Rational points on non-normal hypercubics

TL;DR

This paper extends the known results on counting rational points from a specific cubic surface to a broader class of non-normal integral hypercubics that are not cones, enhancing understanding of rational solutions on these varieties.

Contribution

It generalizes the count of rational points from the Cayley ruled cubic surface to all non-normal integral hypercubics that are not cones.

Findings

Count of rational points extends to all non-normal integral hypercubics not cones

Provides a unified approach for these hypercubics

Enhances understanding of rational solutions on non-normal cubic varieties

Abstract

We show that the count of rational points by de la Bret\`{e}che, Browning and Salberger on the Cayley ruled cubic surface extends to all non-normal integral hypercubics which are not cones.

Equations

$$H:{\mathbb{P}}^n(\mathbb{Q})\to {\mathbb{R}}_{>0},(t_0:\dots :t_n)\mapsto \sqrt{t_0^2+\dots +t_n^2},$$

$$N(W^{norm},B)=\#\{t\in W^{norm}(\mathbb{Q})\mid H(t)\le B\}.$$

$$N(W^{norm},B)=\frac{\pi B^2}{4\zeta (2)}\cdot \left(4+\sum _{\begin{array}{c}(\mu ,\lambda )\in {\mathbb{Z}}_{\mathrm{prim}}^2\\ \mu \ne 0\end{array}}\frac{\gcd (a,\mu )}{\sqrt{f(\mu ,\lambda )}}\right)+O(B^{3/2}(\log B{)}^2),$$

$$N(W^{norm},B)=\frac{\pi B^3}{3\zeta (3)}\sum _{(\mu ,\lambda )\in {\mathbb{Z}}_{\mathrm{prim}}^2}\frac{1}{\sqrt{f(\mu ,\lambda )}}+O(B^2\log B),$$

$$0\to K\to \mathrm{Aut}(\overline{W})\to \mathrm{PGL}(2)\to 0,$$

$$A(t_2)=u_4\cdot (d^2\cdot t_2-cd\cdot t_3+c^2\cdot t_4)-(c{a}_{31}+cd{a}_{41})\cdot t_0-(d{a}_{31}+d^2{a}_{41})\cdot t_1,$$

$$A(t_3)=u_4\cdot (-2d\cdot t_2+(ad+c)\cdot t_3-2ac\cdot t_4)+(a{a}_{31}+(ad-c)a_{41})\cdot t_0+a_{31}\cdot t_1,$$

$$A(t_4)=u_4\cdot (t_2-a\cdot t_3+a^2\cdot t_4)+a{a}_{41}\cdot t_0+a_{41}\cdot t_1,$$

$$A(t_2)=w_4\cdot (d^2\cdot t_2-cd\cdot t_3+c^2\cdot t_4)-(c{a}_{30}+c^2{a}_{40})\cdot t_0-(d{a}_{30}+cd{a}_{40})\cdot t_1,$$

$$A(t_3)=w_4\cdot (d\cdot t_3-2c\cdot t_4)+a_{30}\cdot t_0-d{a}_{40}\cdot t_1,$$

$$A(t_4)=w_4\cdot t_4+a_{40}\cdot t_0,$$

$$V_y=\{(t_0:t_1:t_2:t_3)\in {\mathbb{P}}^3\mid \lambda t_0-\mu t_1=\mu \lambda t_2+({\mu }^2+a{\lambda }^2)t_3=0\}\subset W,$$

$$N(V,B)=\sum _{y\in {\mathbb{P}}^1(\mathbb{Q})}N(V\cap V_y,B).$$

$$N(V\cap V_y,B)=\frac{1}{2}\cdot \#\{({\tau }_0,{\tau }_3)\in {\mathbb{Z}}_{\mathrm{prim}}^2\mid {\tau }_0\ne 0,H(\mu {\tau }_0,\lambda {\tau }_0,({{\mu }_1}^{2}d+a_1{\lambda }^2)\cdot {\tau }_3,({\mu }_1\lambda )\cdot {\tau }_3)\le B\},$$

