ON UNIRATIONAL QUARTIC HYPERSURFACESILYA KARZHEMANOVAbstract. We study the unirationality property of an algebraic variety X(over C) versus the so - called stable birational infinite transitivity of X. We
show that in the case when X is a smooth quartic hypersurface these two
notions do not coincide.

1. Introduction
1.1.

Consider a smooth projective variety X over an algebraically closed fieldk ⊆ C. Recall that X is called unirational if there exists a rational dominant map
PN 99K X from some projective space. In dimensions ≤ 2, unirationality implies
rationality, whereas in higher dimensions this is no longer true (examples are due
to Iskovskikh - Manin, Clemens - Griffiths, Artin - Mumford, Bogomolov, Kollár, andothers). At the same time not a single example of rationally connected X that is
not unirational has been found yet (see [11] for an overview of the current stateof art). In this respect the paper [5] suggested a possible criterion for projective
manifold to be unirational (cf. [6]).
Namely, it was proposed in [5] that unirationality of X is equivalent to the stable birational infinite transitivity (or stable b - inf. trans. for short), with respect
to the group SAut (see 2.8 below for some recollections). The latter means thatthe product X × Pk , k À 1, carries a Zariski open subset U such that the group
SAut(U ) acts infinitely transitively on it. One of the goals, given the stated unirationality criterion, was to find non - unirational Fano hypersurfaces among thosetreated in [10].
The aim of the present paper is to show that the above - mentioned criterion forunirationality does not hold as stated. We prove the following:
Theorem 1.2. There exists a unirational X which is not stably b - inf. trans.
Let us briefly outline the strategy of the proof of Theorem 1.2.
MS 2010 classification: 14M20, 14E05, 14M10.
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1.3.

Recall that in [5, Section 3] smooth cubic (resp. some singular quartic)

hypersurfaces were shown to be stably b - inf. trans. This makes it reasonable totest smooth quartic hypersurfaces on having the same property (compare with [8]).
In Section 3, we prove that the latter does not hold for some smooth quartic X ofa given dimension ≥ 3, and since every such X of sufficiently large dimension isunirational (see e. g. [7, Corollary 3.7]), Theorem 1.2 will follow.
Originally, we wanted to test unirationality criterion from [5] over a field kp of
positive characteristic p, where one finds a very interesting example of unirationalvariety, that is a supersingular K3 surface S. In Section 2, we employ further
the analogy between such S and singular K3 surfaces, defined over k, namely the“torus - like” structure of their Hodge groups Hg.
More precisely, the idea is that once S is stably b - inf. trans., Hg(S) must be
isomorphic (as an algebraic group) to the additive group (kp , +), which is absurd(see 3.8 below). However, while realizing this idea we heavily relied on anotherinterplay between the supersingular and singular cases, which is the deep structural
result proved in [13] (cf. 2.4). It then remains to choose X in such a way that its
mod p reduction will be a cone over S and deduce the stable b - inf. trans. for S
from that for X to get the needed contradiction.
In turn, the “ stable b - inf. trans. conservation ” step is carried in 3.1, 3.4(see also 2.8) and is based on the observation made in Lemma 3.3, together witha standard technique of deforming rational curves in families and the fact that S(when lifted to k) actually contains a rational curve.
Acknowledgments. I am grateful to F. Bogomolov, S. Galkin, and D. Huybrechts for their interest and fruitful conversations. Some parts of the paper wereprepared during my stay at the Oxford University (July 2014) and CIRM, Universit‘a di Trento (August 2014), whose hospitality I happily acknowledge. 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. Hodge group. Let X be an Abelian variety and V := H 1 (X, Q) ⊗ R. The
complex structure on X corresponds to a homomorphism ϕ : S 1 −→ Sp(V ), the2

