RATIONALITY OF AN S6 -INVARIANT QUARTIC 3-FOLD
ILYA KARZHEMANOVAbstract. We complete the study of rationality problem for hypersurfaces
Xt ⊂ P4 of degree 4 invariant under the action of the symmetric group S6 .

1. Introduction
1.1.

Any quartic 3-fold Xt ⊂ P4 with a non-trivial action of the group S6 can begiven by the equations
(1.2)

X

xi = tXX
x4i − (
x2i )2 = 0

in P5 . Here the parameter t ∈ P1 is allowed to vary.
When t = 2 one gets the Burkhardt quartic whose rationality is well-known (seee. g. [10, 5.2.7]). Similarly, t = 4 corresponds to the Igusa quartic, which is again
rational (see [21, Section 3]). On the other hand, it was shown in [1] that for allother t 6= 0, 6, 10/7 the quartic Xt is non-rational.
Example 1.3. Following [4, Section 4], let us blow up an A6 -orbit of 12 lines in
P3 to get a 3-fold that contracts, A6 -equivariantly, onto a quartic threefold with
36 nodes. It follows from Remark in [1] that this (Todd) quartic must be X10/7 .
Hence X10/7 is rational.
Thus, excluding the trivial case of t = 0 it remains to consider only X6 , in orderto determine completely the birational type of all S6 -invariant quartics. Here is theresult we obtain in this paper:
Theorem 1.4. The quartic X := X6 is rational.
Theorem 1.4 is proved in Section 3 by, basically, running the equivariant-MMPtype of arguments as in [22]. (Although the proof also uses some computations
carried in Section 2.) Unfortunately, we were not able to apply the results fromMS 2010 classification: 14E08, 14E30, 14M10.
Key words: quartic 3-fold, ordinary double point, rationality.
1[14], since non-rational Xt all have defect equal 5 (see [1, Lemma 2]), which seems
to contradict either [14, 5.2, Lemma 8] or [14, 5.2, Proposition 3] (compare alsowith [14, Corollary 1] and the list of cases in [14, Main Theorem]).
Conventions. The ground field is C and X signifies the quartic X6 in what follows.
We will be using freely standard notions and facts from [11] and [16] (but we recall
some of them for convenience).
Acknowledgments. Some parts of the paper were prepared during my visits to
AG Laboratory at HSE (Moscow) and Miami University (US). I am grateful to these
Institutions and people there for hospitality. 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. Auxiliary results
2.1.

Consider the subspace P3 ⊂ P5 given by equations
x0 + x2 + x5 = x1 + x3 + x4 = 0.

We have X ∩ P3 = Q1 + Q2 , where the quadric Q1 ⊂ P3 is given by√3
x20 + x0 x2 + x22 + w(x21 + x1 x3 + x23 ) = 0, w := 1,
while the equation of Q2 ⊂ P3 is
x20 + x0 x2 + x22 − (w + 1)(x21 + x1 x3 + x23 ) = 0.
Identify the set {x2 , x0 , x4 , x3 , x1 } with {1, . . . , 5} and consider the corresponding
action of the group S5 . Put τ := (13524) ∈ S4 ⊂ S5 and o := [1 : 1 : w : w : w2 :
w2 ] ∈ Sing(X) (cf. [1]).1) Then the following (evident) assertion holds:
Lemma 2.2. τ c (Qi ) 3 o iff c = 0 or 2.
Consider h := (23451) ∈ S5 . Again a direct computation gives the following:
Lemma 2.3. ha τ b (Qi ) 3 o iff(a, b) ∈ {(0, 0), (3, 0), (4, 0), (0, 2), (3, 3), (1, 2), (4, 2), (1, 1)}. More precisely, we
have1)For the set {1, . . . , n}, any n ≥ 1, symbol (i . . . i ), 1 ≤ i ≤ n, denotes its permutationn1j{i1 , . . . , in } (i. e. 1 7→ i1 and so on). Also, if ij = j for some j, we will identify (in the obviousway) (i1 . . . in ) with permutation of the respective (n − 1)-element set.
2• τ 2 (Qi ) = h4 (Qi ) 3 o and τ 2 (Qi ) 6= Qi ;• h4 τ 2 (Qi ) = h3 (Qi ) 3 o and h4 τ 2 (Qi ) 6= Qi , τ 2 (Qi );• hτ 2 (Qi ) = Qi ;• h3 τ 3 (Qi ) = hτ (Qi ) 3 o and h3 τ 3 (Qi ) 6= Qi , τ 2 (Qi ), h4 τ 2 (Qi ).
2.4.

