian curve $V_{1}$ with center at a point
$P$. Suppose $F\in\mathbb{F}_{q^{2}}[x_{0},x_{1},x_{2},x_{3}]$ be a
homogeneous polynomial of degree $d$. Then $V(F)$ is a surface of degree $d$
in ${\mathbb{P}}^{3}$ defined over $\mathbb{F}_{q^{2}}$. We note that any
plane that does not pass through $P$ intersects $V_{2}^{\prime}$ at a non-
degenerate Hermitian curve, and $V_{2}^{\prime}$ is still a cone over that
Hermitian curve with center at $P$. In this section, we are going to give an
upper bound on the quantity $|V(F)(\mathbb{F}_{q^{2}})\cap V_{2}^{\prime}|$
that will be instrumental in the proof of our main result. Throughout this
section, we will assume that $d\leq q$. Since our main purpose of this article
is to work towards a proof of Conjecture 1, it is enough for us to assume that
$d\leq q$.

######  Lemma 3.1.

If $\Pi$ is a plane defined over $\mathbb{F}_{q^{2}}$ in ${\mathbb{P}}^{3}$
that does not pass through $P$ and $V(F)$ does not contain $\Pi$, then
$|{\mathsf{V}}(F)\cap V_{2}^{\prime}\cap\Pi|\leq d(q+1)$.

###### Proof.

As noted above, the curve $V_{2}^{\prime}\cap\Pi$ is a non-degenerate
Hermitian curve in the plane $\Pi$ and, in particular, an irreducible plane
curve of degree $q+1$. By hypothesis, the curve $V(F)\cap\Pi$ is a plane curve
of degree $d$. The assertion now follows from Bézout’s Theorem. ∎

######  Lemma 3.2.

If $\Pi$ is a plane in ${\mathbb{P}}^{3}$ defined over $\mathbb{F}_{q^{2}}$
such that $P\not\in\Pi$ and $\Pi\subset V(F)$, then
$|V(F)(\mathbb{F}_{q^{2}})\cap V_{2}^{\prime}\cap\Pi^{C}|\leq(d-1)(q+1)q^{2}$,
where $\Pi^{C}={\mathbb{P}}^{3}\setminus\Pi$.

###### Proof.

Since $\Pi\subset{\mathsf{V}}(F)$, it follows that
${\mathsf{V}}(F)\cap\Pi^{C}$ is an affine surface of degree at most $d-1$.
Furthermore, $V_{2}^{\prime}\cap\Pi^{C}$ is also an irreducible affine surface
of degree $q+1$ contained in $\Pi^{C}$. It follows that $(V(F)\cap\Pi^{C})$
and $(V_{2}^{\prime}\cap\Pi^{C})$ have no common components, and consequently,
their intersection is a complete intersection of degree at most $(d-1)(q+1)$.
Consequently, $\dim(V_{2}^{\prime}\cap V(F)\cap\Pi^{C})=1$ and it follows from
Proposition 2.15, that $|V(F)(\mathbb{F}_{q^{2}})\cap
V_{2}^{\prime}\cap\Pi^{C}|\leq(d-1)(q+1)q^{2}$. ∎

We are ready to state and prove the main result of this section.

######  Theorem 3.3.

We have $|{\mathsf{V}}(F)\cap V_{2}^{\prime}|\leq d(q+1)q^{2}+1$.

###### Proof.

Case 1: $V(F)$ does not contain any plane not passing through $P$. We may
assume that $V(F)$ intersects $V_{2}^{\prime}$ at an
$\mathbb{F}_{q^{2}}$-rational point $Q$ other than $P$. Let $\ell$ be a line
in ${\mathbb{P}}^{3}$ defined over $\mathbb{F}_{q^{2}}$ that intersects
$V_{2}^{\prime}$ only at $Q$. It is evident that $P\not\in\ell$. Then there
exists a unique plane, say $\Pi_{0}$, containing $\ell$ that passes through
$P$. Note that $\Pi_{0}\cap V_{2}^{\prime}=\ell^{\prime}$, where
$\ell^{\prime}$ is the line joining $P$ and $Q$. Indeed, the line
$\ell^{\prime}\subseteq\Pi_{0}\cap V_{2}^{\prime}$ since $P,Q\in\Pi_{0}\cap
V_{2}^{\prime}$. We write
$\mathcal{B}(\ell)=\\{\Pi_{0},\Pi_{1},\dots,\Pi_{q^{2}}\\}$, where
$\mathcal{B}(\ell)$ denotes the set of all planes containing $\ell$. It
follows that,

(2) $|{\mathsf{V}}(F)\cap V_{2}^{\prime}\cap\Pi_{0}|\leq q^{2}+1.$

Furthermore, the planes $\Pi_{1},\dots,\Pi_{q^{2}}$ are planes that do not
contain $P$. As a consequence, $\Pi_{j}$ intersects $V_{2}^{\prime}$ at a
nonsingular Hermitian curve for every $j=1,\dots,q^{2}$. Since
$\Pi_{j}\not\subseteq V(F)$, we see that $|V(F)\cap\Pi_{j}|$ is a plane curve
of degree $d$ for all $j=1,\dots,q^{2}$. From Lemma 3.1, we have

(3) $|{\mathsf{V}}(F)\cap V_{2}^{\prime}\cap\Pi_{i}|\leq d(q+1)\ \text{for}\
i=1,\dots,q^{2}.$

Note that $V_{2}^{\prime}$ intersects $\ell$ only at $Q$. Thus,
$|{\mathsf{V}}(F)\cap V_{2}^{\prime}\cap\ell|=1$. We have,

$\displaystyle|{\mathsf{V}}(F)\cap V_{2}^{\prime}|$
$\displaystyle=\sum_{i=0}^{q^{2}}\left(|{\mathsf{V}}(F)\cap
V_{2}^{\prime}\cap\Pi_{i}|-|{\mathsf{V}}(F)\cap
V_{2}^{\prime}\cap\ell|\right)+|{\mathsf{V}}(F)\cap V_{2}^{\prime}\cap\ell|$
$\displaystyle\leq\left((q^{2}+1)-1\right)+\sum_{i=1}^{q^{2}}\left(d(q+1)-1\right)+1$
$\displaystyle=d(q+1)q^{2}+1.$

The inequality above, of course, follows from Equations (2) and (3).

