K3 SURFACES WITH NON-SYMPLECTIC AUTOMORPHISMS
OF PRIME ORDER
MICHELA ARTEBANI, ALESSANDRA SARTI, SHINGO TAKI
(WITH AN APPENDIX BY SHIGEYUKI KONDŌ)
Abstract. In this note we present the classification of non-symplectic automorphisms of prime order on K3 surfaces, i.e. we describe the topological
structure of their fixed locus and determine the invariant lattice in cohomology. In particular, we provide new results for automorphisms of order 5 and
7 and alternative proofs for higher orders. Moreover, for any prime p, we
identify the irreducible components of the moduli space of K3 surfaces with a
non-symplectic automorphism of order p.
1. Introduction
Let X be an algebraic surface over C. If its canonical line bundle KX is trivial
and dim H 1 (X, OX ) = 0 then X is called a K3 surface. In the following, we denote
by SX , TX and ωX the Picard lattice, the transcendental lattice and a nowhere
vanishing holomorphic 2-form on a K3 surface X, respectively.
An automorphism σ of X is symplectic if it acts trivially on CωX . This paper
deals with non-symplectic automorphisms of prime order p. In this case it is known
that σ acts without non zero fixed vectors on the transcendental lattice, so that TX
aquires the structure of a module over Z[ζp ], where ζp = e2πi/p . This easily implies
that p is at most 19.
Non-symplectic automorphisms have been studied by several authors e.g. in
[14], [2], [24], [15], [16], [17], [9]. A complete classification is known for p =
2, 3, 11, 13, 17, 19. The key idea for this purpose is to characterize the fixed locus of
the automorphism in terms of the invariants of the fixed sublattice of H 2 (X, Z).
In this note we classify the orders p = 5, 7, we survey the known results and, for
orders 11, 13, 17, 19, we provide different proofs and examples. The main results in
the paper are the description of the fixed locus of a non-symplectic automorphism
of prime order 5 ≤ p ≤ 19 and
Theorem. Let X be a K3 surface with p-elementary Picard lattice SX , where p
is a prime. Let r be the rank of SX and a be the minimal number of generators
∗
/SX . Then X has a non-symplectic automorphism σ of order p which acts
of SX
trivially on SX if and only if
22 − r − (p − 1)a ∈ 2(p − 1)Z≥0
Date: March 13, 2009.
2000 Mathematics Subject Classification. Primary 14J28; Secondary 14J50, 14J10.
Key words and phrases. K3 surface, non-symplectic automorphism, lattice.
1
2
M. Artebani, A. Sarti, S. Taki
In this case the fixed locus Fix(σ) := {x ∈ X : σ(x) = x} has the form


if SX = U (2) ⊕ E8 (2),
∅
Fix(σ) = C (1) ∐ C (1)
if SX = U ⊕ E8 (2),


(g)
{P1 , . . . , Pn } ∐ C ∐ E1 ∐ · · · ∐ EN −1 otherwise,
where Pj is an isolated point, C (g) is a smooth curve of genus g, Ek is a smooth
rational curve and
g=