$$N(V\cap V_{(0:1)},B)=\frac{1}{2}\cdot \#\{({\tau }_0,{\tau }_3)\in {\mathbb{Z}}_{\mathrm{prim}}^2\mid {\tau }_0\ne 0,\sqrt{{{\tau }_0}^2+{{\tau }_3}^2}\le B\}.$$

$${\lambda }^2+{\mu }^2\le B^2<{\lambda }^2+{\mu }^2+{{\mu }_1}^2{\lambda }^2+({{\mu }_1}^{2}d+a_1{\lambda }^2{)}^2,$$

$$|{\tau }_0|\ll \frac{B}{\sqrt{{\lambda }^2+{\mu }^2}},|{\tau }_3|\ll \frac{B}{\max \{|\lambda |,|\mu |\}}.$$

$$N^{\ast }(V_y,B)=\#\{({\tau }_0,{\tau }_3)\in {\mathbb{Z}}^2\mid {\tau }_0\ne 0,H(\mu {\tau }_0,\lambda {\tau }_0,({{\mu }_1}^{2}d+a_1{\lambda }^2)\cdot {\tau }_3,({\mu }_1\lambda )\cdot {\tau }_3)\le B\},$$

$$N(V\cap V_y,B)=\frac{1}{2}\sum _{k\ll \frac{B}{\sqrt{{\lambda }^2+{\mu }^2}}}\mu (k)\cdot N^{\ast }\left(V_y,\frac{B}{k}\right).$$

$$\begin{gathered} N^{\ast }(V_y,B)=\sum _{|{\tau }_3|\le \frac{B}{\sqrt{({\mu }_1\lambda {)}^2+({{\mu }_1}^{2}d+a_1{\lambda }^2{)}^2}}}2\cdot \left[\frac{\sqrt{B^2-(({\mu }_1\lambda {)}^2+({{\mu }_1}^{2}d+a_1{\lambda }^2{)}^2)\cdot {{\tau }_3}^2}}{\sqrt{{\lambda }^2+{\mu }^2}}\right] \\ =2\cdot \underset{-\frac{B}{\sqrt{({\mu }_1\lambda {)}^2+({{\mu }_1}^{2}d+a_1{\lambda }^2{)}^2}}}{\overset{\frac{B}{\sqrt{({\mu }_1\lambda {)}^2+({{\mu }_1}^{2}d+a_1{\lambda }^2{)}^2}}}{\int }}\frac{1}{\sqrt{{\lambda }^2+{\mu }^2}}\cdot \sqrt{B^2-(({\mu }_1\lambda {)}^2+({{\mu }_1}^{2}d+a_1{\lambda }^2{)}^2)\cdot x^2}\cdot dx+O\left(\frac{B}{\max \{|\mu |,|\lambda |\}}\right) \\ =\frac{\pi B^2}{\sqrt{{\lambda }^2+{\mu }^2}\cdot \sqrt{({\mu }_1\lambda {)}^2+({{\mu }_1}^{2}d+a_1{\lambda }^2{)}^2}}+O\left(\frac{B}{\max \{|\lambda |,|\mu |\}}\right). \end{gathered}$$

$$N(V\cap V_y,B)=\frac{\pi B^2}{2\zeta (2)}\cdot \frac{1}{\sqrt{({\lambda }^2+{\mu }^2)(({\mu }_1\lambda {)}^2+({{\mu }_1}^{2}d+a_1{\lambda }^2{)}^2)}}+O\left(\frac{B\cdot \log B}{\max \{|\lambda |,|\mu |\}}\right).$$

$$N(V\cap V_{(0:1)},B)=\frac{\pi B^2}{2\zeta (2)}+O(B\cdot \log B).$$

$$\begin{gathered} N(V,B)=N(V\cap V_{(0:1)},B) \\ +\frac{1}{2}\sum _{d\mid a}\left(\sum _{\begin{array}{c}(\mu ,\lambda )\in {\mathbb{Z}}_{\mathrm{prim}}^2\\ \mu \ne 0,d=\gcd (a,\mu )\\ {\mu }^2+{\lambda }^2\le B^2<{\mu }^2+{\lambda }^2+({\mu }_1\lambda {)}^2+({{\mu }_1}^{2}d+a_1{\lambda }^2{)}^2\end{array}}1+\sum _{\begin{array}{c}(\mu ,\lambda )\in {\mathbb{Z}}_{\mathrm{prim}}^2\\ \mu \ne 0,d=\gcd (a,\mu )\\ {\mu }^2+{\lambda }^2+({\mu }_1\lambda {)}^2+({{\mu }_1}^{2}d+a_1{\lambda }^2{)}^2\le B^2\end{array}}N(V\cap V_y,B)\right), \end{gathered}$$