Case 2: $V(F)$ contains a plane not passing through $P$. Suppose $V(F)$
contains a plane $\Pi$ not passing through $P$. Then $V_{2}^{\prime}$
intersects $\Pi$ at a nonsingular Hermitian curve $V_{1}$. Since
$\Pi\subset{\mathsf{V}}(F)$, we have ${\mathsf{V}}(F)\cap
V_{2}^{\prime}\cap\Pi=V_{1}$. Note that,

$\displaystyle|{\mathsf{V}}(F)\cap V_{2}^{\prime}|$
$\displaystyle=|{\mathsf{V}}(F)\cap
V_{2}^{\prime}\cap\Pi|+|{\mathsf{V}}(F)\cap V_{2}^{\prime}\cap\Pi^{C}|$
$\displaystyle=|V_{1}|+|{\mathsf{V}}(F)\cap V_{2}^{\prime}\cap\Pi^{C}|$
$\displaystyle\leq q^{3}+1+(d-1)(q+1)q^{2}$ $\displaystyle<d(q+1)q^{2}+1.$

The inequality above follows from Lemma 3.2. ∎

######  Remark 3.4.

It is not difficult to produce a surface of degree $d$ that intersects
$V_{2}^{\prime}$ at exactly $d(q+1)q^{2}+1$ many $\mathbb{F}_{q^{2}}$-rational
points. Take a plane $\Pi$ defined over $\mathbb{F}_{q^{2}}$, not passing
through $P$. As noted above, the plane $\Pi$ intersects $V_{2}^{\prime}$ at a
nonsingular Hermitian curve $V_{1}$. Take a plane curve $C$ defined over
$\mathbb{F}_{q^{2}}$ of degree $d$ in $\Pi$ that intersects $V_{1}$ at exactly
$d(q+1)$ many $\mathbb{F}_{q^{2}}$-rational points. It is now evident that the
cone over $C$, which is a surface of degree $d$ defined over
$\mathbb{F}_{q^{2}}$ with center at $P$ intersects $V_{2}^{\prime}$ at exactly
$d(q+1)q^{2}+1$ many $\mathbb{F}_{q^{2}}$-rational points. While our argument
shows that the bound of the above theorem is strict, it does not classify all
the surfaces of degree $d$ that attain this bound. This problem could be
interesting for further investigation.

##  4\. General results towards Edoukou’s conjecture

Throughout this section, we will denote by $\mathcal{X}$, a threefold of
degree $d$ defined over $\mathbb{F}_{q^{2}}$ contained in ${\mathbb{P}}^{4}$.
That is $\mathcal{X}=V(F)$, the set of all points $P\in{\mathbb{P}}^{4}$ such
that $F(P)=0$, for some homogeneous polynomial
$F\in\mathbb{F}_{q^{2}}[x_{0},x_{1},x_{2},x_{3},x_{4}]$ with $\deg F=d$. Since
the linear sections of Hermitian threefolds are well-understood, we will be
assuming that $1<d\leq q$. Also, as before, the non-degenerate Hermitian
threefold will be denoted by $V_{3}$.

######  Proposition 4.1.

Suppose that $\mathcal{X}$ does not contain a generator. Then

$|\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}|\leq d(q^{5}+1).$

In particular, we have $|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}|<d(q^{5}+q^{2})+q^{3}+1.$

###### Proof.

Let us consider the incidence set

$I=\\{(P,\ell):P\in\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3},\ell\ \text{is a
generator},P\in\ell\\}.$

We will count $|I|$ in two ways. First, we see that,

(4) $|I|=\sum_{P\in\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}}\\#\\{\ell:P\in\ell\\}=|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}|(q^{3}+1).$

The fact that $\\#\\{\ell:P\in\ell,\ell\ \text{is a generator of }\
V_{3}\\}=q^{3}+1$ is already noted in Remark 2.5. On the other hand, using the
fact that $V_{3}$ contains $(q^{5}+1)(q^{3}+1)$ generators, we have

(5) $|I|=\sum_{\ell\subset V_{3}}|\mathcal{X}(\mathbb{F}_{q^{2}})\cap\ell|\leq
d(q^{5}+1)(q^{3}+1).$

It follows from Equations (4) and (5) that
$|\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}|\leq d(q^{5}+1)$. ∎

It is now natural to proceed to determine an upper bound for
$|\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}|$ in the case when $\mathcal{X}$
contains a generator of $V_{3}$, but no planes in ${\mathbb{P}}^{4}$ defined
over $\mathbb{F}_{q^{2}}$. However, we do not have a good bound in the general
case and as a consequence, the complete solution to Conjecture 1 remains open.
We will present some partial results under this hypothesis later in this
section and we will study this case for cubic threefolds in the subsequent
section. For the time being, let us deal with the case when $\mathcal{X}$
contains a plane in ${\mathbb{P}}^{4}$ defined over $\mathbb{F}_{q^{2}}$, but
it does not contain a hyperplane in ${\mathbb{P}}^{4}$ defined over
$\mathbb{F}_{q^{2}}$.

######  Proposition 4.2.

Suppose $\mathcal{X}$ contains a plane defined over $\mathbb{F}_{q^{2}}$, but
no hyperplane defined over $\mathbb{F}_{q^{2}}$. Then

$|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}|\leq(d-1)q^{5}+(d-1)q^{4}+dq^{3}+dq^{2}+1.$

In particular, $|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}|<d(q^{5}+q^{2})+q^{3}+1.$

###### Proof.

Let $\Pi$ be a plane defined over $\mathbb{F}_{q^{2}}$ contained in
$\mathcal{X}$. Let us denote

$\mathcal{B}^{\prime}(\Pi):=\\{\Sigma:\Pi\subset\Sigma,\Sigma\ \text{is a
hyperplane defined over}\ \mathbb{F}_{q^{2}}\subset{\mathbb{P}}^{4}\\}.$

Note that for any plane $\Pi\subset{\mathbb{P}}^{4}$ defined over
$\mathbb{F}_{q^{2}}$, we have $|V_{3}\cap\Pi|\leq q^{3}+q^{2}+1.$ Also, for
any $\Sigma\in\mathcal{B}^{\prime}(\Pi)$, using Proposition 2.15, and the fact
that $\Sigma\not\subset\mathcal{X}$, we may conclude that

$|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}\cap(\Sigma\setminus\Pi)|\leq(d-1)(q+1)q^{2}.$

We have,

$\displaystyle|\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}|$
$\displaystyle=|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}\cap\Pi|+\sum_{\Sigma\in\mathcal{B}^{\prime}(\Pi)}|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}\cap(\Sigma\setminus\Pi)|$ $\displaystyle\leq
q^{3}+q^{2}+1+(q^{2}+1)(d-1)(q+1)q^{2}$
$\displaystyle=(d-1)q^{5}+(d-1)q^{4}+dq^{3}+dq^{2}+1.$

The last inequality follows trivially since $d\leq q$. ∎

As mentioned above, it is difficult to find good bounds for
$|\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}|$ when $\mathcal{X}$ contains a
generator of $V_{3}$ but does not contain a plane defined over
$\mathbb{F}_{q^{2}}$. Some partial results to this end are given below.