symplectic group of V (w. r. t. the symplectic form coming from an ample divisor
on X), so that the usual Riemann positivity conditions are satisfied.
The Hodge group Hg(X) is the smallest algebraic subgroup of Sp(V ) defined over
Q and such that ϕ(S 1 ) ⊆ Hg(X). There is the following criterion for X to have acomplex multiplication (i. e. for the algebra End(X) ⊗ Q to be a product of fields):
Proposition 2.2 (see [15, §2]). X is of CM - type iff Hg(X) is a torus algebraic
group.
More generally, if X is any smooth projective variety, then the Hodge group of X
is the largest subgroup Hg(X) of automorphisms of H ∗ (X, Q) that acts trivially on
all the spaces H k,k ∩ H 2k (X n , Q) (for the natural Hodge filtration on H ∗ (X n , C)),
where k, n are arbitrary and X n := X × |{z}. . . × X. (Note that this definition ofn timesHg(X) coincides with the previous one in the case when X is an Abelian variety.)
Similarly, X is said to be of CM - type, if the group Hg(X) is commutative.
Finally, given a K3 surface X together with the corresponding Kuga – Satake
Abelian variety AX , the group Hg(AX ) is an extension of Hg(X) by Z/2Z. In particular, since the group Hg(AX ) is reductive, we obtain the following (cf. Proposition 2.2):
Corollary 2.3 (see e. g. [17, §3]). In the previous setting, X is of CM - type iff AX
is, iff Hg(X) is a torus algebraic group.
2.4. Supersingular K3 surfaces. Let us now briefly recall a few properties of theK3 surfaces in question (see [18], [2], [21], [20] and [12] for an extensive treatmentof the subject).
Fix an algebraically closed field kp of characteristic p ≥ 5 and consider a supersingular K3 surface S over kp (i. e. rk NS(S) = 22 = b2 (S)) with its Artin invariant
σ0 (for the discriminant of Néron – Severi lattice NS(S) being equal p−2σ0 ). Then,
given a supersingular elliptic curve E with the associated Kummer surface Km(E 2 ),there exist rational dominant maps
Km(E 2 ) 99K S 99K Km(E 2 ),which are purely inseparable and generically finite of degree p2σ0 −2 (see [13]). Thisand unirationality of supersingular Kummer surfaces established in [20] immediately
yield the following:3

Theorem 2.5 (C. Liedtke). Any supersingular K3 surface S is unirational.1)
Example 2.6. In [19], Fermat quartic surface S := (P

x4i = 0) ⊂ P3 was shown

to be supersingular, provided p ≡ 3 mod 4.
Further, through the rest of the paper kp will be the residual field R/m of a
local algebra R ⊂ k with respect to the maximal ideal m ⊂ R, so that k is thefield of fractions of R (cf. [11, Ch. II, 5.10]). Similarly, given any variety X overR, replacing k - scalars by the kp - ones will be referred to (as usual) as the mod p
reduction of X and denoted Xp (variety X in turn will be called the lift of Xp (tok)).
Remark 2.7. Note that for the above supersingular surface S, when identified with∗its model (if any) over R ⊂ k, the action of the group Hg(S) on Hét
(S, Q` ) (here ` 6=

p is another prime number) is induced by the Frobenius action on S. In particular,
we have Hg(S) ' k∗p (as algebraic groups), which suggests further analogy between
supersingular K3 and the singular ones (cf. the discussion in [13, Section 2]).
Indeed, the latter can be lifted to k (see [13, Theorem 2.6]), and so Corollary 2.3
applies.
2.8. The group SAut. Fix a smooth affine k - variety U and consider somederivation D (identified with a vector field on U ) of the algebra k[U ].

Re-

call that there is a 1 - to - 1 correspondence between the locally nilpotent D andregular (k, +) - actions on U (see e. g.

[3, 1.19]).