$${\Sigma }_1=\sum _{\begin{array}{c}(\mu ,\lambda )\in {\mathbb{Z}}_{\mathrm{prim}}^2\\ \mu \ne 0,d=\gcd (a,\mu )\\ {\mu }^2+{\lambda }^2\le B^2\end{array}}1+O(B^{3/2}),$$

$${\Sigma }_2=\frac{\pi B^2}{2\zeta (2)}\sum _{\begin{array}{c}(\mu ,\lambda )\in {\mathbb{Z}}_{\mathrm{prim}}^2\\ \mu \ne 0,d=\gcd (a,\mu )\end{array}}\frac{1}{\sqrt{({\mu }^2+{\lambda }^2)(({\mu }_1\lambda {)}^2+({{\mu }_1}^{2}d+a_1{\lambda }^2{)}^2)}}+O(B^{3/2}\cdot (\log B{)}^2).$$

$$\sum _{\begin{array}{c}(\mu ,\lambda )\in {\mathbb{Z}}_{\mathrm{prim}}^2\\ \mu \ne 0\\ {\mu }^2+{\lambda }^2\le B^2\end{array}}1=\frac{\pi B^2}{\zeta (2)}+O(B\cdot \log B).$$

$$V_y=\{(t_0:t_1:t_2:t_3:t_4)\in {\mathbb{P}}^4\mid \lambda t_0-\mu t_1={\mu }^2{t}_2+{\lambda }^2{t}_3+\mu \lambda t_4=0\}\subset W,$$

$$N(V,B)=\sum _{y\in {\mathbb{P}}^1(\mathbb{Q})}N(V\cap V_y,B).$$

$$N(V\cap V_y,B)=\frac{1}{2}\cdot \#\{({\tau }_0,{\tau }_2,{\tau }_3)\in {\mathbb{Z}}_{\mathrm{prim}}^3\mid {\tau }_0\ne 0,H(\mu {\tau }_0,\lambda {\tau }_0,\lambda {\tau }_2,\mu {\tau }_3,\lambda {\tau }_3+\mu {\tau }_2)\le B\}.$$

$$|c|\le 1/2,{\tau }_0^2+(1-|c|)\cdot ({\tau }_2^2+{\tau }_3^2)\le {\tilde{B}}^2\text{mbox}and{\lambda }^2+{\mu }^2\le B^2.$$

$$N^{\ast }(V_y,B)=\#\{({\tau }_0,{\tau }_2,{\tau }_3)\in {\mathbb{Z}}^3\mid {\tau }_0\ne 0,H(\mu {\tau }_0,\lambda {\tau }_0,\lambda {\tau }_2,\mu {\tau }_3,\lambda {\tau }_3+\mu {\tau }_2)\le B\},$$

$$N(V\cap V_y,B)=\frac{1}{2}\sum _{k\ll \tilde{B}}\mu (k)\cdot N^{\ast }\left(V_y,\frac{B}{k}\right).$$

Full text

1 Introduction

In [2], a precise asymptotic formula for the number of rational points on the Cayley ruled cubic surface was established. Moreover, the leading term was expressed in terms of Tamagawa constants. The purpose of this note is to show that the asymptotic formula of [2] extends to all non-normal integral hypercubics which are not cones. The fibration method, which was one of the methods used in [2], goes through exactly as in [2].

In order to count rational points, we use the following height function:

[TABLE]

where $(t_{0},\ldots,t_{n})\in{\mathbb{Z}}^{n+1}_{\operatorname{prim}}$ and ${\mathbb{Z}}^{n+1}_{\operatorname{prim}}=\{(t_{0},\ldots,t_{n})\in{\mathbb{Z}}^{n+1}\setminus(0,\ldots,0)\;\mid\;\operatorname{gcd}(t_{0},\ldots,t_{n})=1\}$.