######  Proposition 4.3.

Suppose that $\mathcal{X}$ contains no plane in ${\mathbb{P}}^{4}$ defined
over $\mathbb{F}_{q^{2}}$. If there exists a plane $\Pi_{0}$ in
${\mathbb{P}}^{4}$ defined over $\mathbb{F}_{q^{2}}$ such that
$\Pi_{0}\cap\mathcal{X}$ has $d$ generators of $V_{3}$, then

$|\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}|\leq
dq^{5}-q^{4}+dq^{3}+dq^{2}+1<d(q^{5}+q^{2})+q^{3}+1.$

###### Proof.

Let $\Pi_{0}\cap V_{3}$ contain $d$ generators of $V_{3}$. Since a plane
containing a generator of $V_{3}$ can intersect $V_{3}$ at either one line or
$q+1$ concurrent lines, it follows that $V_{3}\cap\Pi_{0}$ is a union of $q+1$
lines $\ell_{0},\dots,\ell_{q}$ passing through a common point, say $P\in
V_{3}$. It is also easy to see that $\mathcal{B}^{\prime}(\Pi_{0})$ contains
exactly one hyperplane that is tangent to $V_{3}$, namely, the tangent
hyperplane to $V_{3}$ at $P$. Indeed, if $\mathcal{B}^{\prime}(\Pi)$ contains
a tangent hyperplane $T_{Q}(V_{3})$ where $Q\neq P$, then
$\Pi_{0}=T_{P}(V_{3})\cap T_{Q}(V_{3})$ and hence the lines
$\ell_{0},\dots,\ell_{q}$ are contained in $T_{Q}(V_{3})$. Consequently, the
$q+1$ lines meet at the point $Q$, implying $P=Q$, a contradiction. If
$\Sigma=T_{P}(V_{3})$, then $\mathcal{X}\cap\Sigma$ is a surface of degree $d$
in $\Sigma$. It now follows from Theorem 3.3 that

$|\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}\cap T_{P}(V_{3})|\leq
d(q+1)q^{2}+1.$

If $\Sigma\in\mathcal{B}^{\prime}(\Pi_{0})$, but $\Sigma\neq T_{P}(V_{3})$, we
conclude from Theorem 2.2 (b), that $\Sigma\cap V_{3}$ is a non-degenerate
Hermitian surface in $\Sigma$. Now Theorem 2.9 yields

$|\Sigma\cap\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}|\leq
d(q^{3}+q^{2}-q)+q+1.$

However, the upper bound above is attained if and only if
$\Sigma\cap\mathcal{X}$ contains $d$ planes that are tangent to $V_{3}$.
Since, in our case, the threefold $\mathcal{X}$ does not contain a plane, a
strict bound [4, Theorem 5.1] applies and we have

$|\Sigma\cap\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}|\leq
dq^{3}+(d-1)q^{2}+1.$

Finally, $\mathcal{X}(\mathbb{F}_{q^{2}})$ does not contain $\Pi_{0}$ and
$\mathcal{X}\cap\Pi_{0}$ contains the union of $d$ generators. It follows that
$\mathcal{X}\cap\Pi_{0}$ is exactly the union of $d$ generators of $V_{3}$.
Thus

$|\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}\cap\Pi_{0}|=dq^{2}+1.$

We may thus conclude that, if $\Sigma\in\mathcal{B}^{\prime}(\Pi_{0})$ and
$\Sigma\neq T_{P}(V_{3})$, then

$|\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}\cap(\Sigma\setminus\Pi_{0})|\leq
dq^{3}+(d-1)q^{2}+1-(dq^{2}+1)=dq^{3}-q^{2}.$

Following the same argument as in Proposition 4.2, we see that

$\displaystyle|\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}|$
$\displaystyle=|\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}\cap
T_{P}(V_{3})|+\sum_{\begin{subarray}{c}\Sigma\in\mathcal{B}^{\prime}(\Pi_{0})\\\
\Sigma\neq T_{P}(V_{3})\end{subarray}}|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}\cap(\Sigma\setminus\Pi_{0})|$ $\displaystyle\leq
d(q+1)q^{2}+1+q^{2}(dq^{3}-q^{2})$
$\displaystyle=dq^{5}-q^{4}+dq^{3}+dq^{2}+1$

The last strict inequality in the assertion follows trivially since $d\leq q$.
∎

######  Proposition 4.4.

Suppose that $\mathcal{X}$ contains no plane in ${\mathbb{P}}^{4}$ defined
over $\mathbb{F}_{q^{2}}$. Furthermore, assume that $\mathcal{X}\cap\Pi$
contains at most one generator for any plane $\Pi$ defined over
$\mathbb{F}_{q^{2}}$ in ${\mathbb{P}}^{4}$. Then

$|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}|\leq(d-1)q^{5}+q^{2}+(d-1)q^{3}+(d-1)q+1<d(q^{5}+q^{2})+q^{3}+1.$

###### Proof.

Suppose that for all plane $\Pi$ defined over $\mathbb{F}_{q^{2}}$ in
${\mathbb{P}}^{4}$, $\mathcal{X}\cap\Pi$ does not contain any generator. This
in particular implies that $\mathcal{X}$ does not contain any generator. Then
Proposition 4.1 applies and proves the assertion. Thus we may assume that
$\mathcal{X}$ contains a generator $\ell$ of $V_{3}$.

Claim: For every $\Pi\in\mathcal{B}(\ell)$, we have $|\mathcal{X}\cap
V_{3}\cap(\Pi\setminus\ell)|\leq(d-1)q$.

Proof of claim: Note that, for any $\Pi\in\mathcal{B}(\ell)$ the intersection
$\Pi\cap V_{3}$ consists of at most $q+1$ generators. Since $\mathcal{X}\cap
V_{3}$ contains no generators of $V_{3}$ other than $\ell$, it follows that
$\mathcal{X}\cap(\Pi\setminus\ell)$ and $V_{3}\cap(\Pi\setminus\ell)$ are
plane curves of degrees $d-1$ and at most $q$ respectively. Furthermore, the
two affine curves have no common components. Using Bézout’s Theorem, we now
conclude that $|\mathcal{X}\cap
V_{3}\cap(\Pi\setminus\ell)|=|\left(\mathcal{X}\cap(\Pi\setminus\ell)\right)\cap\left(V_{3}\cap(\Pi\setminus\ell)\right)|\leq(d-1)q.$
Thus

$\displaystyle|\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}|$
$\displaystyle=|\ell(\mathbb{F}_{q^{2}})|+\sum_{\Pi\in\mathcal{B}(\ell)}|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}\cap(\Pi\setminus\ell)|$
$\displaystyle\leq(q^{2}+1)+(q^{4}+q^{2}+1)(d-1)q$
$\displaystyle=(d-1)q^{5}+q^{2}+(d-1)q^{3}+(d-1)q+1$
$\displaystyle<d(q^{5}+q^{2})+q^{3}+1.$

The first inequality follows from the claim, whereas the strict inequality is
an immediate consequence of the hypothesis $d\leq q$. ∎

Before proceeding further, we show that the results derived above are enough
to prove Edoukou’s bound. To this end, let us first prove the following Lemma.

######  Lemma 4.5.

If $\Sigma_{1}$ be a hyperplane in ${\mathbb{P}}^{4}$ defined over
$\mathbb{F}_{q^{2}}$ that is not a tangent to $V_{3}$, then for any hyperplane
$\Sigma$ in ${\mathbb{P}}^{4}$ defined over $\mathbb{F}_{q^{2}}$, we have
$|\Sigma\cap\Sigma_{1}\cap V_{3}|\geq q^{3}+1$.