Let G := hτ, hi be the group generated by τ and h. Note that the order of G

is divisible by 4 and 5. Then from the classification of subgroups in S5 we deducethat G is the general affine group GA(1, 5). Note also that G = F5 o F∗5 for thefield F5 (here F5 , F∗5 are the additive and multiplicative groups, respectively).
Xγ(Q1 ) and the local class group Clo,X at o. Note
Consider the divisor D :=γ∈Gthat both Qi are smooth because they are projectively equivalent to x20 + x22 +x21 + x23 = 0. In particular, blowing up P4 ⊃ X at Q1 yields a small resolutionof the singularity o ∈ X. Then by the standard properties of (small) extremal
contractions we may identify Q1 with the generator 1 ∈ Clo,X = Z.
With all this set-up we get the following:
Proposition 2.5. rk ClG X > 1 for D ∈ Clo,X being equal to either 4 or 8.
Proof. Let us recall the construction of the group Clo,X . One identifies X =Spec Oo,X and considers various morphisms µ : X −→ X 0 . Here X 0 is any (notnecessarily normal) variety. Then Clo,X is generated by the sheaves OX (Q1 ) andµ∗ OX 0 (H) for all Cartier divisors H on X 0 (note that µ∗ OX 0 (H) may no longer be
a divisorial sheaf for non-flat µ). The group operation “+” on Clo,X is induced by
the usual product of OX -modules.
Further, by construction of τ, h (cf. Lemmas 2.2, 2.3) we have
(2.6)

D=

X

τ a hb (Q1 ) = 2h4 (Q1 ) + 2h3 (Q1 ) + 2Q1 + 2hτ (Q1 )(a,b)∈{(0,0),...,(1,1)}

in Clo,X (we have identified OX (Q1 ) with Q1 ). Now, since h3 (Q1 ), h4 (Q1 ) 3 o,both h3 , h4 act on Clo,X = Z. Indeed, h3 (Q1 ) and h4 (Q1 ) differ from (a power of)
Q1 by some suitable µ∗ H as above.
For h3 = (h4 )2 we get h3 (Q1 ) = 1 = (h3 )3 (Q1 ) = h4 (Q1 ) and hence D = 4 or 8.
This means in particular that the product of OX -modulesI :=Y

OX (−γ(Q1 )),γ∈G
3identified with D as an element in Clo,X , is not invertible (otherwise D will bezero).
Take a G-equivariant resolution r : W −→ X.

Then the sheaf r∗ I be-

comes invertible and the corresponding (effective) divisor is not of the form
[relatively trivial part] + [r-exceptional part]. Indeed, otherwise I will be equalto µ∗ OX 0 (H), with some X 0 and H as earlier, which is impossible for D 6= 0 in
Clo,X .
Applying relative G-equivariant MMP to W (cf. [25, 9.1]) yields a small Gequivariant contraction Y −→ X and a relatively non-trivial G-invariant Cartierdivisor on Y (note that according to [15, Lemma 5.1] “Cartier = Q-Cartier” in thiscase). This shows that rk ClG X > 1 and completes the proof of Proposition 2.5.2)¤
2.7.

Fix some terminal GQ-factorial modification φ : Y −→ X. Here φ is a G-

equivariant birational morphism with 1-dimensional exceptional locus (see Proposition 2.5). Let also ψ : Y −→ Z be a KY -negative G-extremal contraction.
Lemma 2.8. 3-fold Y is Gorenstein.
Proof. This follows from the relation φ∗ ωX = ωY , the fact that φ is small, and thefreeness of | − KX |.

¤Recall that the singular locus of X consists of two S6 -orbits, of length 30 and
10, respectively, where the first orbit contains the point o, while the second one
contains o0 := [−1 : −1 : −1 : 1 : 1 : 1] (see Remark in [1]).
For an appropriate Y we get the following:
Lemma 2.9. Sing Y = ∅ or G · o0 .
Proof. Indeed, the divisor D from 2.4 contains o and the morphism φ makes X
GQ-factorial near o, which means that one may take φ to resolve the singularities
in G · o ⊂ D (run the G-equivariant Q-factorialization procedure from the proof of
Proposition 2.5).
The complement Σ := [the longest S6 -orbit in Sing X] \ G · o is also a G-orbit (oflength 10). Furthermore, we have s(o) 6= o ∈ Σ for s := (21) ∈ S5 (see 2.1), and2)Present definition of Clo,X differs from the usual (algebraic) one that is via the direct limitof groups Cl U/Pic U over all Zariski opens U 3 o on X. A priori there is no natural isomorphismof the latter with Clo,X . At the same time, we have used the fact that 0 6= D ∈ Clo,X in order toconstruct Y as above, thus proving the existence of some G-invariant non-Cartier divisor on X.
4

so the arguments in the proof of Proposition 2.5, with s(Q1 ) = Q1 , apply to showthat X not GQ-factorial near Σ as well. Hence we may assume that φ resolves the
singularities in Σ as well.
Finally, φ either resolves or not the singularities in G · o0 , depending on whether
there is a G-invariant non-Cartier divisor passing through o0 or there is no such.