Given a geometrically integral projective variety $W\subset{\mathbb{P}}^{n}_{\mathbb{Q}}$, we let $W^{norm}\subset W$ denote the locus of geometrically normal points. The counting function for the rational points on $W^{norm}$, as a function of $B\in{\mathbb{R}}_{>0}$, is

[TABLE]

Our main result is the following. All asymptotic formulas are given with respect to $B\to+\infty$.

Theorem 1.1**.**

Let $a\in\mathbb{Z}\setminus\{0\}$ be square-free and $W\subset{\mathbb{P}}^{3}_{\mathbb{Q}}$ be given by the equation $t_{0}t_{1}t_{2}+t_{3}\cdot(t_{0}^{2}+a\cdot t_{1}^{2})=0$. Then

[TABLE]

where $f(\mu,\lambda)=({\lambda}^{2}+{\mu}^{2})({\lambda}^{2}{\mu}^{2}+({\mu}^{2}+a\cdot{\lambda}^{2})^{2})$. 2. 2.

Let $W\subset{\mathbb{P}}^{4}_{\mathbb{Q}}$ be given by the equation $t_{0}^{2}t_{2}+t_{1}^{2}t_{3}+t_{0}t_{1}t_{4}=0$. Then

[TABLE]

where $f(\mu,\lambda)=({\lambda}^{2}+{\mu}^{2})({\lambda}^{2}{\mu}^{2}+{\mu}^{4}+{\lambda}^{4})$.

Our notation and arguments follow closely [2].

2 Classification of non-normal hypercubics over the rationals

Non-normal cubic hypersurfaces over algebraically closed fields are classified in [3]. The same argument also gives the following classification over $\mathbb{Q}$.

Theorem 2.1**.**

(cf. [3], Theorem $3.1$) Let $W\subset{\mathbb{P}}^{n}_{\mathbb{Q}}$ be a geometrically integral and geometrically non-normal hypersurface given by a homogeneous cubic polynomial $F\in\mathbb{Q}[t_{0},\ldots,t_{n}]$. Then either $W$ is a cone or $F$ can be obtained by a linear coordinate change over $\mathbb{Q}$ from one of the following polynomials:

• •

$(t_{0}^{2}+a\cdot t_{1}^{2})t_{2}+t_{1}^{2}(b\cdot t_{0}+c\cdot t_{1})$, $a,b,c\in\mathbb{Z}$, $n=2$,

• •

$t_{0}t_{1}t_{2}+t_{0}^{3}+a\cdot t_{1}^{3}$, $a\in\mathbb{Z}$, $n=2$,

• •

$t_{0}^{2}t_{2}+t_{1}^{2}t_{3}$, $n=3$,

• •

$t_{0}t_{1}t_{2}+t_{3}(t_{0}^{2}+a\cdot t_{1}^{2})$, $a\in\mathbb{Z}\setminus\{0,1\}$ is square-free, $n=3$,

• •

$t_{0}t_{1}t_{2}+t_{3}t_{0}^{2}+t_{1}^{3}$, $n=3$,

• •

$t_{0}^{2}t_{2}+t_{0}t_{1}t_{3}+t_{1}^{2}t_{4}$, $n=4$.

Note that $t_{0}t_{1}t_{2}+t_{3}(t_{0}^{2}+t_{1}^{2})$ transforms to $4(t_{0}^{2}t_{3}+t_{1}^{2}t_{2})$ after substitution $t_{0}\mapsto t_{0}+t_{1}$, $t_{1}\mapsto t_{0}-t_{1}$, $t_{2}\mapsto 2(t_{3}-t_{2})$, $t_{3}\mapsto t_{2}+t_{3}$. Rational points on the cubic surfaces given by equations $t_{0}^{2}t_{2}+t_{1}^{2}t_{3}=0$ and $t_{0}t_{1}t_{2}+t_{3}t_{0}^{2}+t_{1}^{3}=0$ were counted in [2].