###### Proof.

If $\Sigma_{1}=\Sigma$, then $|\Sigma\cap\Sigma_{1}\cap V_{3}|=|\Sigma_{1}\cap
V_{3}|=q^{5}+q^{3}+q^{2}+1>q^{3}+1$. We may thus assume that
$\Sigma\neq\Sigma_{1}$. Thus $\Sigma\cap\Sigma_{1}$ is a plane contained in
the projective space $\Sigma_{1}$. Since $\Sigma_{1}$ is not a tangent to
$V_{3}$, we see that $V_{3}\cap\Sigma_{1}$ is a non-degenerate Hermitian
surface in $\Sigma_{1}$. Note that $\Sigma\cap\Sigma_{1}$ is a plane defined
over $\mathbb{F}_{q^{2}}$ contained in $\Sigma_{1}$. We know that a planar
section of a nondegenerate Hermitian surface is either a plane nonsingular
Hermitian curve or a union of $q+1$ lines passing through a common point. Thus
$|\Sigma\cap\Sigma_{1}\cap V_{3}|\geq q^{3}+1$, proving the assertion. ∎

Now, we are ready to prove Edoukou’s result on the intersection of a non-
degenerate Hermitian threefold and a quadric threefold in ${\mathbb{P}}^{4}$.

######  Corollary 4.6 (Edoukou’s Theorem).

[8, Theorem 4.1] Let $Q$ be a quadric threefold defined over
$\mathbb{F}_{q^{2}}$ in ${\mathbb{P}}^{4}$. Then

$|V_{3}\cap Q(\mathbb{F}_{q^{2}})|\leq 2(q^{5}+q^{2})+q^{3}+1.$

The bound is attained by a quadric $Q$ defined over $\mathbb{F}_{q^{2}}$ if
and only if $Q$ is a union of two hyperplanes $\Sigma_{1}$ and $\Sigma_{2}$
defined over $\mathbb{F}_{q^{2}}$, both not tangents to $V_{3}$, such that the
plane $\Sigma_{1}\cap\Sigma_{2}$ intersects $V_{3}$ at a non-degenerate plane
Hermitian curve.

###### Proof.

We distinguish the proof into two cases:

Case 1: Suppose $Q$ is reducible. Then $Q$ must be a union of hyperplanes
$\Sigma_{1}$ and $\Sigma_{2}$ defined over $\mathbb{F}_{q^{2}}$. From Theorem
2.2, we see that for any hyperplane $\Sigma$ in ${\mathbb{P}}^{4}$ defined
over $\mathbb{F}_{q^{2}}$,

$|\Sigma(\mathbb{F}_{q^{2}})\cap V_{3}|=\begin{cases}q^{5}+q^{3}+q^{2}+1\ \
\text{if}\ \Sigma\ \text{is not a tangent to}\ V_{3}\\\ q^{5}+q^{2}+1\ \
\text{otherwise}\end{cases}.$

If $\Sigma_{1},\Sigma_{2}$ are both tangents to $V_{3}$, then

$|V_{3}\cap
Q(\mathbb{F}_{q^{2}})|\leq|V_{3}\cap\Sigma_{1}|+|V_{3}\cap\Sigma_{2}|\leq
2(q^{5}+q^{2}+1)<2(q^{5}+q^{2})+q^{3}+1.$

So we may assume without loss of generality that $\Sigma_{1}$ is not a tangent
to $V_{3}$. Thus

$\displaystyle|V_{3}\cap Q(\mathbb{F}_{q^{2}})|$
$\displaystyle=|V_{3}\cap\Sigma_{1}|+|V_{3}\cap\Sigma_{2}|-|V_{3}\cap\Sigma_{1}\cap\Sigma_{2}|$
$\displaystyle\leq 2(q^{5}+q^{3}+q^{2}+1)-(q^{3}+1)$
$\displaystyle=2(q^{5}+q^{2})+q^{3}+1.$

The inequality above is a consequence of Lemma 4.5. Note that here equality
occurs if and only if both $\Sigma_{1}$ and $\Sigma_{2}$ are not tangents to
$V_{3}$ and the plane $\Sigma_{1}\cap\Sigma_{2}$ intersects $V_{3}$ at a
nonsingular plane Hermitian curve.

Case 2: Suppose $Q$ is irreducible. Thus $Q$ can not contain a hyperplane
defined over $\mathbb{F}_{q^{2}}$. If $Q$ contains a plane defined over
$\mathbb{F}_{q^{2}}$, Proposition 4.2 applies and establishes the assertion.
We may assume that $Q$ contains no plane defined over $\mathbb{F}_{q^{2}}$. If
$Q$ contains no generators of $V_{3}$, then the assertion follows from
Proposition 4.1. Suppose $Q$ contains a generator of $V_{3}$. If every plane
in ${\mathbb{P}}^{4}$ defined over $\mathbb{F}_{q^{2}}$ contains at most one
generator of $V_{3}$, then Proposition 4.4 applies. Finally, if a plane in
${\mathbb{P}}^{4}$ defined over $\mathbb{F}_{q^{2}}$ that contains two
generators of $V_{3}$ exists, then the assertion inequality is obtained from
Proposition 4.3. We note that all the inequalities in this case are strict.
This completes the proof. ∎

We conclude this section by showing that if $\mathcal{X}$ contains a line
$\ell$ defined over $\mathbb{F}_{q^{2}}$ that is tangent to $V_{3}$, then
$|\mathcal{X}({\mathbb{F}}_{q})\cap V_{3}|$ satisfies the bound in Edoukou’s
conjecture. To this end, we have the following Lemma.

######  Lemma 4.7.

Suppose $P\in V_{3}$ and $\ell_{P}$ is a tangent line to $V_{3}$ passing
through $P$. Then

  1. (a)

there exist $q^{2}+1$ planes containing $\ell_{P}$ that are contained in
$T_{P}(V_{3})$, and

  2. (b)

if $\Pi\in\mathcal{B}(\ell_{P})$ and $\Pi\not\subset T_{P}(V_{3})$, then
$\Pi\cap V_{3}$ does not contain any generator of $V_{3}$.

###### Proof.

To prove part (a), it is enough to show that $\ell_{P}\subset T_{P}(V_{3})$.
But the later is true since any line passing through $P$ that is not contained
in $T_{P}(V_{3})$ intersects $V_{3}$ at $q+1$ points. Suppose
$\Pi\in\mathcal{B}(\ell_{P})$ and $\Pi$ contains a generator, say $\ell$, of
$V_{3}$. Now $\ell$ and $\ell_{P}$ are two distinct lines in $\Pi$ and hence
$\Pi$ is the unique plane containing $\ell$ and $\ell_{P}$. On the other hand,
both the lines $\ell$ and $\ell_{P}$ are contained in $T_{P}(V_{3})$. It
follows that $\Pi\subset T_{P}(V_{3})$. This proves part (b). ∎

######  Proposition 4.8.

Suppose that $\mathcal{X}$ contains a tangent line to $V_{3}$ and
$\mathcal{X}$ does not contain any plane defined over $\mathbb{F}_{q^{2}}$.
Then

$|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}|\leq(d-1)q^{5}+(d-1)q^{4}+dq^{3}+dq^{2}+1<d(q^{5}+q^{2})+q^{3}+1.$

###### Proof.

Suppose $\mathcal{X}$ contains a line $\ell_{P}$ that is tangent to $V_{3}$ at
the point $P\in V_{3}$. Write

