ON ENDOMORPHISMS OF HYPERSURFACESILYA KARZHEMANOVAbstract. For any prime p ≥ 5, we show that generic hypersurface Xp ⊂ Pp
defined over Q admits a non-trivial rational dominant self-map of degree > 1,
defined over Q. A simple arithmetic application of this fact is also given.

1. Introduction
1.1.

Let X be an algebraic variety (smooth, projective, over a field of character-istic 0). Then two groups of symmetries of X, namely Aut(X) (biregular automorphisms) and Bir(X) (birational ones), come for free. The question on whether
these groups differ is classical and very important. For instance, the property
Aut(X) = Bir(X), signifying birational (super)rigidity of X, is an obstruction for
X to be rational (such X had been studied in numerous papers including [16], [24],[9], [17] and [18]).
Typically one has Aut(X) = {id} however. In this regard, it is natural to consider another (more general) class of symmetries of X, namely End(X), consistingof rational dominant endomorphisms of X. Then one may ask (what seems to
be a folklore) whether End(X) = Bir(X)? (Note that the latter property is anobstruction for X to be unirational.) This question gets immediate answer (‘yes’)
when X — with Pic(X) = Z at least — is of general type (as the endomorphismspreserve the spaces H 0 (X, mKX ) for all m ∈ Z). On the other hand, already in
the Calabi-Yau case things are not that straightforward; although still one gets
End(X) = Bir(X) when X is a general K3 surface for example (see [7]).
Presently, we would like to treat rationally connected X, namely X := XN ⊂ PN
being a hypersurface of degree N (see [24], [15], [6] for some other aspects of thegeometry of these XN ). Recall that according to [10] every XN , N ≥ 4, is nonrational, having Bir(XN ) = Aut(XN ).
Our main result is
MS 2010 classification: 14E05, 14D06, 14M20.
Key words: hypersurface, rational self-map, incompressibility.
1Theorem 1.2. For any prime p ≥ 5, (Zariski) general Q-hypersurface Xp ⊂ Pp
admits an endomorphism fXp of degree > 1, defined over Q. More precisely, fXp
has a field of definition — k below, — same for all Xp .
Thus the “non-regular” geometry of X = Xp is pretty much fruitful (comparewith results in [2], [4], [23], accounting for regular self-maps of X). At the sametime, the proof of Theorem 1.2 is not effective, and it would be interesting todescribe (a part of) End(X) explicitly. For example, what is the End(X)-action
on the universal Chow group of X (cf. [25], [3])? It is also plausible to get ridof the degree/ground field/dimension assumption in the formulation of our result.
Say, can one take any Fano manifold X in place of Xp , or at least any (composite)
integer N instead of p?
1.3.

In order to prove Theorem 1.2 we first relate Xp to those hypersurfaces thathave lots of endomorphisms. The latter are Xd ⊂ PN , defined over a given field,that happen to be unirational over this field, provided that N À d is sufficiently
large (see 2.1 below for the precise statements).
Next we employ the degree formula from [26] (cf. [22], [19], [14]). Recall that
this formula relates the connective K-theory classes of two algebraic varieties X andY admitting some rational map f : X 99K Y . However, in order to get something
fruitful in this way (e. g. to show that f with deg f = 0 does not exist) one has
to consider X, Y , etc. to be defined over an algebraically non-closed field k, and(roughly speaking) to have no points over k. This is the reason for the degree (resp.
dimension/ground field) restriction in Theorem 1.2.
Remark 1.4. One of the basic obstructions for applying our method to arbitrary
degree d (Fano) hypersurfaces Xd ⊂ PN is that those usually have points over thefield of definition (cf. [5]). There may also be no such nice congruence relation as
in Corollary 2.5 below. But still, once we find (sufficiently many?) endomorphismsof Xp ⊂ Pp which preserve some projective subspace Γ ⊂ Xp , one may hope(by projecting from Γ) to obtain non-trivial endomorphisms of Xd for some (all?)d ≤ p − 1.
The principal part of our arguments relies on (a part of) the main result in [8]which asserts the existence of X = Xp = Y as indicated above. It is then not
hard to derive Theorem 1.2 for the given Xp and the general case follows easily (seeSection 3 for details). Again we indicate that the initial Xp is very special. (It is2defined over a field k of cohomological dimension ≤ 1 and does not contain points
over the extensions of k whose degrees are coprime with p.)
1.5.