¤

We will assume from now on that Y is as in Lemma 2.9.
Proposition 2.10. If ψ is birational, with exceptional locus E, then ψ(E) is a
curve.
Proof. Firstly, recall that Y is terminal, GQ-factorial (but not necessarily Qfactorial) and Gorenstein (see Lemma 2.8).
Lemma 2.11. Y is Q-factorial with rk Pic Y = 11.
Proof. Note that F5 = hhi is the unique normal subgroup in G = F5 o F∗5 . Then weXhave Qi 6∼ h(Qi ). Indeed, otherwise D ∼ 5γ(Qi ), where D is as in 2.4. Butγ∈hτ iin this case D = 5(Q1 + τ 2 (Q1 )) in Clo,X (see Lemma 2.3), which is either 0 or 10,
thus contradicting Proposition 2.5.
Further, since D is a G-orbit of Q1 , all of its components are linearly indepenP
dent in Cl X ⊗ R. Indeed, otherwise we getγ(Q1 ) = 0, which is an absurd.
This, together with computation of the defect in [1], yields rk Cl X = 11 for Cl X
being generated by KX , a G-invariant class of some Weil divisor Do and by thecomponents of D (the number of these components is 10 because Q1 6∼ h(Q1 )).
Similarly, we find that Cl Y is generated by KY , φ−1∗ Do and by the components0of φ−1∗ D, all being Cartier according to Lemma 2.9 and the fact that D 63 o . Thus

Cl Y = Pic Y and the claim follows.

¤

Now let Ei be the irreducible 2-dimensional components of E. Suppose thatdim ψ(E) = 0. Then we get the following:
Lemma 2.12. E is a disjoint union of Ei .
Proof. Since the divisor −KY is nef and big, it follows from Lemma 2.11 and [24]that the Mori cone N E(Y ) is polyhedral, spanned by extremal rays, so that every
extremal ray on Y is contractible. This implies that some (at least 1-dimensional)
family of curves in every Ei generates an extremal ray because there are no small
KY -negative extremal contractions on Y (see [5] and Lemmas 2.8, 2.11). In particular, Ei do not intersect, since dim ψ(E) = 0 by assumption.
5¤Note that Cl X ' Cl Y as G-modules. This induces a natural G-action on thecone N E(Y ). Consider the G-extremal ray in N E(Y ) corresponding to ψ. By
Lemma 2.12 this is a G-orbit of some KY -negative contractible extremal rays Ri
corresponding to Ei .
It remains to exclude the cases Ei = P1 × P1 or quadratic cone, and Ei = P2 ,both for dim ψ(E) = 0 (cf. [5]). Suppose one of these possibilities does occur. Thenwe getLemma 2.13. Every surface Ei is not preserved by the subgroup hhi ⊂ G.
Proof. Assume the contrary. Then all Ri are invariant with respect to hhi and there
is a subspace P3 ⊂ P4 ⊃ X (with φ(Ei ) ⊆ X ∩ P3 ) invariant under F5 = hhi. Recall
4
X
that h = (23451) permutes x0 , x2 , x1 , x3 , x4 . Thus the equation of P3 isxi = 0.
i=0This implies that X ∩ P3 ∩ Sing X = ∅ and so φ(Ei ) is Cartier. But the latter is
impossible for otherwise φ(Ei ) would intersect all the curves on X negatively.

¤

It follows from Lemma 2.13 that all Ei are linearly independent in Pic Y ⊗ R andtogether with KY they generate Pic Y (argue exactly as in the proof of Lemma 2.11).
Note also that Ei ·C ≥ 0 for all i and any KY -trivial curve C ⊂ Y because otherwise
the class of C belongs to Ri (recall that by our assumption ψ(Ei ) is a point). In
particular, there is such C that any other KY -trivial curve 6= C on Y is numericallyP
equivalent to C + ai Ri for all ai ≥ 0, and so there is just one C. This implies thatevery surface φ(Ei ) ⊆ X ∩P3 (of degree (KY )2 ·Ei ≤ 2) contains a G-orbit of length
at least 30 (see Lemma 2.9). Hence φ(Ei ) together with Ei are all hhi-invariant.3)
The latter contradicts Lemma 2.13 and Proposition 2.10 is completely proved. ¤We conclude by the following simple, although useful in what follows, observation:
Lemma 2.14. G 6⊂ GL(3, C).
Proof. The group G has only one 4-dimensional and four 1-dimensional irreducible
representations. The claim follows by decomposing C3 into the direct sum of irreducible G-modules.