0



 (p − 2)r + 22
n=
p−1


(p − 2)r − 2



p−1
22 − r − (p − 1)a
,
2(p − 1)
if p = 2,
if p = 17, 19 ,
otherwise
r − a + 2




2
N= 0


2p + r − (p − 1)a


2(p − 1)
if p = 2
if p = 17, 19
otherwise.
As a consequence of this result, we determine the maximal components of the
moduli space of K3 surfaces with a non-symplectic automorphisms of order p for
any prime p. Moreover, we show that the topological structure of the fixed locus
of σ determines uniquely the action of σ on H 2 (X, Z).
The plan of the paper is the following.
Section 2 introduces some background material on lattices. In section 3 we recall
some basic properties of non-symplectic automorphisms and their fixed locus. As
a first result (Theorem 3.4), we determine the number of isolated fixed points (and
the local action at them), as a function of the rank of the invariant lattice.
Sections 4 and 5 briefly review the known results for p = 2 and 3 respectively.
Figures 1 and 2 represent all possible configurations of the fixed locus, related to
the invariants of the invariant lattice.
In section 6 we classify non-symplectic automorphisms of order 5. The classification theorem is resumed in Table 6.4. The topological structure of the fixed
locus gives a natural stratification in 7 families. Two of them, of dimensions 3 and
4, are the maximal irreducible components of the moduli space (as we will show in
section 10). We provide projective models for the generic member of each family.
In section 7 we give a similar classification and description for p = 7, see Table
7.3. In this case there are two maximal components of dimension 2.
In section 8 and 9 we provide an alternative view of the classification for p =
11, 13, 17, 19. In case p = 11 it is known that the moduli space has two maximal
1-dimensional components, while the pair (X, σ) is unique for p > 11.
In section 10 we deal with moduli spaces. First, we recall the structure of the
moduli space of pairs (X, σ) where X is a K3 surface and σ ∈ Aut(X) is nonsymplectic of order p with a given representation in H 2 (X, Z) ∼
= LK3 . This space
is known to be isomorphic to the quotient of an open dense subset of either an
Hermitian symmetric domain of type IV (for p = 2) or of a complex ball (for
p > 2). For any prime p we identify the irreducible components of the moduli space
of K3 surfaces with a non-symplectic automorphism of order p.
Non-symplectic automorphisms on K3 surfaces
3
In the appendix, S. Kondō shows that the moduli space of pairs (X, σ), where
σ is non-symplectic of order 7 having only isolated fixed points, is a ball quotient
isomorphic to the Naruki K3 surface. A similar example for p = 5 is given in [7].
2. Lattices
A lattice is a finitely generated free abelian group equipped with a non-degenerate
symmetric quadratic form with integer values. If the signature of the lattice is
(1, r − 1) then it is called hyperbolic. We will work with even lattices i.e. such that
the quadratic form takes values in 2Z.
The quadratic form on L determines a canonical embedding L ⊂ L∗ = Hom(L, Z).
We denote by AL the factor group L∗ /L which is a finite abelian group.
Let p be a prime number. A lattice L is called p-elementary if AL ≃ (Z/pZ)a . If
a = 0 the lattice is called unimodular. All p-elementary even lattices are classified
by the following result.
Theorem 2.1 ([20]). An even, indefinite, p-elementary lattice of rank r for p 6= 2
and r ≥ 2 is uniquely determined by the integer a.
For p 6= 2 a hyperbolic lattice with invariants a, r exists if and only if the following
conditions are satisfied: a ≤ r, r ≡ 0 (mod 2) and
(
for a ≡ 0 (mod 2),
for a ≡ 1 (mod 2),
r ≡ 2 (mod 4)
p ≡ (−1)r/2−1 (mod 4).
Moreover r > a > 0, if r 6≡ 2 (mod 8).
If p = 2 an even indefinite lattice is determined by r, a and a third invariant
δ ∈ {0, 1}, see [14].
Notation. We will denote by U the unique even unimodular lattice of signature
(1, 1) and by Am , Dn , El the even, negative definite lattices associated with the
Dynkin diagrams of the corresponding types (m ≥ 1, n ≥ 4, l = 6, 7, 8). Moreover,
L(m) denotes the lattice whose bilinear form is the one on L multiplied by m.
Examples.
− The lattices U and E8 are unimodular. Any even unimodular lattice of signature
(3, 19) is isometric to LK3 = U ⊕3 ⊕ E8⊕2 ([11], [21]).
− If p is prime, the lattice Ap−1 is p-elementary.
− The lattice E7 is 2-elementary with a = 1.
− If p ≡ 3 ( mod 4), then the lattice
−(p + 1)/2 1
Kp =
1
−2
is negative definite, p-elementary, with a = 1. Note that K3 ∼
= A2 .
− If p ≡ 1 ( mod 4) then the lattice
(p − 1)/2 1
Hp =
1
−2
is hyperbolic, p-elementary, with a = 1.
4
M. Artebani, A. Sarti, S. Taki
− The lattice

−4
1
1
 1 −4
1
∗
A4 (5) = 
 1
1 −4
1
1
1
is negative definite, 5-elementary with a = 3.
− The lattice

1
1 

1 
−4

−2
1
0
1
 1 −2
0
0 

L17 = 
 0
0 −2
1 
1
0
1 −4
is negative definite, 17-elementary with a = 1.