3 Geometry of the non-normal cubic threefold

In this section we consider the hypercubic $\overline{W}\subset{\mathbb{P}}_{\overline{\mathbb{Q}}}^{4}$ given by the equation $t_{0}^{2}t_{2}+t_{0}t_{1}t_{3}+t_{1}^{2}t_{4}=0$. The normalization $\nu\colon X\to\overline{W}$ is the projection of the Segre cubic threefold scroll $X={\mathbb{P}}_{\overline{\mathbb{Q}}}^{1}\times{\mathbb{P}}_{\overline{\mathbb{Q}}}^{2}\subset{\mathbb{P}}_{\overline{\mathbb{Q}}}^{5}$ from a point $P\in{\mathbb{P}}_{\overline{\mathbb{Q}}}^{5}\setminus X$. [3]

Theorem 3.1**.**

The automorphism group of $\overline{W}$ fits into the short exact sequence of groups

[TABLE]

where $K={\overline{\mathbb{Q}}}^{*}\oplus\overline{\mathbb{Q}}\oplus\overline{\mathbb{Q}}$ with the product $(a,b,c)\cdot(a^{\prime},b^{\prime},c^{\prime})=(aa^{\prime},a^{\prime}b+b^{\prime},a^{\prime}c+c^{\prime})$.

Proof.

By the Lefschetz theorem, $\operatorname{Pic}(\overline{W})=\mathbb{Z}\mathcal{O}(1)$. Hence any automorphism of $\overline{W}$ is induced by an automorphism of ${\mathbb{P}}_{\overline{\mathbb{Q}}}^{4}$, i.e. by $A\in\operatorname{PGL}(5)$ which preserves (up to a scalar multiple) $t_{0}^{2}t_{2}+t_{0}t_{1}t_{3}+t_{1}^{2}t_{4}$. Any such $A$ also preserves the non-normal locus $\{t_{0}=t_{1}=0\}$ and so induces an automorphism of ${\mathbb{P}}_{\overline{\mathbb{Q}}}(\overline{\mathbb{Q}}t_{0}\oplus\overline{\mathbb{Q}}t_{1})\cong{\mathbb{P}}_{\overline{\mathbb{Q}}}^{1}$. The resulting map $\operatorname{Aut}(\overline{W})\rightarrow\operatorname{PGL}(2)$ is surjective.

Explicitly, suppose $A(t_{0})=at_{0}+t_{1}$, $A(t_{1})=ct_{0}+dt_{1}$, $ad-c\neq 0$. Then

[TABLE]

where $u_{4}\neq 0$ and $a_{31},a_{41}$ are arbitrary. If $A(t_{0})=t_{0}$, $A(t_{1})=ct_{0}+dt_{1}$, $d\neq 0$, then

[TABLE]

where $w_{4}\neq 0$ and $a_{30},a_{40}$ are arbitrary. ∎

The following Corollary was inspired by the arguments in [2].

Corollary 3.2**.**

$\overline{W}$* is not toric.*

Proof.

(cf. [2]) The maximal torus in $\operatorname{Aut}(\overline{W})$ has dimension $2<\operatorname{dim}(\overline{W})$.∎

4 Rational points on $t_{0}t_{1}t_{2}+t_{3}\cdot(t_{0}^{2}+a\cdot t_{1}^{2})=0$

In this section we prove Theorem 1.1, part $1$. The argument and notation follow [2] closely. Let $W\subset{\mathbb{P}}^{3}_{\mathbb{Q}}$ be given by the equation $t_{0}t_{1}t_{2}+t_{3}\cdot(t_{0}^{2}+a\cdot t_{1}^{2})=0$, where $a\in\mathbb{Z}\setminus\{0\}$ is square-free. Let $V=W\setminus\{t_{0}=t_{1}=0\}$ and

[TABLE]

where $y=(\mu:\lambda)\in{\mathbb{P}}^{1}$.