¤

3)As there are no G-invariant curves in P3 ∩ S ∩ S for two different surfaces S of degree ≤ 212icontaining common G-orbit of length 30 (cf. Lemma 2.14).
6

3. Proof of Theorem 1.4
3.1.

We retain the notation and results of Section 2. Consider some G-extremal

contraction ψ : Y −→ Z. Let us assume for a moment that ψ is birational withexceptional locus E. Recall that E is a union of (generically) ruled surfaces Eicontracted by ψ onto some curves (see Proposition 2.10).
Lemma 3.2. E ∩ Sing Y = ∅.
Proof. Over the general point of ψ(Ei ) morphism ψ coincides with the blow-up of acurve (see [5]). Then for any ruling C ⊂ Ei contracted by ψ we have KY · C = −1.
Hence the surfaces φ(Ei ) ⊂ X are swept out by the lines φ(C).
Note further that C corresponds to a contractible extremal face of N E(Y ) (cf.
the proof of Lemma 2.12). In particular, one may assume that C generates a KY negative extremal ray, which shows that C is Cartier on Ei because all scheme¯
fibers of ψ ¯ are smooth (lines) and C varies in a flat family.
Ei

Recall that all divisors Ei are Cartier (cf. Lemmas 2.8, 2.11 and [15, Lemma5.1]). Now, if Ei ∩ Sing Y 6= ∅, then φ(C) is a singular curve for some C as above,
which is impossible. Hence Ei ∩Sing Y = ∅. But then E∩Sing Y = ∅ as well because
Ej ∩ Sing Y ⊂ Ei for all surfaces Ej from the corresponding extremal face.

¤

Remark 3.3. We have h1,2 = 0 for a resolution of Y according to Remark in [1].
Then it follows from [5] and Lemma 3.2 that ψ(Ei ) = P1 for all i.
Lemma 3.4. We have KY = ψ ∗ KZ + E (hence Z is Gorenstein), KY · C = −1
for any ruling C ⊂ Ei contracted by ψ, and Z is smooth near ψ(E).
Proof. One obtains the first two identities by exactly the same argument as in the
proof of Lemma 3.2. Further, since the linear system |− KY | is basepoint-free (withKY = φ∗ KX ), generic surface S ∈ | − KY | passing through a given point on Y issmooth. Then, for S · C = 1, we find that the surface ψ(S) ∈ | − KZ | is smooth aswell, so that Z is smooth near ψ(E).

¤Now let ψ be the result of running a G-MMP on Y .
Lemma 3.5. In the above setting, ψ is a birational contraction that maps its exceptional loci onto 1-dimensional centers, so that the corresponding 3-folds are smoothnear these centers. In particular, all these 3-folds are Q-factorial Gorenstein andterminal, with nef and big −K, and ψ is composed of blow-downs onto smooth
rational curves.
7

Proof. It follows from Lemmas 2.8, 2.11, 3.2, 3.4 and [23, Corollary 4.9] that eachstep of ψ produces a Q-factorial Gorenstein terminal 3-fold, with a G-action andnef and big −K, unless all exceptional Ei = P2 on this step. One can easily see theproper transform of such Ei on X will be a plane. Moreover, arguing as at the endof the proof of Proposition 2.10 we find that this plane will be hhi-invariant, which
contradicts Lemma 2.13.
Further, arguing as in the proof of Corollary 3.9 below one computes that whenever Ei = quadric or P2 , contracted to a point in both cases, its proper transformon Y (hence on X as well) will also have degree ≤ 2 w. r. t. −K. This leads tocontradiction as earlier.
Thus on each step ψ can contract Ei to curves only. Applying the same arguments as in the proof of Proposition 2.10 to each step of ψ gives the claim (the lastassertion of lemma follows from [5]).

¤Let, as above, E be the ψ-exceptional locus. Note that Y contains the G-orbitof 20 curves Cj contracted by φ (see Lemma 2.9). In particular, G induces a
non-trivial action on the set of these Cj , which leads to the next
Lemma 3.6. E can not consist of only one (connected) surface.
Proof. Indeed, otherwise we have (E = Ei )∩Cj 6= ∅ for all j, which yields a faithful
G-action on the base of the ruled surface E. Hence we get G ⊂ PGL(2, C). On theother hand, we have G 6⊂ A5 , S4 (see Lemma 2.14), a contradiction.