The next result was motivated by the paper [1]:Corollary 1.6. Let Xp ⊂ Pp be as in Theorem 1.2. Then there exists a possibly
larger number field K ⊇ Q such that Zariski closure of the fXp -orbit of the set
Xp (K) of K-rational points on Xp has dimension ≥ 2.
One may consider Corollary 1.6 as a generalization of [13, Theorem 1.4]. Yet,
unfortunately, our conclusion is weaker and it would be interesting to establish
potential density of the set Xp (Q) in Xp (e. g. by refining the arguments of Section 4
below).
Acknowledgments. I would like to thank F. Bogomolov and K. Zainoulline
for their interest and fruitful conversations. The results of this paper have been
discussed with K. Oguiso, S. Okawa, and B. Poonen during my visits to Osaka
University and MIT in Spring 2015; I am grateful to these people for hospitality.
Also comments by anonymous referee have helped to improve the exposition. The
work was supported by World Premier International Research Initiative (WPI),
MEXT, Japan, and Grant-in-Aid for Scientific Research (26887009) from Japan
Mathematical Society (Kakenhi).

2. Preliminaries
2.1.

Fix some integers d and r ≥ 1. Recall that every smooth hypersurface

Xd ⊂ PN of degree d contains a projective subspace Λ ' Pr when N À 1 (see [12],[20]). More precisely, Xd corresponds to (generic) point in the incidence subvariety
Z := {(Xd , Λ) | Λ ⊂ Xd } ⊂ P( d ) × Gr(r + 1, N + 1),
N +dwith dominant projections Z −→ P, Gr.
Further, if Λ is given over an arbitrary field k0 ⊂ C by equations lN −r+1 =. . . = lN = 0 for some linear forms li , then Xd can be chosen to be defined√over k := k0 ( −1). Indeed, any Xd passing through Λ has the defining equaNXtion
φi li = 0, for some (varying) forms φi of degree d − 1. Hence, sincei=N −r+1

the set of k-points is obviously dense in P( d−1 ) (w. r. t. the complex analytic
N +d−1

3

topology), one can approximate φi by such k-forms of degree d − 1 that Xd remains
smooth.
Thus, since one can always find generic Λ ∈ Gr(r, N ) to be defined over k, theabove discussion provides a smooth hypersurface Xd and a projective subspaceΛ ⊂ Xd , both over k, for any given d, r and sufficiently large N = N (d, r). From
[12, Corollary 3.7] we obtain
Theorem 2.2 (Harris, Mazur, Pandharipande). Xd is k-unirational.
2.3.

Let X, Y be smooth projective geometrically irreducible k0 -varieties of di-mension d = dim X = dim Y . Assume that there is a rational k0 -map f : X 99K Y .
The degree deg f equals 0 if f is non-dominant; otherwise deg f := [k0 (X) :f ∗ k0 (Y )].
We recall the next result from [26]:Theorem 2.4 (Zainoulline). χ(OX ) · τd−1 ≡ deg f · χ(OY ) · τd−1 mod nY , whereχ(·) is the Euler characteristic,
τd−1 :=Yp [ p−1 ]d−1

p primeis the (d − 1)-st Todd number, and nY is the g. c. d. of degrees of all closed points
on Y .
An immediate consequence of Theorem 2.4 is
Corollary 2.5 (cf. [26, Example 6.4]). In the previous notation, if nX = p,
X = Y = Xp ⊂ Pp (i. e. d = p − 1 and N = p) for some prime p ≥ 3, and (moregenerally) f is a k-map, then deg f > 0. (Such X is called incompressible overk.)1)
Proof. Regard X as a hypersurface over k. Then, since [k : k0 ] ≤ 2, we get nX = p
again. The claim now follows from Theorem 2.4 (with k0 replaced by k) and thefact that χ(OX ) = 1 = (τp−2 , p).