Let us form the subgroup

SAut(U ) ⊆ Aut(U ) generated by various additive groups (k, +) =: Ga parameterized by all D.
It is not difficult to show that once D1 , . . . , DN are non - trivial locally nilpotent
derivations of k[U ] spanning the tangent space TU,ζ at some point ζ ∈ U , thenSAut(U ) acts on U with the dense orbit SAut(U ) · ζ (compare with [5, Theorem2.2]).
Let further U be defined over R (cf. the end of 2.4). In this case, after multiplying by k - scalars, one may assume D and all Di to be derivations of the algebra
R[U ]. Then D (resp. Di ) descends again to a locally nilpotent derivation for U

1) This theorem is stated here only for the completeness of our present exposition and will not

be used anywhere in the text.
4

replaced by Up . However, even though the k - group Ga corresponding to D coincides with exp(tD), t ∈ k, we have failed to establish a similar description for themod p reduction of Ga acting on Up .
In particular, it is not at all clear a priori whether the preceding Di define atleast non - trivial (kp , +) - actions on Up , not to mention the dense orbit property(for different Di may become linearly dependent modulo p). The latter obstacle
will be partially circumvented below by an additional argument (see 3.4) and theformer one is dealt with by the next
Lemma 2.9. In the previous setting, after possibly replacing U with a smalleraffine subset, there exists an R - scheme A1 × Z for some quasi - projective variety
Z such that U ' A1 × Z over k. In particular, if U is flat over R, there is a faithful(kp , +) - action on Up , which is the mod p reduction of the Ga - action on U . The
same holds for all the groups exp(tDi ), t ∈ k, 1 ≤ i ≤ N .
Proof. It follows from [9, Proposition 3.5] that there is a k - isomorphism U ' A1 ×Z
for some quasi - projective variety Z.
More precisely, the natural projection U −→ Z is actually an R - morphism,
corresponding to the inclusion of algebras Ker D ⊂ R[U ] (recall that both U and
D are defined over R). Furthermore, there exists an element g ∈ R[U ] with Dg 6= 0
and D2 g = 0, which yields s := g/Dg ∈ k[U ] such that Ds = 1.2) Multiplying by
k - scalars, one may assume that s ∈ R[U ], satisfying Ds ∈ R \ 0. This shows that
A1 × Z = Spec Ker D[s](via Slice Theorem) and the inclusion Ker D[s] ⊆ R[U ] induces a k - isomorphism
U ' A1 × Z.
Suppose now that U is flat over R. Replacing Z with another affine model one
may assume the R - flatness for Z as well. Then it follows from [16, Ch. III, §10,
Proposition 1] that U = A1 × Z over R. The same continues to hold for Up also,
i. e. Up = A1 × Zp with fiberwise (kp , +) - action, and lemma follows.

¤

2) Here one possibly needs to replace U with a smaller affine subset. Note however that D

remains nilpotent due to D2 g = 0.
5

3. Proof of Theorem 1.2
3.1.

Let us now return to the strategy outlined in 1.3. Namely, we consider asmooth unirational quartic hypersurface X ⊂ PN , N À 1, whose mod p reduction
is a cone X0 over supersingular (quartic) K3 surface S, with defining equation of
X beingp·Q+