¤

Proposition 3.7. E 6= ∅ unless Y is rational.
Proof. Let E = ∅. Then we get rk PicG Y = 2 and N E(Y ) is generated by (G-orbitsof) the classes of Cj and an extremal ray corresponding to some G-Mori fibration
ϕ : Y −→ S (dim S > 0).
Lemma 3.8. Let dim S = 1. Then Y is minimal over S unless it is rational.
Proof. Suppose there is a surface Ξ which is exceptional for some (relative) KY negative extremal contraction on Y /S. Then Ξ necessarily contains one of Cj .
Indeed, otherwise Ξ intersects all curves on Y non-negatively by the structure of
N E(Y ), which is impossible. In particular, we find that Ξ must be a minimal ruled
surface (same argument as in the proof of Lemma 2.12), with the negative section
equal some Cj .
8

We may assume KY2η ≤ 4 for generic fiber Yη of ϕ – otherwise Y is rational (see[9], [18]). Moreover, we have KY2η 6= 1, since otherwise the group G ⊆ Aut(Yη ) must
act faithfully on elliptic curves from | − Kη |, which is impossible (cf. Lemma 2.14).
One also has KY2η 6= 2 because the order of the group of automorphisms of del Pezzo
surfaces of degree 2 is not divisible by 5 (see e. g. [7, Table 8.9]).
Further, if KY2η = 4, then contracting Ξ we arrive at a del Pezzo fibration ofdegree 5, so that Y is rational.
Now, if KY2η = 3, then all smooth fibers of ϕ are isomorphic and have Aut Yη = S5(see [7, Table 9.6]). Away from the singular fibers ϕ defines a locally trivial (inanalytic topology) fibration on smooth cubic surfaces Yη . Two charts, Yη × S 0 andYη × S 00 , say (for some analytic subsets S 0 , S 00 ⊆ S), are glued together via an
automorphism t ∈ Aut Yη , which preserves the elements in the G-orbit of Ξ andsatisfies tGt−1 = G. Since G is not a normal subgroup in S5 , one gets t ∈ G, andthe letter is impossible, once t 6= 1, by the way G acts on Ξ (a. k. a. on Cj ). Thust = 1 and ϕ induces a locally trivial fibration in the Zariski topology, so that Y isrational, and the proof is complete.

¤

Note further that the subgroup hhi ⊂ G must act faithfully on Pic Y . Indeed,otherwise Qi ∼ ha (Qi ) for all a, i, which implies that Qi contains the orbit hhi · o,a contradiction. In particular, if dim S = 1, then from Lemma 3.8 we deduce thateither Pic Y = Z2 (this contradicts Lemma 2.11), or ϕ contains a fiber with ≥ 5irreducible components (interchanged by hhi). In the latter case, we get KY2η ≥ 5
for generic fiber Yη of ϕ, and rationality of Y follows from [9], [18].
Finally, one excludes the case when ϕ is a G-conic bundle exactly as in the proof
of Lemma 3.12 below, and Proposition 3.7 is completely proved.

¤

Here is a refinement of Lemma 3.6 and Proposition 3.7:
Corollary 3.9. E is a disjoint union of G-orbits (length ≥ 2), corresponding to
extremal faces of N E(Y ), unless Y is rational.
Proof. Let E, Ẽ be two ψ-exceptional orbits in question. Choose some connectedcomponents Ej ⊂ E, Ẽj ⊂ Ẽ and suppose they intersect. One may assume both
Ej , Ẽj to be ruled surfaces that can be contracted by the blow-downs, one for eachsurface (cf. Lemma 3.5 and the proof of Lemma 3.2).
Let ψj : Y −→ Yj be the contraction of Ej . Then, given that Ej ∩ Ẽj 6= ∅, there
is a ψ-exceptional curve C ⊂ Ẽj such that Ej · C ≥ 0. On the other hand, we
9have KY = ψj∗ KYj + Ej and KYj · ψj (C) = −1 (for ψ blows down ψj (Ẽj )), which
gives either KY · C = −1 or KY · C = 0 (recall that −KY is nef). The latter caseis an absurd by construction of ψ. In the former case, we get Ej · C = 0 and soψ∗ (Ej ∩ Ẽj ) = ψ∗ C = 0, which is impossible for the ruled surfaces Ej 6= Ẽj , sincethen 0 = Ei · C = (C 2 ) < 0 on Ei , a contradiction.
3.10.