Then $V=\coprod\limits_{y\in{\mathbb{P}}^{1}}V\cap V_{y}$, and so

[TABLE]

Claim. (cf. [2], Lemma $3.1$) If $\mu\neq 0$, $(\mu,\lambda)\in{\mathbb{Z}}^{2}_{\operatorname{prim}}$, $y=(\mu:\lambda)\in{\mathbb{P}}^{1}(\mathbb{Q})$, then

[TABLE]

where $d=\operatorname{gcd}(a,\mu)\geq 1$, $\mu=\mu_{1}d$, $a=a_{1}d$.

[TABLE]

Proof.

[2] Let $\mu\lambda\neq 0$ and take $(t_{0},t_{1},t_{2},t_{3})\in{\mathbb{Z}}^{4}_{\operatorname{prim}}\cap V_{y}$. Then $t_{0}=\mu\tau_{0}$, $t_{1}=\lambda\tau_{0}$, $t_{3}=(\mu_{1}\lambda)\cdot\tau_{3}$, $t_{2}=-({\mu_{1}}^{2}d+a_{1}{\lambda}^{2})\cdot\tau_{3}$. ∎

Note that $N(V\cap V_{y},B)=0$ unless ${\lambda}^{2}+{\mu}^{2}\leq B^{2}$. Moreover, if $\mu\neq 0$ and

[TABLE]

then $N(V\cap V_{y},B)=1$. In particular, if $N(V\cap V_{y},B)>1$, then $\lvert\mu\lambda\rvert\ll B$ and so $\operatorname{min}\{\lvert\mu\rvert,\lvert\lambda\rvert\}\ll\sqrt{B}$. Also, for $(\tau_{0},\tau_{3})\in{\mathbb{Z}}^{2}_{\operatorname{prim}}$ contributing to $N(V\cap V_{y},B)$,

[TABLE]

Suppose $\mu\neq 0$. If

[TABLE]

then

[TABLE]

By Euler’s summation formula,

[TABLE]

Hence

[TABLE]

A similar calculation gives

[TABLE]

In the expression

[TABLE]

denote the first and the second sums over $(\mu,\lambda)$ by $\Sigma_{1}$ and $\Sigma_{2}$ respectively. Then

[TABLE]

and

[TABLE]

The same calculation as above gives

[TABLE]

All together this gives the asymptotic formula in Theorem 1.1, part $1$.

5 Rational points on $t_{0}^{2}t_{2}+t_{0}t_{1}t_{3}+t_{1}^{2}t_{4}=0$

In this section we prove Theorem 1.1, part $2$. The argument and notation follow [2] closely. Let $W\subset{\mathbb{P}}^{4}_{\mathbb{Q}}$ be given by the equation $t_{0}^{2}t_{2}+t_{1}^{2}t_{3}+t_{0}t_{1}t_{4}=0$. Let $V=W\setminus\{t_{0}=t_{1}=0\}$ and

[TABLE]

where $y=(\mu:\lambda)\in{\mathbb{P}}^{1}$.

Then $V=\coprod\limits_{y\in{\mathbb{P}}^{1}}V\cap V_{y}$, and so

[TABLE]

Claim. (cf. [2], Lemma $3.1$) Assume $(\mu,\lambda)\in{\mathbb{Z}}^{2}_{\operatorname{prim}}$, $y=(\mu:\lambda)\in{\mathbb{P}}^{1}(\mathbb{Q})$. Then

[TABLE]

Proof.

[2] Let $\mu\lambda\neq 0$ and take $(t_{0},t_{1},t_{2},t_{3},t_{4})\in{\mathbb{Z}}^{5}_{\operatorname{prim}}\cap V_{y}$. Then $t_{0}=\mu\tau_{0}$, $t_{1}=\lambda\tau_{0}$, $t_{2}=\lambda\tau_{2}$, $t_{3}=\mu\tau_{3}$, $t_{4}=-\mu\tau_{2}-\lambda\tau_{3}$. ∎

Let $\tilde{B}=B/\sqrt{{\lambda}^{2}+{\mu}^{2}}$, $c=\lambda\mu/({\lambda}^{2}+{\mu}^{2})$. Then in the calculation we may assume that

[TABLE]

If

[TABLE]

then

[TABLE]

By Euler’s summation formula,

[TABLE]

Hence

[TABLE]

After summing over $y\in{\mathbb{P}}^{1}(\mathbb{Q})$, this gives the result.