¤

Existence of X = Xp as in Corollary 2.5 is guaranteed by the following result(see [8, Theorem 8]):1) Existence of similar hypersurfaces X of general type, as suggested by [26, Example 6.4], is

not clear in the current setting because the condition nX = p need not be satisfied (compare with
Proposition 2.6 below).
4

Proposition 2.6 (Colliot-Thélène). For every prime p ≥ 5, there is a smooth (overQ) hypersurface X ⊂ Pp , given (over Q) by the equation
xp1 + lxp2 + . . . + lp−1 xpp − αxp0 = 0
for some integers l, α, so that nX = p.
Remark 2.7. Let t be a transcendental parameter, Xd ⊂ PN some Q(t)hypersurface, and d = N = p. One may assume (e. g. by identifying Xd with
an appropriate pencil of degree d = p hypersurfaces) that this Xd specializes via
t 7→ 0 to X from Proposition 2.6. It follows then that generic such Xd also satisfies
nXd = p. (Here “generic hypersurface” means a point in a Zariski open subset of
N +d
the Q(t)-variety P( d ) parameterizing all hypersurfaces of degree d in PN definedover the field Q(t).) Indeed, by definition of nY (see Theorem 2.4) and specialization t 7→ 0 one finds that nXd divides p = nX , and once nXd = 1 we get (by thesame reasoning) that nX = 1 as well, a contradiction.

3. Proof of Theorem 1.2
We keep on with notation of Section 2.
3.1.

b ⊂ PN over X of sufficiently
Let X be as in Proposition 2.6. Consider a cone X

large dimension and a family X of degree p hypersurfaces (⊂ PN ) over Q thatb We may regard (the general fiber of) X assmooths out the singularities of X.
a smooth k0 -hypersurface of degree p in PN for some purely transcendental fieldk0 ⊃ Q.
Next, using the (dominant) projections Z −→ P, Gr one can see by the same
argument as in 2.1 that the set of all k-points (Xp , Λ) ∈ Z is dense on Z in thecomplex analytic topology, and so X can be approximated by k-hypersurfaces Xp(for X , Xp treated as points on P, with k ⊂ C). The hypersurfaces X and Xp can
f1 ⊂ Z ofactually be put on an affine line A1 ⊂ P in such a way that the preimage Akk

A1k has all fibers = some projective spaces and generic fiber isomorphic to Λ (apply
the reasoning from 2.1 to this family over A1k considered as a degree p hypersurface
over k0 (t)).
Lemma 3.2. In the previous setting, the hypersurface X contains a projective
subspace ⊂ PN , isomorphic to Λ and defined over k.
5

Proof. It suffices to show that there exists a k-point on Z whose projection to P
f1 −→ A1 is Gal(k̄/
coincides with X . Indeed, the preceding natural projection Akkk)-equivariant, hence its fiber over X is a projective k-space.

¤

Without loss of generality we will assume that k0 = Q(t) in what follows.
Lemma 3.3. There exists a k-endomorphism f : X 99K X such that deg f > 1.
Proof. It follows from Lemma 3.2 and Theorem 2.2 (applied to Xd := X ) that X
is k-unirational. This yields two rational dominant k-maps φ : PN −1 99K X andψ : X 99K PN −1 . We may assume one of φ, ψ to have degree > 1. Then it remains
to take f := φ ◦ ψ.
3.4.
of P

N

¤

Let f be as in Lemma 3.3. Note that f is induced by a rational self-map
b Indeed, since p ≥ 5 and thus Pic(X ) = Z (Lefschetz), the map f⊃ X , X.

is given by such a linear system on X that is obtained via restriction of a linear
system from PN .
b of the family X (recall thatPut f0 to be the specialization of f at the fiber X
k0 = Q(t)). More precisely, if Lt is a (movable) linear system defining f , then its
b may acquire some divisorial components in the base locus.2)
specialization L0 to X
Subtracting all these we arrive at a linear system which we set to define f0 .
b (for an appropriate f ).
Lemma 3.5. f0 is not induced by a self-map of Sing(X)In other words, if x0 = . . . = xp = 0 are the equations of the singular locusb ⊂ X,b then f0 6= id, 0 on the subspace Pp ⊂ PN complementary to Sing(X).
bSing(X)
Proof. Let the notation be as in the proof of Lemma 3.3. One can choose a k-pointo ∈ X such that ψ (resp. φ) is unramified at o (resp. ψ(o)).
Take f := φ ◦ σ ◦ ψ for some σ ∈ PGL(N, k) (to be specified further) in such a
way that f (o) = o. Namely, since ψ, φ induce isomorphisms of the tangent spaces
To = Tφ(ψ(o)) , Tψ(o) , and (adjoint of) σ can act transitively on the N -tuples of kvectors in Tψ(o) , we arrive at such f whose Jacobian Jaco (f ) is a k-matrix withpairwise distinct eigen values and all defined over Q. Then, considering f0 as arational self-map of PN (cf. the discussion at the beginning of 3.4), we obtain that
all non-zero eigen values of the matrix Jaco0 (f0 ) are pairwise distinct as well. Hereb is the specialization of o.3)
o0 ∈ X
2) The latter stems from the fact that f may not be defined (in codimension 2) on X.
b3) Note that o0 is not contained in the indeterminacy locus of f