¤

We will assume from now on that E 6= ∅ is as in Corollary 3.9. It follows

from Lemma 3.5 that Z is Q-factorial Gorenstein and terminal. Note also that
−KZ is nef and big by [23, Corollary 4.9].
Lemma 3.11. We have φ−1∗ Qj 6⊂ E for some j.
Proof. Note that ψ∗ KY = KZ because Z has rational singularities. This gives the−1claim as −KY = φ−1∗ Q1 + φ∗ Q2 .

¤Let us treat the case when Z admits a G-Mori fibration.
Lemma 3.12. Z is not a G-conic bundle.
Proof. Suppose we are given a G-conic bundle structure on Z with generic fiber
C = P1 . Then if φ−1∗ Q1 6⊂ E, say (see Lemma 3.11), it follows from the definitionof Qi and G in 2.1 that the G-orbit of Q1 (hence also of φ−1∗ Q1 ) has length ≥ 10
(cf. the proof of Lemma 2.11). This yields a faithful G-action on C which in turncontradicts Lemma 2.14.

¤

Lemma 3.13. Z is not a G-del Pezzo fibration unless Z is rational.
Proof. Argue exactly as in the del Pezzo case from the proof of Proposition 3.7.
3.14.

¤

We will assume from now on that Z is a GQ-Fano (cf. Lemmas 3.12 and3.13). Note that any two components of exceptional locus of ψ can intersect only
along the fibers. Then it follows from Remark 3.3, Lemma 3.4 and Corollary 3.9that either(3.15)−KZ3 = 4 + 2k(−KZ · P1 + 1)

for some even k ≤ 10 or(3.16)−KZ3 = 24 − 20KZ · P1 − 10k 0

for some k 0 ≤ −KZ · P1 (recall that rk Pic Y = 11 by Lemma 2.11 and the subgrouphhi ⊂ G acts faithfully on Pic Y ).
10Lemma 3.17. The linear system | − KZ | is basepoint-free.
Proof. Assume the contrary. Then it follows from [12] that Z is a G-equivariant
double cover of the cone over a ruled surface (note that −KZ3 ≥ 12 is divisible by4). This easily gives G ⊂ PGL(2, C) and contradiction with Lemma 2.14.

¤

Lemma 3.18. The morphism defined by | − KZ | is an embedding.
Proof. Assume the contrary. Then it follows from [3, Theorem 1.5] that Z is a Gequivariant double cover of either a rational scroll or the cone over a ruled surface. In
both cases, arguing similarly as in the proof of Lemma 3.12, one gets contradiction.
¤
Lemmas 3.17 and 3.18 allow one to identify Z with its anticanonical model3
Z2g−2 ⊂ Pg+1 (here g := −KX/2 + 1 is the genus of Z).

Lemma 3.19. Z is singular unless it is rational.
Proof. Suppose that Z is smooth. Then rationality of Z follows from the fact thath1,2 (Z) = 0 (see Remark 3.3) and [11, §§12.2 – 12.6].

¤

According to Lemmas 3.19, 2.9 and 3.5 we may reduce to the case when|Sing Z| = |Sing Y | = 10, with the locus Sing Z being some G-orbit.
Proposition 3.20. g ≤ 9.
Proof. Let g > 9. Note that the linear span of any G-orbit in Sing Z has dimension≤ 9. Hence we can consider a G-invariant hyperplane section S ∈ |−KZ | (satisfying
S ∩ Sing Z 6= ∅).
Further, since G 6⊂ GL(3, C), the group G acts on Z without smooth fixed points.
On the other hand, since Z is G-isomorphic to X near Sing Z by construction, we
obtain that G does not have fixed points on Z at all.
Lemma 3.21. There are no G-invariant smooth rational curves on Z.
Proof. Indeed, otherwise the action G ª P1 ⊂ Z is cyclic, which gives a G-fixed
point ∈ P1 , a contradiction.

¤

Lemma 3.22. The pair (Z, S) is plt.
Proof. Lemma 3.21 and the proof of [22, Lemma 4.6] show that the pair (Z, S) islog canonical. Moreover, if (Z, S) is not plt, the same argument as in [22] reduces
11the claim to the case when S is a ruled surface over an elliptic curve, say B. On
the other hand, since |S ∩ Sing Z| = 10, we get either G ⊂ PGL(2, C) or a faithful
G-action on B, a contradiction.

