# Algebraic Gromov’s ellipticity of cubic hypersurfaces

Shulim Kaliman and Mikhail Zaidenberg  University of Miami, Department of
Mathematics, Coral Gables, FL 33124, USA<EMAIL_ADDRESS>Univ.
Grenoble Alpes, CNRS, IF, 38000 Grenoble, France  mikhail.zaidenberg@univ-
grenoble-alpes.fr

###### Abstract.

We show that every smooth cubic hypersurface $X$ in ${\mathbb{P}}^{n+1}$,
$n\geq 2$ is algebraically elliptic in Gromov’s sense. This gives the first
examples of non-rational projective manifolds elliptic in Gromov’s sense. We
also deduce that the punctured affine cone over $X$ is elliptic.

###### Key words and phrases:

spray, Gromov’s ellipticity, unirationality, stable rationality, cubic
threefold, cubic hypersurface, affine cone.

2020 _Mathematics Subject Classification._ Primary 14J30, 14J70, 14M20;
Secondary 14N25, 32Q56.

###### Contents

  1. 1 Introduction
  2. 2 Criteria of ellipticity
  3. 3 Ellipticity of cubic hypersurfaces

##  1\. Introduction

Gromov’s ellipticity appeared (and was extremely useful) in complex analysis
within the Oka-Grauert theory, see [Gro89], [For17a] and [For23]. In the same
paper [Gro89] Gromov developed an algebraic version of this notion. We deal
below exceptionally with algebraic ellipticity. Thus, all varieties and vector
bundles in this paper are algebraic and “ellipticity” means the algebraic
ellipticity.

Let ${\mathbb{K}}$ be an algebraically closed field of characteristic zero and
${\mathbb{A}}^{n}$ resp. ${\mathbb{P}}^{n}$ be the affine resp. projective
$n$-space over ${\mathbb{K}}$.

Given a smooth algebraic variety $X$, a _spray of rank $r$_ over $X$ is a
triple $(E,p,s)$ where $p\colon E\to X$ is a vector bundle of rank $r$ with
zero section $Z$ and $s\colon E\to X$ is a morphism such that $s|_{Z}=p|_{Z}$.
A spray $(E,p,s)$ is _dominating at $x\in X$_ if the restriction
$s|_{E_{x}}\colon E_{x}\to X$ to the fiber $E_{x}=p^{-1}(x)$ is dominant at
the origin $0_{x}\in E_{x}\cap Z$ of the vector space $E_{x}$ i.e.
$ds(T_{0}E_{x})=T_{x}X$.

One says that the variety $X$ is _elliptic_ if it admits a spray $(E,p,s)$
which is dominating at each point $x\in X$, see [Gro89, 3.5A]. It is immediate
from the definition that every elliptic manifold is unirational. An algebraic
variety $X$ is said to be _uniformly rational_ (or _regular_ in the
terminology of [Gro89, 3.5D]) if every point $x\in X$ has a neighborhood
isomorphic to an open subset of ${\mathbb{A}}^{n}$. It is called _stably
uniformly rational_ if $X\times{\mathbb{A}}^{k}$ is uniformly rational for
some $k\geq 0$. A complete stably uniformly rational variety is elliptic, see
[AKZ23, Theorem 1.3 and Corollary 3.7].

It is shown in [BB14, Example 2.4] that a rational smooth cubic hypersurface
$X$ in ${\mathbb{P}}^{n+1}$, $n\geq 2$, is uniformly rational (cf. also
[Gro89, Remark 3.5E′′′]). Hence $X$ is elliptic. There are examples of such
hypersurfaces $X$ of any even dimension $n=2k\geq 2$. Another class of
uniformly rational (and hence elliptic) projective varieties is provided by
smooth complete intersections of two quadrics in ${\mathbb{P}}^{n+2}$, $n\geq
3$, see [BB14, Example 2.5]; cf. also [AKZ23, Examples 4.8].

Gromov asked in [Gro89, 3.5B′′] whether any rational smooth projective variety
is elliptic; the answer is still unknown. Gromov also discussed in [Gro89,
3.5B′′] a conjectural equivalence between ellipticity and unirationality. In
the opposite direction, the question arises whether the ellipticity implies
the (stable, uniform) rationality.