3X

x4i = 0,i=0say, for some generic quartic form Q ∈ R[x0 , . . . , xN ] (cf. Example 2.6). One easily
sees via Bertini theorem that such X is smooth.
Remark 3.2. Note that X is flat over R because all of its (scheme) fibers are reduced.
Suppose that X is actually stably b - inf. trans. In this case, for Xk := X × Pk ,k À 1, and various subgroups Gi := (k, +) ⊂ SAut(U ), i ∈ I, one defines therational curves Ci ⊂ Xk as Zariski closures in Xk of generic Gi - orbits on an inf.
trans. open subset U ⊂ Xk (cf. 2.8).
Lemma 3.3. The group H2 (Xk , Q) is generated by the classes of Ci .
Proof. Let C1 , C2 be the two curves for G1 , G2 as above, together with induced
rational fibrations fi : Xk 99K Bi having Ci as generic fibers (one for each i).
Suppose the classes of Ci are proportional in H2 (Xk , Q).
Choose some common resolution f : W −→ Xk for both f1 , f2 , so that the maps
gi := fi ◦ f : W −→ Bi become regular. Note that according to our assumptionf ∗ C1 ≡ af ∗ C2 (numerically on W ) for some a ∈ Q. At the same time, we havef ∗ Ci · gi∗ OBi (1) = 0 by construction, for both i ∈ {1, 2} and some very ample line
bundles OBi (1). Altogether this gives f ∗ C1 · g1∗ OB1 (1) = f ∗ C1 · g2∗ OB2 (1) = 0, i. e.
g1 = g2 (hence f1 = f2 as well), a contradiction.
Thus the classes of C1 , C2 are not proportional in H2 (Xk , Q). Furthermore, since
Xk is a Fano manifold, we have H2 (Xk , Q) ' Pic(Xk ) ⊗ Q = Q2 , which implies
that H2 (Xk , Q) = Q2 by construction. In particular, the classes of Ci generate
H2 (Xk , Q), as wanted.
3.4.

¤

Next we employ the standard technique (see [11], [1], [4], [14]) of lifting

rational curves from S = X0 ∩ (x4 = . . . = xN = 0) to Xk .
Consider an immersion φ : P1 −→ S ⊂ X0 whose image C := φ(P1 ) is a rational
curve from an arbitrary ample class on S (see e. g. [4, Proposition 17]). Regard Xk
and X0,k := X0 × Pk as fibers — generic and the special one, respectively, — of6

an algebraic family X over Spec R. Let pr : X0,k −→ X0 (resp. π : X0,k 99K S)
be the natural projection (resp. the composition of pr with projection X0 99K S
to the base of the cone). Choose also a point o ∈ Pk and denote the composition
φ : P1 −→ S ' S × o ⊂ X0,k again by φ.
Lemma 3.5. The images of deformations of φ : P1 −→ X0,k do not form a dominant family of rational curves on X0,k (resp. are not contained in the fibers ofpr).
Proof. Suppose C varies in a family {Ct } ⊂ X0,k . Then, since φ ◦ π = φ and C
does not intersect the indeterminacy locus of π, we obtain a family {π(Ct )} on S
of immersed rational curves. Thus {Ct } can not be dominant because S is notseparably uniruled. Finally, since pr∗ C = C 6= 0 by construction, we get pr∗ Ct 6= 0
as well. This proves the claim.

¤

Recall that C ⊂ X0,k \ Sing X0,k by construction. Then Lemma 3.5 implies thate ⊂ Xk as long as
the curve C deforms in the preceding family X to a curve C¯ ¢¡χ C, TX0,k ¯C ≥ −KX0,k · C + dim X + k ≥≥ dim X0,k + dim PGL(2, k) = dim X0,k + 3(cf. [1, Theorem 15]). But the needed estimate is evident because −KX0,k ·C ≥ N −3
and so¯ ¢¡χ C, TX0,k ¯C ≥ 2N − 4 + k ≥ N + k + 2 = dim X0,k + 3.3)

Let Ci be as in Lemma 3.3. Then for the mod p reductions (Ci )p ⊂ X0,k we get
the following:
Lemma 3.6. There exists j ∈ I such that the cycle pr∗ (Cj )p 6= 0.
Proof. Suppose the contrary. Thus we have pr∗ (Ci )p = 0 for all i.
e of C to Xk .
Let C ⊂ S be as above. We have seen that there exists a lift C
e in H 2(N +k−1)−2 (Xk , Q` ) via Ci
Choose a prime ` 6= p and express the class of Cét(cf. Lemma 3.3). Taking the mod p reduction (a. k. a. specializing to X0,k ) we2(N +k−1)−2find an expression for the class of C in Hét

(X0,k , Q` ) via (Ci )p . But thenpr∗ C = pr∗ Ci = 0 by assumption. This contradiction concludes the proof.