3. Non-symplectic automorphisms on K3’s
Let X be a K3 surface i.e. a simply connected smooth complex projective surface with a nowhere vanishing holomorphic 2-form ωX . The cohomology group
H 2 (X, Z), equipped with the cup product, is known to be an unimodular lattice
isometric to LK3 . The Picard lattice (or Néron-Severi group) SX and the transcendental lattice TX are the following sublattices of LK3
SX = {x ∈ H 2 (X, Z) : (x, ωX ) = 0},
⊥
TX = SX
.
An automorphism σ of X is called non-symplectic if its action on the vector space
H 2,0 (X) = CωX is not trivial. Observe that by [12, Theorem 3.1] K3 surfaces with
a non-symplectic automorphism of finite order are always algebraic. In this paper
we are interested in non-symplectic automorphisms of prime order i.e.
σ p = id
and
σ ∗ (ωX ) = ζp ωX ,
where ζp is a primitive p-th root of unity.
The automorphism σ induces an isometry σ ∗ on H 2 (X, Z) which preserves both
SX and TX . We will consider the invariant lattice and its orthogonal
S(σ) = {x ∈ H 2 (X, Z) : σ ∗ (x) = x}, T (σ) = S(σ)⊥ .
Theorem 3.1. Let X be a K3 surface and σ be a non-symplectic automorphism of
X of prime order p. Then
a) S(σ) ⊂ SX and TX ⊂ T (σ).
b) T (σ) and TX are free modules over Z[ζp ] via the action of σ ∗ .
c) S(σ) and T (σ) are p-elementary lattices with a ≤ rank(T (σ))/(p − 1).
Proof. The statements a), b) and the first claim in c) are proved in [12] or [9]. For
the inequality in c), we will generalize the proof of Claim 3.4 in [9] as follows. By
point b), T (σ) ∼
= Z[ζp ]m as a Z[ζp ]-module. Let e1 , . . . , em be a basis of T (σ) over
Z[ζp ] and
{bij : i = 1, . . . , m, j = 0, . . . , p − 2}
be the corresponding Z basis of T (σ). Since T (σ) is p-elementary, then any y ∈
T (σ)∗ is of the form
1X
y=
yij bij .
p
Non-symplectic automorphisms on K3 surfaces
5
Moreover, since σ ∗ = id on AT (σ) , then modulo T (σ)
0 ≡ σ ∗ (y) − y =
p−2
m p−3
X
1X X
(
yij bi j+1 − yi p−2 (bi0 + · · · + bi p−2 ) −
yij bij ).
p i=1 j=0
j=0
From the vanishing of the coefficients of the bij ’s one deduces that
p−1
X
j
yij bij = yi1 [
1X
k(
p
k=1
pk
X
bij )] = yi1 Bi .
j=p(k−1)+1
This implies that AT is generated by B1 , . . . , Bm , hence a ≤ m.
Let σ be an order p non-symplectic automorphism of a K3 surface and Fix(σ)
be its fixed locus. The action of σ can be locally linearized and diagonalized at a
fixed point x ∈ Fix(σ) (see §5, [12]). Since σ acts on ωX as the multiplication by
ζp , its possible local actions are
k+1
0
ζp
Ap,k =
, k = 0, . . . , p − 2.
0
ζpp−k
If k = 0 then x belongs to a smooth fixed curve for σ, otherwise x is an isolated
fixed point. Thus, the fixed locus of σ is the disjoint union of smooth curves and
isolated points.