The answer to the latter question is negative. Indeed, for a certain natural
number $n$ there exists a finite subgroup $F$ of
$\operatorname{SL}(n,{\mathbb{C}})$ such that the quotient
$Y=\operatorname{SL}(n,{\mathbb{C}})/F$ is stably irrational, see e.g. [Sal84,
Theorem 3.6], [Bog87], [Pop11, Example 1.22] and [Pop13]. Being an affine
homogeneous manifold of a semisimple group, $Y$ is flexible, that is, the
subgroup $\operatorname{SAut}(Y)\subseteq\operatorname{Aut}(Y)$ generated by
all one-parameter unipotent subgroups of $\operatorname{Aut}(Y)$ acts highly
transitively on $Y$, see [AFKKZ13, Proposition 5.4]. The latter implies that
$Y$ is elliptic, see e.g. [AFKKZ13, Proposition A.3]. Thus, $Y$ is an example
of a stably irrational elliptic affine manifold.

Our aim is to find an irrational smooth projective variety $X$ which is
Gromov’s elliptic. It is not clear if one could explore to this end the above
example. Indeed, given a smooth elliptic affine variety $Y$, we are lacking
the construction of a smooth elliptic completion $X$ of $Y$.

Nevertherless, we will establish the ellipticity of every smooth cubic
threefold $X\subseteq{\mathbb{P}}^{4}$, see Theorem 1.1. By the celebrated
Clemens-Griffiths theorem [CG72] such a threefold defined over ${\mathbb{C}}$
is irrational. According to Murre [Mur73] the same holds for every smooth
cubic threefold defined over an algebraically closed field of characteristic
different from 2.

Our main result is the following theorem.

######  Theorem 1.1.

A smooth cubic hypersurface $X\subseteq{\mathbb{P}}^{n+1}$ of dimension $n\geq
2$ is elliptic in the sense of Gromov.

######  Remarks 1.2.

1\. It is not known whether there exist a stably (uniformly) rational smooth
cubic threefold, see e.g. [CT19]. Also, no example of an irrational smooth
cubic hypersurface of dimension $\geq 4$ is known. Recall that every smooth
cubic hypersurface of dimension $\geq 2$ is unirational, see [Kol02] or
[Huy23, Remark 5.13].

2\. A smooth cubic threefold $X$ is far from being homogeneous. Indeed, due to
Matsumura–Monsky’s theorem (see [MM63]) $\operatorname{Aut}(X)$ is a finite
group.

3\. Notice that any rational smooth projective surface $X$ has a covering by
open subsets isomorphic to ${\mathbb{A}}^{2}$. Hence it is uniformly rational
and also elliptic. In the sequel we deal with cubic hypersurfaces of dimension
$\geq 3$.

Let us recall the following open question, cf. [Gro89, Remark 3.5E′′′]:

Question. _Does the Gromov ellipticity of a smooth complete variety survive
the birational maps generated by blowups and contractions with smooth
centers?_

There are some partial results concerning this question. The uniform
rationality is preserved under blowups with smooth centers, see [BB14,
Proposition 2.6] (cf. [Gro89, 3.5E-E′′]). Hence, also the ellipticity of a
complete uniformly rational variety is preserved, see [AKZ23, Corollary 1.5].
For not necessary complete or uniformly rational elliptic varieties, the
ellipticity is preserved under blowups with smooth centers under some
additional assumptions, see [Gro89, Corollary 3.5.D′′], [LT17, Corollary 2],
[KKT18] and [KZ23b, Theorem 0.1]. The preservation of ellipticity under
blowdowns with smooth centers is unknown.

Forstnerič [For17b] proved that every elliptic projective manifold $X$ of
dimension $n$ admits a surjective morphism from ${\mathbb{C}}^{n}$, whose
restriction to an open subset of ${\mathbb{C}}^{n}$ is smooth and also
surjective. Due to Kusakabe [Kus22] this remains true for not necessary
complete elliptic algebraic manifold $X$ over ${\mathbb{K}}$, provided one
replaces ${\mathbb{C}}^{n}$ by ${\mathbb{A}}^{n+1}_{{\mathbb{K}}}$. Exploring
these facts and also [KZ23b, Corollary 3.8 and Proposition 6.1] and Theorem
1.1, we deduce the following immediate corollary.