Thus f0 is defined at o0 and satisfies f0 (o0 ) = o0 .
60 for the indicated o and σ.

b
Now suppose that f0 is induced by some rational endomorphism of Sing(X).
Again we regard f0 as a self-map of PN . Then f0 is regular at some point o0 ∈ PN ,f0 (o0 ) = o0 and the non-zero eigen values of Jaco0 (f0 ) are all different. On the otherb a contradiction.
hand, f0 = id on the subspace Pp ⊂ PN complementary to Sing(X),Same argument shows that f0 has non-trivial components on Pp , i. e. f0 6= 0
there, which concludes the proof.

¤b isomorphic X (cf.
Note that generic subspace Pp ⊂ PN cuts out a subvariety on X
b so that the restriction
the beginning of 3.1). Identify X with such a section Pp ∩ X¯¯f0 ¯X is a well-defined rational map. Let fX : X 99K X be the composition of f0 ¯X
b 99K X from Sing(X).
band the linear projection X
Lemma 3.6. fX 6= id and is dominant.
Proof. Indeed, f0 induces a non-identical map on X by Lemma 3.5, and it remains√
to apply Corollary 2.5 to the Q( −1)-map fX .
¤Remark 3.7. For the last part of Lemma 3.6, observe that the fact deg fX 6= 0 is
not immediate from the proof of Lemma 3.5, and hence one needs an additional
argument (results of the second half of Section 2 for instance) in order to proceed.
Theorem 1.2 (for the given X) now follows from
Proposition 3.8. deg fX > 1.
Proof. Regard X as a (flat) family of hypersurfaces Xt ⊂ PN × t ⊂ PN × P1 (withb Let π : X −→ P1 be the natural projection (so that π −1 (t) = Xt ). Let
X0 := X).
also Lt be as in the second paragraph of 3.4.
b (unless deg fX > 1).
Lemma 3.9. The linear system L0 is non-trivial on X
Proof. Suppose the contrary (i. e. L0 = {0}). Then the π-map f : X 99K X has
b Resolve these by some π-blow-up σ : Y −→ X . Let
indeterminacies along X.
ab

Y −→ Z −→ X be the Stein factorization of the resolved f . Here a, b are some πmorphisms, with a inducing a birational isomorphism between Y and Z. Moreover,b the proper transform σ∗−1 X
b of Xb on Y belongs tosince f is not defined on X,
the exceptional locus of a. Composing further with b yields a rational self-map
b 99K Xb of degree 0.
µ:X
b 99K X from Sing(X)bNotice that µ does not coincide with the projection X
b is a non-multiple fiber of π (so that
because deg b > 1 (cf. Lemma 3.3) and X0 = X
7b is not a branching divisor of b). Thus one may assume X = Pp ∩ X
b intersectsX¯
b 99K X either gives a selfgeneric fiber of µ at > 1 points. Composing µ¯ with X
X

map X 99K X of degree > 1 (which we take for fX ), or that X is not incompressible,
in contradiction with Corollary 2.5.

¤

b = X0 and the
Let Y, σ, a, . . . be as above. It follows from Lemma 3.9 that X
scheme Z0 := b−1 (0) are birationally isomorphic via a ◦ σ −1 (recall that b is finite).
¯
In particular, Z0 is not a ramification divisor of b, which implies that deg b¯Zt =¯deg b¯Z0 for all t ∈ C close to 0.
¯¯
Now, since deg b¯ > 1 by construction, we deduce that deg f0 = deg b¯ > 1 as
Zt

Z0

well. The latter also gives deg fX > 1. Indeed, otherwise restricting L0 to genericb we get fX ∈ Aut(X) according to Lemma 3.6 and [10]. But then,
X = Pp ∩ X,¯
b 99K X to the base of the cone,
since fX is composed of f0 ¯X and projection Xb
we obtain that f0 must be induced by some projective transformation of PN ⊃ X.
The latter obviously contradicts deg f0 > 1 and the proof of Proposition 3.8 isfinished.