$\mathcal{B}(\ell_{P})=\\{\Pi_{0},\dots,\Pi_{q^{2}},\Pi_{q^{2}+1},\dots,\Pi_{q^{4}+q^{2}}\\},$

where $\Pi_{0},\dots,\Pi_{q^{2}}\subset T_{P}(V_{3})$. Note that

(6) $|\ell_{P}\cap\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}|+\sum_{i=0}^{q^{2}}|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}\cap(\Pi\setminus\ell_{P})|=|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}\cap T_{P}(V_{3})|\leq d(q+1)q^{2}+1.$

Note that the last inequality follows from Theorem 3.3. Using Lemma 4.7, we
conclude that, for $j=q^{2}+1,\dots,q^{4}+q^{2}$, the plane $\Pi_{j}$
intersects $V_{3}$ at a plane non-degenerate Hermitian curve. Also the curve
$\mathcal{X}\cap(\Pi_{j}\setminus\ell_{P})$ is a plane curve of degree at most
$d-1$. From Bézout’s theorem,

(7) $|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}\cap(\Pi\setminus\ell_{P})|\leq(d-1)(q+1).$

Finally, using equations (6) and (7), we see that,

$\displaystyle|\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}|$
$\displaystyle=|\ell_{P}\cap\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}|+\sum_{j=0}^{q^{4}+q^{2}}|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}\cap(\Pi\setminus\ell_{P})|$
$\displaystyle=|\mathcal{X}(\mathbb{F}_{q^{2}})\cap V_{3}\cap
T_{P}(V_{3})|+\sum_{j=q^{2}+1}^{q^{4}+q^{2}}|\mathcal{X}(\mathbb{F}_{q^{2}})\cap
V_{3}\cap(\Pi\setminus\ell_{P})|$ $\displaystyle\leq
d(q+1)q^{2}+1+q^{4}(d-1)(q+1)$
$\displaystyle=(d-1)q^{5}+(d-1)q^{4}+dq^{3}+dq^{2}+1<d(q^{5}+q^{2})+q^{3}+1.$

This completes the proof. ∎

##  5\. Intersection of cubic threefolds with Hermitian threefold

The main goal of this section is to determine the maximum number of
$\mathbb{F}_{q^{2}}$-rational points on the intersection of a cubic threefold
and Hermitian threefolds in ${\mathbb{P}}^{4}$. To this end, we will denote by
$\mathcal{C}$ a cubic threefold in ${\mathbb{P}}^{4}$ defined over
$\mathbb{F}_{q^{2}}$ while we keep using $V_{3}$ for denoting the non-
degenerate Hermitian threefold in ${\mathbb{P}}^{4}$. Note that, we have dealt
with threefolds that

  1. (a)

contains no generators of $V_{3}$ and

  2. (b)

contains a plane defined over $\mathbb{F}_{q^{2}}$, but no hyperplanes in
${\mathbb{P}}^{4}$ defined over $\mathbb{F}_{q^{2}}$

in the previous section. Thus, we will need to study the cases when

  1. (i)

$\mathcal{C}$ contains generators of $V_{3}$ but no planes defined over
$\mathbb{F}_{q^{2}}$ in ${\mathbb{P}}^{4}$ and

  2. (ii)

$\mathcal{C}$ contains a hyperplane of ${\mathbb{P}}^{4}$ defined over
$\mathbb{F}_{q^{2}}$

in this section. For ease of reading, we will divide the section into two
subsections.

###  5.1. Cubic threefolds containing a hyperplane

In this subsection, we will make use of Edoukou’s result (Proposition 2.8) to
show that if $\mathcal{C}$ contains a hyperplane in ${\mathbb{P}}^{4}$ defined
over $\mathbb{F}_{q^{2}}$, then the bound in Edoukou’s conjecture holds in the
affirmative.

######  Proposition 5.1.

Let $\mathcal{C}$ be a cubic threefold in ${\mathbb{P}}^{4}$ defined over
$\mathbb{F}_{q^{2}}$ that contains a hyperplane $\Sigma$ defined over
$\mathbb{F}_{q^{2}}$. Then

$|\mathcal{C}(\mathbb{F}_{q^{2}})\cap V_{3}|\leq 3(q^{5}+q^{2})+q^{3}+1.$

###### Proof.

Suppose that $\mathcal{C}(\mathbb{F}_{q^{2}})$ contains a hyperplane $\Sigma$
defined over $\mathbb{F}_{q^{2}}$. It follows immediately that
$\mathcal{C}=\Sigma\cup Q$, where $Q$ is a quadric threefold defined over
$\mathbb{F}_{q^{2}}$. Now, $|\Sigma\cap V_{3}|\leq q^{5}+q^{3}+q^{2}+1$. We
distinguish four cases.

Case 1: $Q$ is of Type I. Then $Q=\Sigma_{1}\cup\Sigma_{2}$, where
$\Sigma_{1}$ and $\Sigma_{2}$ are hyperplanes in ${\mathbb{P}}^{4}$ defined
over $\mathbb{F}_{q^{2}}$, neither of them tangent to $V_{3}$. Using
Proposition 2.8 we derive that $|Q\cap V_{3}|=2(q^{5}+q^{2})+q^{3}+1$. We
have,

$\displaystyle|\mathcal{C}\cap V_{3}|$ $\displaystyle=|\Sigma\cap
V_{3}|+|Q(\mathbb{F}_{q^{2}})\cap
V_{3}|-|(\Sigma_{1}\cup\Sigma_{2})\cap\Sigma\cap V_{3}|$
$\displaystyle\leq|\Sigma\cap V_{3}|+|Q(\mathbb{F}_{q^{2}})\cap
V_{3}|-|(\Sigma_{1}\cap\Sigma)\cap V_{3}|$
$\displaystyle\leq(q^{5}+q^{3}+q^{2}+1)+2(q^{5}+q^{2})+q^{3}+1-(q^{3}+1)$
$\displaystyle=3(q^{5}+q^{2})+q^{3}+1.$

The last inequality above follows from Lemma 4.5.

Case 2: $Q$ is of Type II. We have $Q=\Sigma_{1}\cup\Sigma_{2}$, where
$\Sigma_{1}$ and $\Sigma_{2}$ are hyperplanes in ${\mathbb{P}}^{4}$ defined
over $\mathbb{F}_{q^{2}}$, neither of them tangent to $V_{3}$. Using
Proposition 2.8 we derive that $|Q(\mathbb{F}_{q^{2}})\cap
V_{3}|=2q^{5}+q^{2}+q^{3}+1$. We have,

$\displaystyle|\mathcal{C}(\mathbb{F}_{q^{2}})\cap V_{3}|$
$\displaystyle=|\Sigma\cap V_{3}|+|Q(\mathbb{F}_{q^{2}})\cap
V_{3}|-|(\Sigma_{1}\cup\Sigma_{2})\cap\Sigma\cap V_{3}|$
$\displaystyle\leq|\Sigma\cap V_{3}|+|Q(\mathbb{F}_{q^{2}})\cap
V_{3}|-|(\Sigma_{1}\cap\Sigma)\cap V_{3}|$
$\displaystyle\leq(q^{5}+q^{3}+q^{2}+1)+2q^{5}+q^{2}+q^{3}+1-(q^{3}+1)$
$\displaystyle=3q^{5}+q^{3}+2q^{2}+1<3(q^{5}+q^{2})+q^{3}+1.$

The last inequality above follows from Lemma 4.5.