¤

It follows from Lemma 3.22 and [25, Corollary 3.8] that S is either normal or reducible. But in the latter case, −KZ ∼ [disconnected surface], which is impossible.
Thus the surface S is normal with at most canonical singularities. Let us identify
S with its (G-equivariant) minimal resolution. In particular, we may assume that
S contains a G-invariant collection of disjoint (−2)-curves Ci , 1 ≤ i ≤ 10.
From G ⊆ Aut S one obtains a G-action on the space H 2,0 (S) = C[ωS ]. In­ ®
particular, the subgroup τ 2 ⊂ G preserves the 2-form ωS , which implies that­ ®
the quotient Sτ := S/ τ 2 has at worst canonical singularities. Note also that
τ 2 (Ci ) = Ci and h(Ci ) 6= Ci for all i.
Let C̃i be the image of Ci on Sτ .
Lemma 3.23. |C̃i ∩ Sing Sτ | = 2 for all i.
Proof. This follows from the fact that (C̃i2 ) = −1 by the projection formula.

¤

Let Sτ0 be the minimal resolution of Sτ . From Lemma 3.23 we obtain that Sτ0contains ≥ 20 disjoint (−2)-curves. This contradicts h1,1 (Sτ0 ) = 20 and finishes theproof of Proposition 3.20.

¤

According to Proposition 3.20 and (3.15), (3.16) we may assume that −KZ3 ∈{12, 16}.4)
Remark 3.24. Actually, since Z = Z16 ⊂ P10 and the projective G-action is induced
from the linear one on C11 = H 0 (Z, −KZ ), one gets a pencil on Z consisting of
G-invariant hyperplane sections. In particular, there is such S intersecting Sing Z,
so that the arguments in the proof of Proposition 3.20 apply and exclude the case
−KZ3 = 16.
Proposition 3.25. rk Pic Z 6= 2.
Proof. Suppose that rk Pic Z = 2 and consider a 1-parameter family s : Z −→ ∆over a small disk ∆ ⊂ C of smooth Fano 3-folds Zt , t 6= 0, deforming to Z0 = Z(see Lemma 3.5 and [20]). Since H i (Zt , nKZt ) = 0 for all n ≤ 0, i > 1 and t, we
deduce that the sheaf s∗ (−KZ ) is locally free.
4)Note that the case k = 10 yields rk Pic Z = 1 and can be excluded exactly as in the proof of

Proposition 3.27 below.
12Similarly to Y , the cone N E(Z) is polyhedral, with contractible extremal rays
(cf. the proof of Lemma 2.12). Let H be a nef divisor on Z that determines oneof these contractions. Then [13] and [17, Proposition 1.4.13] imply that H varies
in the family Ht of nef divisors on Zt . It follows from the condition rk PicG Z = 1
that both of the extremal contractions on each Zt must be either birational or Morifibrations. Now [11, §12.3] (cf. Remark 3.24) shows that Z can only be a divisorin P2 × P2 of bidegree (2, 2).
Lemma 3.26. Z is smooth.
Proof. Let xi (resp. yi ) be coordinates on the first (resp. second) P2 -factor of
P2 × P2 . Let also f (x, y) = 0 be the equation of Z (so that it defines a conic in P2
whenever x := [x0 : x1 : x2 ] or y is fixed).
Note that projections to the P2 -factors induce conic bundle structures on Z.
­®
These are interchanged by G (because of rk PicG Z = 1) and are h, τ 2 -invariant.
One may assume that Sing Z belongs to the affine chart x0 = y0 = 1 on P2 × P2 .
Then, after a coordinate change, we obtain that f (x, y) = x1 x2 y1 y2 +x1 x2 +y1 y2 +1
in this chart, for h acting diagonally on xi and yi .
Now, differentiating f (x, y) by x1 , x2 we get xi = −y1 y2 , and similarly yi =−x1 x2 . This gives x1 = x2 , y1 = y2 ∈ {−1, −w}, which contradicts f (x, y) = 0.
Lemma 3.26 contradicts |Sing Z| = 10 and Proposition 3.25 follows.