6 Tamagawa numbers

In this section we express, following [2], the leading term of the asymptotic formula in Theorem 1.1, part $2$, via Tamagawa numbers [4], [1].

Let $W\subset{\mathbb{P}}_{\mathbb{Q}}^{4}$ be defined by the equation $t_{0}^{2}t_{2}+t_{0}t_{1}t_{3}+t_{1}^{2}t_{4}=0$, $V=W\setminus\{t_{0}=t_{1}=0\}$ and $\nu\colon X\to W$ be the normalization. Explicitly, we take $X=\{\;\operatorname{rank}\begin{pmatrix}t_{0}&t_{5}&-t_{4}\\ t_{1}&t_{2}&(t_{3}+t_{5})\end{pmatrix}\leq 1\}\subset{\mathbb{P}}^{5}$ and $P=(0:0:0:0:0:1)$. Then $\nu$ is the projection from $P$ into ${\mathbb{P}}^{4}=\{t_{5}=0\}$.

We use terminology and notation from [1]. Let $\mathcal{L}=\mathcal{O}(1)$ be the chosen ample invertible sheaf on $V$ metrized as in [2]. The following Lemma is proven exactly as in [2].

Lemma 6.1**.**

(cf. [2]) $X$ is the $\mathcal{L}$-closure of $V$. $V$ is weakly $\mathcal{L}$-saturated, not $\mathcal{L}$-primitive and contains no strongly $\mathcal{L}$-saturated Zariski open dense subvariety. The fibration $V\to{\mathbb{P}}^{1}$, which was used to count rational points on $V$, extends to an $\mathcal{L}$-primitive fibration $X={\mathbb{P}^{1}}\times{\mathbb{P}^{2}}\to{\mathbb{P}}^{1}$, which is the projection onto the first factor. In particular, $V\cap V_{y}$ is $\mathcal{L}$-primitive and ${\alpha}_{\mathcal{L}}(V\cap V_{y})={\alpha}_{\mathcal{L}}(V)=3$. Moreover,

[TABLE]

The Tamagawa number ${\tau}_{\mathcal{L}}(V\cap V_{y})$, defined in [1], coincides with the Tamagawa number ${\tau}_{H}(\mathbb{P}^{2})$ defined in [4] with respect to the adelic metric on ${\omega}_{\mathbb{P}^{2}}^{-1}$ chosen as in [2]. The projection $\mathbb{P}^{2}\to\mathbb{P}^{4}$ corresponding to the point $y=(\mu:\lambda)\in\mathbb{P}^{1}$ is given by $({\tau}_{0}:{\tau}_{1}:{\tau}_{2})\mapsto(\mu{\tau}_{0}:\lambda{\tau}_{0}:\lambda{\tau}_{1}:(-\mu{\tau}_{1}-\lambda{\tau}_{2}):\mu{\tau}_{2})$. Following [4], Lemma $2.1.2$, one computes

[TABLE]

Hence

[TABLE]

Thus, the leading coefficient of the asymptotic formula in Theorem 1.1, part $2$, confirms the prediction of Batyrev and Tschinkel [1] in this case, up to a factor of $1/3$ coming from ${\gamma}_{\mathcal{L}}(V\cap V_{y})$.

Remark 6.2**.**

The discrepancy with the conjectural form of the leading term in [1] will be resolved if one redefines the constant ${\gamma}_{\mathcal{L}}(V)$ in general as follows:

[TABLE]

See [1], Definition $2.3.16$. Such a modification is justified by the observation that the constant $c_{{\mathcal{L}}^{k}}(V)$, if defined as in [1], section $3.4$, grows linearly with $k$. After this modification, $c_{{\mathcal{L}}^{k}}(V)$ grows as $k^{1-{\beta}_{\mathcal{L}}(V)}$, as needed for the compatibility of the conjecture in [1] with the equality $N(V,{\mathcal{L}}^{k},B)=N(V,\mathcal{L},B^{1/k})$.

Acknowledgement

The author is grateful to Beijing International Center for Mathematical Research, the Simons Foundation and Peking University for support, excellent working conditions and encouraging atmosphere.