¤

To complete the proof of Theorem 1.2 we simply apply Remark 2.7 and the factthat all the preceding arguments go verbatim with Q replaced by Q(t). It remains
to set t := t0 ∈ Q — a given general parameter value — to obtain hypersurfaces
as in the statement of Theorem 1.2. Furthermore, in the generic setting one canreplace the argument with fX ∈ Aut(X) at the end of the proof of Proposition 3.8
by that with fX = id (cf. [21, Theorem 5]), thus getting a contradiction with
Lemma 3.6.4)

4. Proof of Corollary 1.6
4.1.

We proceed with applying the constructions of Section 3 to study the arith-

metics of hypersurfaces Xp ⊂ Pp (the notation is as earlier).5)
Let again X be as in Proposition 2.6.
Lemma 4.2. fX is non-periodic.
4) This argument may be considered as another way to prove Theorem 1.2.
5) Note that one can not specialize the (abundance of) k-points on X to “many” Q-points onb (as we did with (some of) endomorphisms) because nX = p and so all the points on
the cone Xbb
X we obtain this way are concentrated on Sing(X).
8

k−1
k
Proof. Indeed, otherwise we have fX= id for some k, so that both fX , fX
areinvertible. But this contradicts Proposition 3.8.

¤k
Fix fX as in Lemma 4.2. Then after possibly replacing fX by fX, k À 1, we

obtain a point o ∈ X(K) such that fX (o) = o and fX is defined at o (see [11]). We
also have det Jaco (fX ) 6= 0 because fX is dominant.
Lemma 4.3. There exists a k-cube ¤o ⊂ X (i. e. ¤o is given by linear inequalities
with coefficients in k) centered at o and invariant under fX .
Proof. Let ¤ ⊂ X be some cube containing o and defined over Q. Then the set[k
X \ fX(¤) is not everywhere dense in X. Indeed, otherwise there would exist ak>0
ki
subsequence fX(¤) → o for i → ∞, which implies that o ∈ X(Q) and contradicts\knX = p. It remains to take fX -invariant ¤o ⊆
fX
(¤).
¤
k>0

It follows from Lemma 4.3 (and the implicit function theorem) that the eigen
values λi of the matrix Jaco (fX ) are algebraic numbers (from K), all having norms|λi | = 1, 1 ≤ i ≤ p − 1. Furthermore, making if necessary a coordinate change on
Pp ⊃ X of the form xj 7→ αj xj , 0 ≤ j ≤ p (i. e. rescaling the metric on X), forsome αj ∈ Q∗ , we may assume λi to be algebraic integers (from OK ).
Lemma 4.4. The matrix Jaco (fX ) is semi-simple. Moreover, there is j < p−2 suchthat λ1 = . . . = λj = ±1, while λj+1 , . . . , λp−1 are multiplicatively independent.
Proof. The first claim follows from the fact that fX linearizes on ¤o . Now, if
2
λi = ±1 (or equivalently λi ∈ R) for all i, then fX= id, a contradiction.

Further, Dirichlet’s unit theorem yields λj+1 . . . λp−1 = |λj+1 . . . λp−1 | = 1, λj =∗λ−1
k as the only relations between λj+1 , . . . , λp−1 ∈ OK . In particular, since j <∗p − 1, λj+1 , . . . , λp−1 generate a free subgroup ⊆ Zp−j−1 ⊂ OK.

Suppose that j = p − 2. Then λj+1 , . . . , λp−1 are all equal to λ±1
j+1 say. On theother hand, characteristic polynomial of Jaco (fX ) is defined over k (by constructionof ¤o in Lemma 4.3), and hence the minimal polynomial of λj+1 divides it. All
together, this implies that λ2i = ±1 for all i ≥ j + 1, a contradiction.
Thus we get j < p − 2 and the claim follows.

¤

The arguments in [1, Section 2] and Lemma 4.4 imply that Zariski closure of the
fX -orbit of the locus X(K) has dimension ≥ p − j − 1 > 1.
9

Finally, to complete the proof of Corollary 1.6 one replaces Q by Q(t), as at theend of Section 3, and repeats the previous arguments.

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Kavli IPMU (WPI), The University of Tokyo, 5 - 1 - 5 Kashiwanoha,
Kashiwa, 277 - 8583, Japan
Moscow Institute of Physics and Technology, 9 Institutskiy per.,Dolgoprudny, Moscow Region, 141701, Russia
E-mail address: ilya.karzhemanov@ipmu.jp

11