Case 3: $Q$ is of Type III. Here $Q=\Sigma_{1}\cup\Sigma_{2}$, where
$\Sigma_{1}$ and $\Sigma_{2}$ are hyperplanes in ${\mathbb{P}}^{4}$ defined
over $\mathbb{F}_{q^{2}}$. Furthermore, the hyperplane $\Sigma_{2}$ is not a
tangent to $V_{3}$. Using Proposition 2.8 we derive that
$|Q(\mathbb{F}_{q^{2}})\cap V_{3}|=2q^{5}+2q^{2}+q^{3}+1$. We have,

$\displaystyle|\mathcal{C}(\mathbb{F}_{q^{2}})\cap V_{3}|$
$\displaystyle=|\Sigma\cap V_{3}|+|Q(\mathbb{F}_{q^{2}})\cap
V_{3}|-|(\Sigma_{1}\cup\Sigma_{2})\cap\Sigma\cap V_{3}|$
$\displaystyle\leq|\Sigma\cap V_{3}|+|Q(\mathbb{F}_{q^{2}})\cap
V_{3}|-|(\Sigma_{2}\cap\Sigma)\cap V_{3}|$
$\displaystyle\leq(q^{5}+q^{3}+q^{2}+1)+2q^{5}+2q^{2}+1-(q^{3}+1)$
$\displaystyle=3q^{5}+3q^{2}+1<3(q^{5}+q^{2})+q^{3}+1.$

Case 4: $Q$ is not of Type I, Type II or Type III. In this case, from
Proposition 2.8, we obtain that $|Q\cap V_{3}|\leq 2q^{5}+q^{2}+1$. Thus

$\displaystyle|\mathcal{C}\cap V_{3}|$ $\displaystyle\leq|\Sigma\cap
V_{3}|+|Q\cap V_{3}|$ $\displaystyle\leq q^{5}+q^{3}+q^{2}+1+2q^{5}+q^{2}+1$
$\displaystyle=3q^{5}+2q^{2}+q^{3}+2<3(q^{5}+q^{2})+q^{3}+1.$

This completes the proof. ∎

######  Remark 5.2.

For the sake of proving the inequality in the above Proposition, it is not
really necessary to distinguish Case 1, Case 2, and Case 3. In each of these
three cases, the presence of one nontangent hyperplane in the conic plays a
pivotal role in establishing the required upper bound. However, the above
distinction shows us that the upper bound could be attained by a cubic
threefold $\mathcal{C}$ in ${\mathbb{P}}^{4}$ defined over
$\mathbb{F}_{q^{2}}$ and containing a hyperplane in ${\mathbb{P}}^{4}$ defined
over $\mathbb{F}_{q^{2}}$ will attain the upper bound if and only if the
following conditions are satisfied:

  1. (a)

$\Sigma_{1},\Sigma_{2},\Sigma_{3}$ are distinct hyperplanes in
${\mathbb{P}}^{4}$ defined over $\mathbb{F}_{q^{2}}$.

  2. (b)

$\Sigma_{1},\Sigma_{2},\Sigma_{3}$ intersect at a common plane $\Pi$ in
${\mathbb{P}}^{4}$ such that $\Pi\cap V_{3}$ is a plane nonsingular Hermitian
curve.

In fact, we will establish at the end of this article that if $q\geq 7$, then
a cubic threefold $\mathcal{C}$ defined over $\mathbb{F}_{q^{2}}$ in
${\mathbb{P}}^{4}$ attains the upper bound of Edoukou’s conjecture if and only
if $\mathcal{C}$ satisfies the conditions mentioned above.

###  5.2. Cubic threefolds containing no hyperplanes

######  Lemma 5.3.

Let $\mathcal{C}^{\prime}$ be an irreducible plane cubic curve defined over
$\mathbb{F}_{q^{2}}$. Then

  1. (a)

If $\mathcal{C}^{\prime}$ is absolutely irreducible, then

$||\mathcal{C}^{\prime}(\mathbb{F}_{q^{2}})|-(q^{2}+1)|\leq 2q.$

  2. (b)

If $\mathcal{C}^{\prime}$ is not absolutely irreducible, then
$|\mathcal{C}^{\prime}(\mathbb{F}_{q^{2}})|\leq 1$.

###### Proof.

  1. (a)

Follows from Theorem 2.14.

  2. (b)

Suppose that $\mathcal{C}^{\prime}$ is a plane cubic curve defined over
$\mathbb{F}_{q^{2}}$ that is irreducible over $\mathbb{F}_{q^{2}}$ but
reducible in $\overline{{\mathbb{F}}}_{q}$, the algebraic closure of
${\mathbb{F}}_{q}$. From [11, Theorem 3.2], we see that if $Z$ is an
absolutely irreducible component of $\mathcal{C}^{\prime}$, then
$\deg(Z)\mid\deg\mathcal{C}^{\prime}$. Since $\deg\mathcal{C}^{\prime}=3$, it
follows that $\mathcal{C}^{\prime}$ is a union of three lines passing through
a common point, say $P$. Thus Proposition 2.16 implies that
$\mathcal{C}^{\prime}(\mathbb{F}_{q^{2}})\subseteq\\{P\\}$ and hence the
assertion is established.

This completes the proof. ∎

######  Lemma 5.4.

Let $q\geq 7$ and $\mathcal{C}$ be a cubic threefold defined over
$\mathbb{F}_{q^{2}}$ and $P\in\mathcal{C}(\mathbb{F}_{q^{2}})\cap V_{3}$. If
$\mathcal{C}\cap T_{P}(V_{3})$ is a cone over a plane cubic absolutely
irreducible curve $\mathcal{C}^{*}$ with center at $P$, then $\mathcal{C}$
contains a line that is tangent to $V_{3}$ at $P$.

###### Proof.

Since $\mathcal{C}^{*}$ is a plane absolutely irreducible curve defined over
$\mathbb{F}_{q^{2}}$, it follows from Lemma 5.3 that
$|\mathcal{C}^{*}(\mathbb{F}_{q^{2}})|\geq q^{2}+1-2q$. Suppose that
$\mathcal{C}^{*}\subset\Pi$, where $\Pi$ is a plane, not passing through $P$,
defined over $\mathbb{F}_{q^{2}}$ which is contained in $T_{P}(V_{3})$. Now
$V_{3}\cap\Pi$ is a plane non-degenerate Hermitian curve and hence Bézout’s
theorem implies $|(V_{3}\cap\Pi)\cap C^{*}|\leq 3(q+1)$. Since $q\geq 7$, it
follows that $|\mathcal{C}^{*}(\mathbb{F}_{q^{2}})|>|(V_{3}\cap\Pi)\cap
C^{*}|$ and hence there is a point $P^{\prime}\in\mathcal{C}^{*}$ that is not
contained in $V_{3}$. Since $\mathcal{C}$ is a cone over $C^{*}$ with center
at $P$, the line $\ell$ joining $P$ and $P^{\prime}$ is contained in
$\mathcal{C}$. Since $\ell$ is a line defined over $\mathbb{F}_{q^{2}}$
passing through $P$, the line $\ell$ must be either a tangent line or a
generator. Since $P^{\prime}\not\in V_{3}$, it follows that $\ell$ can not be
a generator and the assertion follows. ∎

######  Lemma 5.5.

Let $P\in V_{3}$ and $\ell$ be a generator of $V_{3}$ containing $P$. Let
$\mathcal{C}$ be a cubic threefold defined over $\mathbb{F}_{q^{2}}$ such that
for any plane $\Pi$ defined over $\mathbb{F}_{q^{2}}$, the threefold
$\mathcal{C}$ contains at most two generators of $V_{3}$. If $\ell$ is
contained in $\mathcal{C}$, then one of the following holds:

  1. (a)

$T_{P}(V_{3})\cap\mathcal{C}$ contains a cone over a plane curve with center
at $P$,

  2. (b)

$|T_{P}(V_{3})\cap\mathcal{C}\cap V_{3}|\leq 2q^{3}+6q^{2}-3q-4$.