¤3) Alternatively, one may set Q ∈ R[x , . . . , x ] in the equation for X, so that any curve from4N

the class OPN (1) S deforms to Xk (this fact can again be applied to prove Lemmas 3.6 and 3.7
e etc. covers more general situation.
below). However, the presented technique with C, C,
7

It follows from the construction (cf. 3.1) that there exist infinitely many curves(Cj )p as in Lemma 3.6 which are not contained in the indeterminacy locus of
π : X0,k 99K S. Then the following holds:
Lemma 3.7. dim π((Cj )p ) = 1.
Proof. Suppose dim π((Cj )p ) = 0 for all Cj . Let the notation be as in the proof of2(N −1)−2

Lemma 3.6. One can express the class of C = pr(C) ⊂ X0 in Hét

(X0 , Q` )via pr∗ (Cj )p . Then resolving π if necessary, we find that π(C) = C must be a point,
which is absurd.
3.8.

¤

We work over the field kp = R/m in what follows. In particular, Up ⊂ X0,k isan affine open subset, carrying a non - trivial (but possibly not inf. trans.) action of
the group SAut(Up ) (see Lemma 2.9 and Remark 3.2). One may obviously assume
that S ∩ Up 6= ∅.
Now, given any g ∈ Ga ⊂ SAut(Up ), composing the induced map S ' S × o 99Kg∗ (S × o) (onto the proper birational transform of S × o ⊂ X0,k under g) with¯π¯: g∗ (S × o) 99K S, yields a rational dominant endomorphism gS of S (hereg∗ (S×o)o ∈ Pk is as in 3.4).
Lemma 3.9. gS has degree > 1 for a generic choice of Ga ⊂ SAut(Up ).
Proof. Suppose the contrary. Then we have gS = id because S is not ruled andhence Ga 6⊂ Aut(S).
Note that by construction (cf. 3.1) a general Ga - orbit on Up coincides with oneof the curves (Cj )p as in Lemma 3.6. Thus, for gS = id, we obtain that π((Cj )p ) isa point. But this contradicts Lemma 3.7.

¤

By the discussion in 2.4 one may assume that S := Km(E 2 ). Let ψ : E 2 99K S
be the natural 2 : 1 - map.
Lemma 3.10. There exists a lift of gS to a rational self - map gE of E 2 such thatgS ◦ ψ = ψ ◦ gE .
Proof. Note that ψ ramifies precisely at the locus of 2 - torsion points on E 2 . Thisimplies that the field extension kp (E 2 ) ⊃ ψ ∗ (kp (S)) is generated by an element θ
whose minimal polynomial has coefficients only in kp . Then we can set gE to act
on kp (E 2 ) = ψ ∗ (kp (S))[θ] as follows:¯• gE ¯ψ∗ (kp (S)) = gS ;8

• gE (θ) = θ.
¤Note that gE constructed in Lemma 3.10 is regular because E 2 does not containrational curves. Further, by varying gS in Ga we obtain an algebraic family ofgraphs of the corresponding self - maps of E 2 , all isomorphic to E 2 .
2
In particular, the corresponding family (over Ga ) of groups Hét
(Q` ) is a trivial fi2
bration, which yields a 1 - parameter family of automorphisms (gS )∗ of Hét
(E 2 , Q` ),mapping the lattice Pic(E 2 ) to itself. This is only possible when all gS are Frobenius twists (cf. Remark 2.7). Moreover, gS 6= id according to Lemma 3.9, andhence we get k∗p ' Ga as affine curves, a contradiction.
Theorem 1.2 is proved.
Remark 3.11. In general, algebraic family of rational endomorphisms of a manifold
2
X need not lead to (again algebraic) family of automorphisms of Hét
(X, Q` ), forthe corresponding family of graphs need not admit a natural trivialization.

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Kavli IPMU (WPI), The University of Tokyo, 5-1-5 Kashiwanoha,
Kashiwa, 277-8583, Japan
E-mail address: ilya.karzhemanov@ipmu.jp

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