######  Corollary 1.3.

Let $X\subseteq{\mathbb{P}}^{n+1}$, $n\geq 2$ be a smooth cubic hypersurface
and $Y$ be a punctured generalized affine cone over $X$ defined by an ample
polarization of $X$. Then the following hold.

  * •

$Y$ is elliptic in Gromov’s sense.

  * •

There exist surjective morphisms ${\mathbb{A}}^{n+1}\to X$ resp.
${\mathbb{A}}^{n+2}\to Y$ which are smooth and surjective on appropriate open
subsets $U\subseteq{\mathbb{A}}^{n+1}$ resp. $V\subseteq{\mathbb{A}}^{n+2}$.

  * •

If ${\mathbb{K}}={\mathbb{C}}$, then there exists a surjective morphism
${\mathbb{A}}^{n}\to X$ which is smooth and surjective on an appropriate open
subset $U\subseteq{\mathbb{A}}^{n}$.

  * •

The monoid of endomorphisms $\operatorname{End}(Y)$ acts $m$-transitively on
$Y$ for any natural number $m$.

Recall that a _generalized affine cone_ $\hat{Y}$ over a smooth projective
variety $X$ defined by an ample ${\mathbb{Q}}$-divisor $D$ on $X$ is the
affine variety

$\hat{Y}=\operatorname{Spec}\left(\bigoplus_{k=0}^{\infty}H^{0}\left(X,{\mathcal{O}}_{X}(\lfloor
kD\rfloor)\right)\right),$

see, e.g., [KPZ13, Sec. 1.15]. In the case where $D$ is a hyperplane section
of $X\subseteq{\mathbb{P}}^{n}$ the cone $\hat{Y}$ is the usual affine cone
over $X$. The corresponding punctured cone $Y$ over $X$ is obtained from
$\hat{Y}$ by removing the vertex of $\hat{Y}$; so $Y$ is a smooth quasiaffine
variety.

##  2\. Criteria of ellipticity

For the proofs of the following proposition see [Gro89, 3.5B], [For17a,
Propositions 6.4.1 and 6.4.2], [LT17, Remark 3] and [KZ23b, Appendix B].

######  Proposition 2.1 (Gromov’s extension lemma).

Let $X$ be a smooth complete variety and $(E,p,s)$ be a spray on an open
subset $U\subseteq X$ with values in $X$, that is, $p\colon E\to U$ is a
vector bundle on $U$ with zero section $Z$ and $s\colon E\to X$ is a morphism
such that $s|_{Z}=p|_{Z}$. Then for each $x\in U$ there is a smaller
neighborhood $V\subseteq U$ of $x$ and a spray
$(E^{\prime},p^{\prime},s^{\prime})$ on $X$ such that
$(E^{\prime},p^{\prime},s^{\prime})|_{V}=(E,p,s)|_{V}$.

Recall that a variety $X$ is said to be _locally elliptic_ if for any $x\in X$
there is a local spray $(E_{x},p_{x},s_{x})$ defined in a neighborhood $U$ of
$x$ and dominating at $x$ such that $s_{x}\colon E\to X$ takes values in $X$.
The variety $X$ is called _subelliptic_ if there is a family of sprays
$\\{(E_{i},p_{i},s_{i})\\}_{i\in I}$ on $X$ which is dominating at each point
$x\in X$, that is,

$T_{x}X={\rm span}({\rm d}s_{i}(T_{0_{i,x}}E_{i,x})\,|\,i\in I)\quad\forall
x\in X.$

The following corollary on the equivalence of local and global ellipticity is
indicated in [Gro89, 3.5B′]; see also [KZ23a, Theorem 1.1 and Corollary 2.4].

######  Corollary 2.2 (Gromov’s Localization Lemma).

The ellipticity of a smooth variety is equivalent to its local ellipticity.

###### Sketch of the proof.

Choose a finite covering of $X$ equipped with sprays with values in $X$
dominating on the elements of this covering. We may suppose that the vector
bundles of these sprays are trivial, and so can be decomposed in a direct sum
of linear bundles. By Proposition 2.1 one can extend the resulting rank 1
sprays to sprays defined on the whole $X$, and then compose them in order to
get a spray dominating on $X$. ∎

Likewise, we have the following fact.