¤¤

Proposition 3.27. rk Pic Z 6= 1.
Proof. Let rk Pic Z = 1. Then we have Zt ⊂ P8 (in the notation from the proof of
Proposition 3.25) are Fano 3-folds of the principal series.
Note that there is a G-invariant surface S ∈ | − KZ |, since P8 = P(C9 ) ⊃ Z,
similarly as in Remark 3.24.
Lemma 3.28. The pair (Z, S) is plt.
Proof. As in the proof of Lemma 3.22, it suffices to exclude the case when (thenormalization of) the surface S is ruled, over a base curve B of genus ≤ 1.
Note that any line L passing through two points from Sing Z is contained in Z (asZ is an intersection of quadrics). In particular, we have S ·L > 0 for > 10 of such L,
which yields either G ⊂ PGL(2, C) or a faithful G-action on B, a contradiction. ¤It follows from Lemma 3.28 that S is normal and connected. Further, we havek ≤ 2 and −KZ · P1 ≤ 2 in (3.15), which means (cf. Lemma 3.21) that the13

exceptional locus of ψ : Y −→ Z consists of two disjoint surfaces, say E1 , E2 , sothat Li := ψ(Ei ) are two lines on Z. In particular, there is a G-invariant subspaceP3 ⊂ P8 , with Z ∩ P3 = L1 ∪ L2 , such that X is obtained from Z via the linear
projection from P3 (recall that both X and Z are anticanonically embedded).
We may assume that Z ∩ P3 ⊂ S (otherwise there is a pencil as in Remark 3.24).
Hence S contains the (−2)-curve L1 (we have identified S with its minimal resolution). Note that L1 is preserved by the group hhi.
Consider the quotient Sh := S/ hhi. Then the image of L1 on Sh has selfintersection = −2/5 by projection formula. On the other hand, this self-intersection∈ Z[0.5] (for Sh has at most canonical singularities due to h∗ (ωSh ) = ωSh ), a
contradiction.
Proposition 3.27 is completely proved.

¤

It follows from Propositions 3.25, 3.27, Remark 3.24, (3.15), (3.16) and[20], [13], [11, §§12.4 – 12.6] that Z is a deformation of either P1 ×[del Pezzo surface of degree 2] or of a double cover of P1 × P1 × P1 , ramified along
a divisor of tridegree (2, 2, 2). In both cases, Z is hyperelliptic (cf. the beginningof the proof of Proposition 3.25), which contradicts Lemma 3.18.
The proof of Theorem 1.4 is finished.

4. Concluding discussion
4.1.

Equations (1.2) and the results of [6] show that any S6 -invariant quartic Xt

is not Q-factorial. In turn, as we saw in Section 2, it is indispensable to compute
the group Cl Xt = H4 (Xt , Z) (e. g. for the arguments of Section 3 to carry on).
This amazing interrelation between topology and (birational) geometry of Xt
provides one with a hint for studying the birational type of Xt by “topological”
means. In this regard, let us give a sketch of an argument, showing that Xt isunirational for generic t ∈ R, hence for (again generic) t ∈ C (cf. [8, Proposition2.3]).
Namely, differentiating (1.2) one interprets this system of equations as the graphof a Morse function F : RP4 −→ R, so that XtR = F −1 (t) are smooth level sets fort 6∈ {∞, 0, 10/7, 2, 4, 6}, while the rest of t 6∈ {0, 4} correspond to critical level setsof (maximal) index 3 (here XtR denotes the real locus of Xt ).
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We may replace RP4 by its double cover S 4 . Then F lifts to a Morse function on
S 4 and thus all smooth XtR are homotopy RP3 . In fact general XtR is diffeomorphic
to RP3 (note that this XtR is smooth and connected).
Further, XtR is contained in an affine space RN , some N , because

Px4i 6= 0

over R. Then the function Fp := dist(·, p) defines a Morse function on XtR for very
general points p ∈ RN . (Here dist(x, y) := kx − yk2 is the standard Euclidean
distance.)
The layers of Fp yield a vector field on XtR , which is non-degenerate and normal
to these layers outside two points, where this field vanishes. We thus obtain a(Hopf) fibration on XtR with a section Fp−1 (o) \ {2 points o1 , o2 } = RP2 such that
Fp−1 (o) ⊂ XtR as an algebraic subset. It remains to apply a diffeomorphism overFp−1 (o) \ {o1 , o2 } which makes XtR \ {Fp−1 (o1 ), Fp−1 (o2 )} = RP1 × Fp−1 (o) \ {o1 , o2 }as algebraic varieties.
The upshot of the above discussion is that XtR (hence Xt ) admits many cancellations in the sense of [2]. This implies that Xt is unirational.
4.2.

We conclude with the following questions:• What is the Fano 3-fold which the quartic X6 is G-birationally isomorphic
to (cf. Section 3)?
• Are there non-trivial G-birational modifications of X6 for other subgroups
G ⊂ S6 ?
• Is Xt unirational over a number field field? 5)• Does the set of Q-points on Xt satisfy the potential density property?
• Does Xt carry a pencil of (birationally) Abelian surfaces? 6)

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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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