###### Proof.

Suppose (a) does not hold. Then, in view of Theorem 2.17, we see that
$T_{P}(V_{3})\cap\mathcal{C}$ contains at most six lines passing though $P$.
Let

$\mathcal{S}:=\\{\Pi\in\mathcal{B}_{T_{P}(V_{3})}(\ell):\mathcal{C}\cap\Pi\
\text{contains two generators of}\ V_{3}\\}\ \ \text{and}\ \ s=|\mathcal{S}|.$

It follows that $s\leq 5$ and for any $\Pi\in\mathcal{S}$, we have $|\Pi\cap
V_{3}\cap(\mathcal{C}\setminus\ell)|\leq q^{2}+q-1$. On the other hand, for
any $\Pi\in\mathcal{B}_{T_{P}(V_{3})}(\ell)\setminus\mathcal{S}$, using
Bézout’s Theorem, we see that $|\Pi\cap
V_{3}\cap(\mathcal{C}\setminus\ell)|\leq 2q$. Thus

$\displaystyle|T_{P}(V_{3})\cap C\cap V_{3}|$
$\displaystyle\leq|\ell|+\sum_{\Pi\in\mathcal{S}}|\Pi\cap
V_{3}\cap(\mathcal{C}\setminus\ell)|+\sum_{\Pi\in\mathcal{B}_{T_{P}(V_{3})}(\ell)\setminus\mathcal{S}}|\Pi\cap
V_{3}\cap(\mathcal{C}\setminus\ell)|$
$\displaystyle\leq(q^{2}+1)+s(q^{2}+q-1)+(q^{2}+1-s)(2q)$
$\displaystyle=2q^{3}+q^{2}+2q+1+s(q^{2}-q-1)$ $\displaystyle\leq
2q^{3}+q^{2}+2q+1+5(q^{2}-q-1)\ \ (\text{since}\ q^{2}-q-1>0)$
$\displaystyle=2q^{3}+6q^{2}-3q-4.$

This completes the proof. ∎

######  Proposition 5.6.

Let $q\geq 7$. Let $\mathcal{C}$ be a cubic threefold defined over
$\mathbb{F}_{q^{2}}$ that contains a generator of $V_{3}$ but contains no
plane defined over $\mathbb{F}_{q^{2}}$. Moreover, assume that for any plane
$\Pi$ defined over $\mathbb{F}_{q^{2}}$, the intersection $\mathcal{C}\cap\Pi$
contains at most two generators of $V_{3}$. Then

$|\mathcal{C}(\mathbb{F}_{q^{2}})\cap V_{3}|<3(q^{5}+q^{2})+q^{3}+1.$

###### Proof.

Suppose $\mathcal{C}$ contains a generator $\ell$ of $V_{3}$. Now, as
discussed in Remark 2.18, for every $P\in\ell$, either the cubic surface
$\mathcal{C}\cap T_{P}(V_{3})$ is a cone over a plane cubic curve
$\mathcal{C}_{P}^{\prime}$ with center at $P$ or $\mathcal{C}\cap
T_{P}(V_{3})$ contains at most $6$ lines passing through $P$.

Let $L=\\{P\in\ell:\mathcal{C}\cap T_{P}(V_{3})\ \text{is a cone over a plane
cubic}\ \mathcal{C}^{\prime}_{P}\ \text{with center at}\ P\\}.$ Observe that
for each $P\in L$, if $\mathcal{C}^{\prime}_{P}$ is reducible, then
$\mathcal{C}^{\prime}_{P}$ contains a line defined over $\mathbb{F}_{q^{2}}$,
and as a consequence, $\mathcal{C}$ contains a plane defined over
$\mathbb{F}_{q^{2}}$. We may thus assume that for each $P\in L$, the plane
curve $\mathcal{C}^{\prime}_{P}$ is irreducible. If $\mathcal{C}^{\prime}_{P}$
is absolutely irreducible, then it follows from Lemma 5.4, that $\mathcal{C}$
contains a line that is tangent to $V_{3}$. Here, Proposition 4.8 applies, and
we obtain the assertion. We may thus assume from now that for every $P\in L$,
the plane curve $\mathcal{C}^{\prime}_{P}$ is irreducible but not absolutely
irreducible. It now follows from Lemma 5.3 (b), that for each $P\in L$, we
have $T_{P}(V_{3})\cap\mathcal{C}\cap V_{3}=\ell.$ Consequently, we have the
following:

(8) $|V_{3}\cap T_{P}(V_{3})\cap(\mathcal{C}\setminus\ell)|=\begin{cases}0\ \
&\text{if}\ P\in L\\\ \leq 2q^{3}+5q^{2}-3q-5\ \ &\text{if}\ P\in\ell\setminus
L.\end{cases}$

The last inequality follows from Lemma 5.5. Since
${\mathbb{P}}^{4}=\displaystyle{\bigcup_{P\in\ell}}T_{P}(V_{3})$, we have

$\displaystyle|C(\mathbb{F}_{q^{2}})\cap V_{3}|$
$\displaystyle=|\ell|+\sum_{P\in L}|V_{3}\cap
T_{P}(V_{3})\cap(\mathcal{C}\setminus\ell)|+\sum_{P\in\ell\setminus
L}|V_{3}\cap T_{P}(V_{3})\cap(\mathcal{C}\setminus\ell)|$
$\displaystyle\leq(q^{2}+1)+|L|\cdot 0+(q^{2}+1-|L|)(2q^{3}+5q^{2}-3q-5)$
$\displaystyle=q^{2}+1+2q^{5}+5q^{4}-q^{3}-3q-5-|L|(2q^{3}+5q^{2}-3q-5)$
$\displaystyle\leq 2q^{5}+5q^{4}-q^{3}+q^{2}-3q-4$
$\displaystyle<3(q^{5}+q^{2})+q^{3}+1.$

The last inequality is true since $q\geq 7$, as per hypothesis. This completes
the proof. ∎

######  Theorem 5.7.

Let $F\in\mathbb{F}_{q^{2}}[x_{0},\dots,x_{4}]$ be a homogeneous polynomial of
degree three and $V_{3}$ be a non-degenerate Hermitian threefold in
${\mathbb{P}}^{4}(\mathbb{F}_{q^{2}})$. If $q\geq 7$, then

$|V(F)(\mathbb{F}_{q^{2}})\cap V_{3}|\leq 3(q^{5}+q^{2})+q^{3}+1.$

Moreover, this bound is attained by a cubic threefold $V(F)$ in
${\mathbb{P}}^{4}$, if and only if $V(F)$ is a union of three non-tangent
hyperplanes $\Sigma_{1},\Sigma_{2}$ and $\Sigma_{3}$ defined over
$\mathbb{F}_{q^{2}}$, such that each of the hyperplanes contains a common
plane $\Pi$ defined over $\mathbb{F}_{q^{2}}$ and $\Pi$ intersects $V_{3}$ at
a non-degenerate plane Hermitian curve.