######  Proposition 2.3 ([KZ23a, Theorem 1.1]).

The ellipticity of a smooth variety is equivalent to its subellipticity.

Given a spray $(E,p,s)$ with values in $X$ and a point $x\in X$, the
constructive subset $O_{x}:=s(E_{x})\subseteq X$ is called the _$s$ -orbit_ of
$x$. The proof of Proposition 2.3 leads to the following lemma, cf. [KZ23a].

######  Lemma 2.4.

A smooth variety $X$ of dimension $n$ is elliptic if for every $x\in X$ there
exist local rank 1 sprays $(E_{i},p_{i},s_{i})$, $i=1,\ldots,n$ with values in
$X$ defined on respective neighborhoods $U_{i}$ of $x$ such that the
$s_{i}$-orbits $O_{i,x}$ of $x$ are curves with local parameterizations
$s\colon(E_{i,x},0)\to(O_{i,x},x)$ étale at $x$ whose tangent vectors at $x$
span $T_{x}X$.

###### Sketch of the proof.

Shrinking $U_{i}$ we may suppose that $(E_{i},p_{i},s_{i})$ is defined on the
whole $X$ for $i=1,\ldots,n$, see Proposition 2.1. Since the tangent lines
$ds(T_{0}E_{i,x})$ at $x$ of the $s_{i}$-orbits $O_{i,x}$, $i=1,\ldots,n$ span
$T_{x}X$, the composition of these extended sprays is a spray of rank $n$ on
$X$ dominating at $x$. This domination spreads to a neighborhood of $x$ in
$X$. It follows that $X$ is locally elliptic (and also subelliptic). By
Gromov’s Localization Lemma, see Corollary 2.2 (or alternatively by
Proposition 2.3) $X$ is elliptic. ∎

In our proof of Theorem 1.1 in the next section we construct on any smooth
cubic hypersurface $X$ of dimension $n\geq 3$ such an $n$-tuple of independent
rank 1 sprays $(E_{i},p_{i},s_{i})$ dominating in the orbit directions.

##  3\. Ellipticity of cubic hypersurfaces

We use the following notation.

######  Notation 3.1.

For a subset $M\subseteq{\mathbb{P}}^{n}$ we let $\langle M\rangle$ be the
smallest projective subspace which contains $M$. Given a smooth hypersurface
$X\subseteq{\mathbb{P}}^{n+1}$ and a point $x\in X$ we let
$S_{x}=\mathbb{T}_{x}X\cap X$, where $\mathbb{T}_{x}X$ stands for the
projective tangent space of $X$ at $x$. Let ${\mathcal{C}}_{x}$ stand for the
union of lines on $X$ through the point $x\in X$. Clearly,
${\mathcal{C}}_{x}\subseteq S_{x}$. Recall that for $n\geq 3$ every smooth
cubic hypersurface $X\subseteq{\mathbb{P}}^{n+1}$ is covered by projective
lines (this follows, for instance, from [CG72, Corollary 8.2]). Therefore,
${\mathcal{C}}_{x}$ has positive dimension for every $x\in X$.

Given a cubic threefold $X\subseteq{\mathbb{P}}^{4}$, the equality
${\mathcal{C}}_{x}=S_{x}$ holds if and only if $x$ is an Eckardt point of $X$.
Recall that a point $x\in X$ is called an _Eckardt point_ if there is an
infinite number of lines on $X$ passing through $x$. In the latter case
${\mathcal{C}}_{x}$ is the cone with vertex $x$ over a plane elliptic cubic
curve. Notice that a general cubic threefold has no Eckardt points, a cubic
threefold can contain at most 30 Eckardt points (see [CG72, p. 315]), and the
maximal number of 30 Eckardt points is attained only for the Fermat cubic
threefold, see [CG72, p. 315], [Rou09] and [Huy23, Chapter 5, Remark 1.7].

######  Lemma 3.2.

Given a smooth cubic hypersurface $X\subseteq{\mathbb{P}}^{n+1}$ where $n\geq
3$, the tangent space $T_{x}X$ at a general point $x\in X$ is spanned by some
$n$ tangent lines to projective lines on $X$ passing through $x$.

###### Proof.

Let us start with the case $n=3$, that is, let $X$ be a smooth cubic
threefold. Through a general point $x\in X$ pass exactly 6 lines, see e.g.
[AK77, Proposition (1.7)] or [Huy23, Chapter 5, Exercise 1.4]. More precisely,
the lines of the second type on $X$ sweep a proper closed subset, say
$Y\subseteq X$, and through every point $x\in X\setminus Y$ pass exactly 6
distinct lines on $X$, see [Mur72, Lemma (1.19)]. These 6 lines form the cone
${\mathcal{C}}_{x}$. No four of them are coplanar, hence the tangent space
$T_{x}X$ is spanned by the tangent lines to some three of these projective
lines through $x$.

Let us show now that the latter property holds as well in higher dimensions.
Given a smooth cubic hypersurface $X\subseteq{\mathbb{P}}^{n+1}$, $n\geq 4$
and a point $x\in X$, consider the projective variety

${\mathbb{P}}{\mathcal{C}}_{x}:=\\{T_{x}l\,|\,x\in
l\subseteq{\mathcal{C}}_{x}\\}\subseteq{\mathbb{P}}T_{x}X$

where $l$ stands for a line on $X$. Clearly, $T_{x}X$ is spanned by the
tangent lines $T_{x}l$ for $l\subseteq{\mathcal{C}}_{x}$ if and only if
${\mathbb{P}}{\mathcal{C}}_{x}$ is linearly nondegenerate in
${\mathbb{P}}T_{x}X\cong{\mathbb{P}}^{n-1}$. There exists an open dense subset
$U\subseteq X$ such that for every $x\in U$ the projective cone
${\mathcal{C}}_{x}$ has codimension 2 in $\mathbb{T}_{x}X$. The dimension
$\dim\,\langle{\mathbb{P}}{\mathcal{C}}_{x}\rangle$ is lower semicontinuous on
$U$ and attains its maximal value, say $m$ on an open dense subset
$U_{0}\subseteq U$. Let $\mathcal{V}\subseteq{\mathbb{P}}TX|_{U_{0}}$ be the
subvariety swept out by the $\langle{\mathbb{P}}{\mathcal{C}}_{x}\rangle$ for
$x\in U_{0}$.

Suppose that $m<n-1$ (contrary to the assertion of the lemma), so that
$\mathcal{V}$ is a proper closed subset of ${\mathbb{P}}TX|_{U_{0}}$. Choose a
general linear section $Y$ of $X$ by a $4$-plane
${\mathbb{P}}^{4}\subseteq{\mathbb{P}}^{n+1}$. Then $Y$ is a smooth cubic
threefold. Furthermore, for a general $x\in Y\cap U_{0}$ the proper subspaces
${\mathbb{P}}T_{x}Y\cong{\mathbb{P}}^{2}$ and
$\langle{\mathbb{P}}{\mathcal{C}}_{x}\rangle\cong{\mathbb{P}}^{m}$ of
${\mathbb{P}}T_{x}X\cong{\mathbb{P}}^{n-1}$ are transversal. On the other
hand, $T_{x}Y$ is spanned by the tangent lines to the lines on $Y$ passing
through $x$. Thus, we have
${\mathbb{P}}T_{x}Y\subseteq\langle{\mathbb{P}}{\mathcal{C}}_{x}\rangle$. This
contradiction ends the proof. ∎

######  Notation 3.3.

For a general point $u\in X$ we let $S^{*}_{u}$ be the set of points $x\in X$
such that $u\in\mathbb{T}_{x}X$. Let $x\in S^{*}_{u}$ be a general point and
$u^{*}=\mathbb{T}_{u}X$ resp. $x^{*}=\mathbb{T}_{x}X$ be the corresponding
points of the dual hypersurface $X^{*}\subseteq({\mathbb{P}}^{n+1})^{\vee}$.
Then $u\in\mathbb{T}_{x}X$ if and only if $x^{*}\in\mathbb{T}_{u^{*}}X^{*}$,
that is, $x\in S^{*}_{u}$ if and only if $x^{*}\in S_{u^{*}}$. It follows that
$S^{*}_{u}$ is a hypersurface in $X$ passing through $u$. It is easily seen
that ${\mathcal{C}}_{u}\subseteq S_{u}^{*}$, and so
${\mathcal{C}}_{u}\subseteq S_{u}\cap S_{u}^{*}$.