###### Proof.

Throughout, we will denote by $\mathcal{C}$ the threefold given by the
vanishing locus of $F$, that is $\mathcal{C}=V(F)$. We distinguish the proof
into two cases.

Case 1: $\mathcal{C}$ is reducible over $\mathbb{F}_{q^{2}}$. In this case,
the threefold $\mathcal{C}$ contains a hyperplane defined over
$\mathbb{F}_{q^{2}}$. From Proposition 5.1 we see that
$|V(F)(\mathbb{F}_{q^{2}})\cap V_{3}|\leq 3(q^{5}+q^{2})+q^{3}+1$. Also it
follows from Remark 5.2 that the bound is attained in this case if and only if
$V(F)$ is a union of three non-tangent hyperplanes $\Sigma_{1},\Sigma_{2}$ and
$\Sigma_{3}$ defined over $\mathbb{F}_{q^{2}}$, such that each of the
hyperplanes contains a common plane $\Pi$ defined over $\mathbb{F}_{q^{2}}$
and $\Pi$ intersects $V_{3}$ at a non-degenerate plane Hermitian curve.

Case 2: $\mathcal{C}$ is irreducible over $\mathbb{F}_{q^{2}}$. In this case,
$\mathcal{C}$ does not contain any hyperplane defined over
$\mathbb{F}_{q^{2}}$. If $\mathcal{C}$ does not contain any generator of
$V_{3}$, then Proposition 4.1 shows that $|\mathcal{C}(\mathbb{F}_{q^{2}})\cap
V_{3}|<3(q^{5}+q^{2})+q^{3}+1$. If $V(F)$ contains a plane defined over
$\mathbb{F}_{q^{2}}$, then Proposition 4.2 applies and shows that
$|\mathcal{C}(\mathbb{F}_{q^{2}})\cap V_{3}|<3(q^{5}+q^{2})+q^{3}+1$. So, we
may assume that $\mathcal{C}$ contains a generator of $V_{3}$ but no plane
defined over $\mathbb{F}_{q^{2}}$. It follows that for any plane $\Pi$ defined
over $\mathbb{F}_{q^{2}}$, the plane curve $V(F)\cap\Pi$ can contain at most
three generators of $V_{3}$. If there exists a plane $\Pi_{0}$ in
${\mathbb{P}}^{4}$ defined over $\mathbb{F}_{q^{2}}$ such that
$V(F)\cap\Pi_{0}$ has exactly three generators of $V_{3}$, then Proposition
4.3 implies that $|V(F)(\mathbb{F}_{q^{2}})\cap
V_{3}|<3(q^{5}+q^{2})+q^{3}+1$. Furthermore, if for any plane $\Pi$ defined
over $\mathbb{F}_{q^{2}}$, the intersection $V(F)\cap\Pi$ contains at most two
generators of $V_{3}$, then Proposition 5.6 shows that
$|V(F)(\mathbb{F}_{q^{2}})\cap V_{3}|<3(q^{5}+q^{2})+q^{3}+1$. Thus, we see
that a strict bound always holds in this case. This completes the proof. ∎

Acknowledgements: We are grateful to the anonymous referees for their
important suggestions in making the article better than the previous version.

## References

  * [1] Y. Aubry and M. Perret, A Weil theorem for singular curves. Arithmetic, Geometry and Coding Theory (Luminy, 1993) (eds. R. Pellikaan, M. Perret and S. G. Vladut) (de Gruyter, Berlin, 1996), 1–7.
  * [2] R.C. Bose and I.M. Chakravarti, Hermitian varieties in a finite projective space $PG(N,q^{2})$. _Canad. J. Math._ 18 (1966), 1161 – 1182.
  * [3] P. Beelen and M. Datta, Maximum number of points on intersection of a cubic surface and a non-degenerate hermitian surface. _Moscow Math. J._ , 20 (2020), 453–474.
  * [4] P. Beelen, M. Datta and M. Homma, A proof of sørensen’s conjecture on hermitian surfaces. _Proc. Amer. Math. Soc._ 149 (2021), 1431–1441.
  * [5] I.M. Chakravarti, Some properties and applications of Hermitian varieties in a finite projective space $PG(N,q^{2})$ in the construction of strongly regular graphs (two-class association schemes) and block designs. _J. Combinatorial Theory Ser. B_ 11 (1971), 268 – 283.
  * [6] M. Datta, Maximum number of ${\mathbb{F}}_{q}$-rational points on nonsingular threefolds in ${\mathbb{P}}^{4}$. _Finite Fields Appl._ 59, (2019), 86–96.
  * [7] F.A.B. Edoukou, Codes defined by forms of degree 2 on Hermitian surfaces and Sørensen’s conjecture. _Finite Fields Appl._ 13 (2007), no. 3, 616 – 627.
  * [8] F.A.B. Edoukou, Codes defined by forms of degree 2 on non-degenerate Hermitian varieties in $\mathbb{P}^{4}(\mathbb{F}_{q^{2}})$. _Des. Codes Cryptogr._ 50 (2009), no. 1, 135–146.
  * [9] J. Harris, Algebraic geometry. A first course. Graduate Texts in Mathematics, 133. Springer-Verlag, New York, 1992.
  * [10] M. Homma and S. J. Kim, Number of points of a nonsingular hypersurface in an odd-dimensional projective space.(English summary), _Finite Fields Appl._ 48 (2017), 395–419.
  * [11] G. Lachaud and R. Rolland, On the number of points of algebraic sets over finite fields. _J. Pure Appl. Algebra_ 219 (2015), no. 11, 5117 – 5136.
  * [12] J.-P. Serre, Lettre à M. Tsfasman (French, with English summary), _Journées Arithmétiques_ , 1989 (Luminy, 1989), Astérisque 198 –200 (1991), 11, 351–353 (1992).
  * [13] A.B. Sørensen, Projective Reed-Muller codes, _IEEE Trans. Inform. Theory_ 37 (1991), no. 6, 1567–1576.
  * [14] A.B. Sørensen, Rational points on hypersurfaces, Reed–Muller codes and algebraic-geometric codes, Ph.D. thesis, Aarhus, Denmark, 1991.
  * [15] O. Zariski and P. Samuel, Commutative algebra. Vol. II. Reprint of the 1960 edition. Graduate Texts in Mathematics, Vol. 29. Springer-Verlag, New York-Heidelberg, 1975.