######  Lemma 3.4.

For a smooth cubic hypersurface $X\subseteq{\mathbb{P}}^{n+1}$, $n\geq 1$ one
has $S_{u}\cap S_{u}^{*}={\mathcal{C}}_{u}$ for every $u\in X$.

###### Proof.

If $x\in S_{u}\cap S_{u}^{*}$ is different from $u$ then $l:=\langle
x,u\rangle\subseteq\mathbb{T}_{x}X\cap\mathbb{T}_{u}X$. By the Bézout theorem,
the bitangent line $l$ of $X$ is contained in $X$. It follows that
$x\in{\mathcal{C}}_{u}$. Hence we have $S_{u}\cap
S_{u}^{*}\subseteq{\mathcal{C}}_{u}$. Since also ${\mathcal{C}}_{u}\subseteq
S_{u}\cap S_{u}^{*}$ one has $S_{u}\cap S_{u}^{*}={\mathcal{C}}_{u}$. ∎

Let again $X$ be a smooth cubic hypersurface in ${\mathbb{P}}^{n+1}$, $n\geq
1$. Following [BB14, Example 2.4] and [BKK13] let us consider the birational
self-map $\tau_{u}\colon X\dasharrow X$ which sends a general point $x\in X$
to the third point $y=\tau(x)$ of intersection of the line $\langle
x,u\rangle$ with $X$. Thus, $X$ cuts out on the line $\langle x,u\rangle$ the
divisor $x+u+y$. In fact, $\tau_{u}$ is well defined unless $x=u$ or the line
$\langle x,u\rangle$ is contained in $X$. Thus, $\tau_{u}\colon X\dasharrow X$
is regular on $X\setminus{\mathcal{C}}_{u}$.

######  Lemma 3.5.

Let $X\subseteq{\mathbb{P}}^{n+1}$, $n\geq 1$ be a smooth cubic hypersurface
and $x,u\in X$ be such that $x\in X\setminus{\mathcal{C}}_{u}$. Then
$\tau_{u}(x)=\tau_{u}(x^{\prime})$ for a point $x^{\prime}\in X$ if and only
if either $u\in\mathbb{T}_{x}X$ (i.e. $x\in S_{u}^{*}$) and $x^{\prime}=x$, or
$x\in\mathbb{T}_{u}X$ (i.e. $x\in S_{u}$) and $x^{\prime}=u$.

###### Proof.

Notice that by our assumption $x\neq u$ and $\langle x,u\rangle\not\subseteq
X$. Suppose that $\tau_{u}(x)=\tau_{u}(x^{\prime})$ holds. Then we have
$\langle x,u\rangle=\langle x^{\prime},u\rangle$. So the points
$x,x^{\prime},u$ are alined. If $x^{\prime}\notin\\{x,u\\}$, then
$\tau_{u}(x)=x^{\prime}$ and $\tau_{u}(x^{\prime})=x$. Applying $\tau_{u}$
once again we obtain $x^{\prime}=\tau_{u}(x)=\tau_{u}(x^{\prime})=x$, a
contradiction. Thus, $x^{\prime}\in\\{x,u\\}$, and so the divisor cut out by
$X$ on the line $\langle x,u\rangle$ is either $2x+u$, or $2u+x$. We have
$x^{\prime}=x$ in the former case and $x^{\prime}=u$ in the latter case. Now
the assertion follows. ∎

The following corollary is immediate.

######  Corollary 3.6.

  * (a)

The morphism $\tau_{u}\colon X\setminus{\mathcal{C}}_{u}\to X$ contracts the
hyperplane section $S_{u}\setminus{\mathcal{C}}_{u}$ to the point $u$ and
fixes pointwise the variety $S_{u}^{*}\setminus
S_{u}=S_{u}^{*}\setminus{\mathcal{C}}_{u}$ (see Lemma 3.4).

  * (b)

The indeterminacy points of $\tau_{u}$ are contained in ${\mathcal{C}}_{u}$.

  * (c)

The affine threefold $X\setminus S_{u}\subseteq X\setminus{\mathcal{C}}_{u}$
is invariant under $\tau_{u}$ and $\tau_{u}|_{X\setminus S_{u}}$ is a
biregular involution with mirror $S_{u}^{*}\setminus{\mathcal{C}}_{u}$ which
acts freely on $X\setminus(S_{u}\cup S_{u}^{*})$.

In the proof of Theorem 1.1 we use the following construction of a rank 1
spray on a smooth cubic hypersurface.

######  Lemma 3.7.

Let $X\subseteq{\mathbb{P}}^{n+1}$, $n\geq 3$ be a smooth cubic hypersurface,
$y\in X$ be an arbitrary point and $x\in X$ be a general point. Let $u\in
X\cap\langle x,y\rangle$ be different from $x$ and $y$. Choose a line
$l\subseteq{\mathcal{C}}_{x}$ on $X$ through $x$ and let
$C^{*}=\tau_{u}(l\setminus S_{u})$. Then $C^{*}\cong{\mathbb{A}}^{1}$.
Furthermore, there exists a rank $1$ spray $(E,p,s)$ on $X$ such that the
$s$-orbit $O_{y}$ of $y$ coincides with $C^{*}$ and $ds(T_{0}E_{y})$ is the
tangent line to $C^{*}=O_{y}$ at $y$.

###### Proof.

Since $X$ cuts out on the line $\langle x,y\rangle$ the reduced divisor
$x+u+y$ we have

$x,y\in U_{u}:=X\setminus(S_{u}\cup S_{u}^{*})\quad\text{and}\quad u,y\in
U_{x}:=X\setminus(S_{x}\cup S_{x}^{*}),$

see Lemma 3.5. In particular, $x\notin\mathbb{T}_{u}X$. Hence the line
$l\subseteq X$ through $x$ meets $\mathbb{T}_{u}X$ in a single point, say
$z\in S_{u}=\mathbb{T}_{u}X\cap X$. Fix an isomorphism
$f\colon{\mathbb{P}}^{1}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}l$
sending $0$ to $x$ and $\infty$ to $z$. Then $f$ embeds
${\mathbb{A}}^{1}={\mathbb{P}}^{1}\setminus\\{\infty\\}$ onto
$l\setminus\\{z\\}\subseteq X\setminus S_{u}$. Since $\tau_{u}|_{X\setminus
S_{u}}\in\operatorname{Aut}(X\setminus S_{u})$, see Corollary 3.6(c), the map
$\varphi_{u}:=\tau_{u}\circ f|_{{\mathbb{A}}^{1}}\colon{\mathbb{A}}^{1}\to
C^{*}$ is an isomorphism.

By Corollary 3.6(c) $\tau_{x}|_{U_{x}}$ is a biregular involution acting
freely on $U_{x}$ and interchanging $u$ and $y$. Since the projective line
$\langle x,u\rangle$ is not tangent to $X$, we can choose a neighborhood
$V_{u}\subseteq U_{x}$ of $u$ such that also the line $\langle
x,u^{\prime}\rangle$ is not tangent to $X$ for each $u^{\prime}\in V_{u}$.
Letting $y^{\prime}=\tau_{u^{\prime}}(x)\in\langle x,u^{\prime}\rangle\cap X$
we have $\tau_{x}(u^{\prime})=y^{\prime}$. Letting
$V_{y}=\tau_{x}(V_{u})\subseteq U_{x}$ note that the restriction
$\tau_{x}|_{V_{u}}\colon
V_{u}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}V_{y}$ is biregular, and
so $V_{y}$ is a neighborhood of $y=\tau_{u}(x)$ in $X$.

Consider the morphism

$\varphi\colon V_{u}\times{\mathbb{A}}^{1}\to
X,\quad(u^{\prime},t)\mapsto\varphi_{u^{\prime}}(t)=\tau_{u^{\prime}}(f(t)),$

and let

$s=\varphi\circ(\tau_{x}\times\operatorname{id}_{{\mathbb{A}}^{1}})\colon
V_{y}\times{\mathbb{A}}^{1}\to
X,\,\,\,(y^{\prime},t)\mapsto\varphi_{u^{\prime}}(t)\,\,\,\text{where}\,\,\,u^{\prime}=\tau_{x}(y^{\prime})\in
V_{u}.$

Consider also the trivial line bundle $p\colon
E=V_{y}\times{\mathbb{A}}^{1}\to V_{y}$ where $p$ is the first projection, and
let $Z=V_{y}\times\\{0\\}$ be its zero section. We have
$s(y^{\prime},0)=y^{\prime}$, that is $s|_{Z}=p|_{Z}$. Thus, the triplet
$(E,p,s)$ is a spray of rank 1 on $V_{y}$ with values in $X$. The $s$-orbit
$O_{y}$ of $y$ coincides with $\varphi_{u}({\mathbb{A}}^{1})=C^{*}$ and the
map $ds|_{T_{0}E_{y}}\colon T_{0}E_{y}\to T_{y}O_{y}$ is onto. Due to
Proposition 2.1 $(E,p,s)$ can be extended to a spray on $X$. ∎

######  Remark 3.8.

We claim that the general $s$-orbits $O_{y^{\prime}}$ in $X$ are smooth affine
conics whose closures are fibers of the conic bundle
$\hat{X}\to{\mathbb{P}}^{2}$ resulting from the blowup of $\hat{X}\to X$ with
center $l$. Indeed, let $y^{\prime}\in V_{y}$ be a general point. Then
$\langle x,y^{\prime}\rangle$ being a general projective line in
${\mathbb{P}}^{4}$ through $y^{\prime}$, $X$ cuts out the plane
$L^{\prime}:=\langle l,y^{\prime}\rangle$ along the line $l$ passing through
$x$ and the residual smooth conic $C_{y^{\prime}}$ passing through
$u^{\prime}$ and $y^{\prime}$. So, the plane $L^{\prime}$ is
$\tau_{u^{\prime}}$-invariant and $\tau_{u^{\prime}}$ interchanges $l$ and
$C_{y^{\prime}}$ fixing their intersection points, which are the points of
$l\cap S_{u^{\prime}}^{*}$, see Corollary 3.6(c). Furthermore,
$C_{y^{\prime}}$ is the closure of the $s$-orbit
$O_{y^{\prime}}=\tau_{u^{\prime}}(l\setminus\\{z\\})$.

We are ready now to prove Theorem 1.1.

###### Proof of Theorem 1.1.

Since every complete smooth rational surface is elliptic, see [KZ23b, Theorem
1.1], we may suppose that $n\geq 3$. Due to Corollary 2.2 it suffices to show
that $X$ is locally elliptic. Fixing an arbitrary $y\in X$, choose a general
point $x\in X$ and $n$ projective lines $l_{1},\ldots,l_{n}$ on $X$ through
$x$ such that their tangent lines $T_{x}l_{i}$ span $T_{x}X$. Applying Lemma
3.7 to each of the $l_{i}$ yields rank 1 sprays $(E_{i},p_{i},s_{i})$,
$i=1,\ldots,n$, on $X$ dominating at $y$ along their respective orbits
$O_{i,y}$. We claim that the collection of sprays
$\\{(E_{i},p_{i},s_{i})\\}_{i=1,\ldots,n}$ is dominating at $y$. The latter
implies that $X$ is subelliptic, and so elliptic, see Corollary 2.2 (cf. also
Lemma 2.4). The above domination is equivalent to the fact that the tangent
lines $T_{y}O_{i,y}$ span the tangent space $T_{y}X$. By the construction in
Lemma 3.7 we have

$T_{y}O_{i,y}=d\varphi_{i,u}(T_{x}l_{i})\,\,\,\text{where}\,\,\,\varphi_{i}=\tau_{u}\circ
f_{i}\,\,\,\text{with}\,\,\,f_{i}\colon({\mathbb{A}}^{1},0)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(l_{i}\setminus\\{z_{i}\\},x).$

Therefore, we have

${\rm span}(T_{y}O_{1,y},\ldots,\,T_{y}O_{n,y})=d\tau_{u}({\rm
span}(T_{x}l_{1},\ldots,\,T_{x}l_{n}))=d\tau_{u}(T_{x}X)=T_{y}X,$

which proves our claim. The domination at $y$ spreads to a neighborhood of $y$
in $X$, which gives the local ellipticity (and the subellipticity), hence also
the ellipticity. ∎

Acknowledgments. It is a pleasure to thank Yuri Prokhorov for valuable
discussions, references, and useful comments on a preliminary version of this
paper.

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