In what follows, we will denote by n the number of isolated fixed points of σ.
Moreover, we will say that x is of type k if the local action at x is given by Ap,k
and we will denote by nk the number of isolated points of σ of type k.
Lemma 3.2. If σ has order p > 3 and X admits a σ-stable elliptic fibration, then a
curve in the fixed locus of σ is either rational or elliptic. In particular this happens
if rank S(σ) ≥ 5.
Proof. Assume that σ has a fixed curve C of genus > 1, then C is transversal to
any fiber of the elliptic fibration. Hence any fiber has an order p > 3 automorphism
with a fixed point. This is a contradiction, since there exists no elliptic curve with
such automorphism. If rank S(σ) ≥ 5, then X has a σ-stable elliptic fibration since
S(σ) ⊂ SX represents zero (e.g [9, Lemma 1.5], [19, Corollary 3, §3]).
Proposition 3.3. The Euler characteristic of the fixed locus of an order p nonsymplectic automorphism σ is given by
p(r + 2) − 24
χ(Fix(σ)) =
,
p−1
where r is the rank of S(σ).
Proof. By the topological Lefschetz formula:
χ(Fix(σ)) =
4
X
i=0
(−1)i tr(σ ∗ |H i (X, R)).
We can calculate the right-hand side as follows:
4
X
i=0
(−1)i tr(σ ∗ |H i (X, R)) = 2 + tr(σ ∗ |S(σ)) + tr(σ ∗ |T (σ)) =
p(r + 2) − 24
.
p−1
6
M. Artebani, A. Sarti, S. Taki
If C is a σ-fixed curve, let X (C) be its Euler characteristic and let
1 X
X (C).
α=
2
C⊂Fix(σ)
Theorem 3.4. Let σ be a non-symplectic automorphism of prime order p of a K3
surface and r be the rank of S(σ). Then the types of the isolated fixed points of σ
and the invariant α are functions of r as given in Table 1.
p
α
2
r−10
3
5
7
11
13
17
19
r−8
2
r−6
4
r−4
6
r−2
10
r+2
12
r−6
16
r−4
18
n1
n2
n3
n4
n5
n6
n7
n8
n9
n
0
α+3
α+3
2α+3
1+α
3α+4
2α+2
1+2α
α
1+2α
2α
2α
1+2α
α
1+2α
1+2α
2α
2α−1
2α−2
α−1
2α
2α
2α
2α
2α+1
2α+2
2α+3
α+1
2α
2α
2α
2α+1
2α+2
2α+1
2α+1
2α
5α+3
9α+2
11α−2
15α+7
α
17α+5
Table 1. Isolated fixed points
Proof. The holomorphic Lefschetz formula allows to compute the holomorphic Lefschetz number L(σ) in two ways. First we have that
L(σ) =
2
X
i=0
tr(σ ∗ |H i (X, OX )).
By Serre duality H 2 (X, OX ) ≃ H 0 (X, OX (KX ))∨ , this imples
L(σ) = 1 + ζpp−1 .
(1)
Moreover we have
L(σ) =
p−2
X
k=0
where
(2)
a(k) =
nk a(k) +
X
cg b(g)
g
1
1
1
=
=
,
∗
k
det(I − σ |Tk )
det(I − Ap,k )
(1 − ζ )(1 − ζ p−k+1 )
where Tk is the tangent space of X at a point of type k;
(3)
b(g) =
1−g
(1 + ζp )(1 − g)
ζp (2g − 2)
=
,
−
1 − ζp
(1 − ζp )2
(1 − ζp )2
where cg is the number of curves of genus g in the fixed locus.
Non-symplectic automorphisms on K3 surfaces
7
P
Note that α = g cg (1 − g). Combining (1), (2) and (3) we get the values of the
types appearing in Table 1.
Since χ(Fix(σ)) = n + 2α we get the values of α in Table 1 by applying the
formula in Proposition 3.3.
4. Order 2
We briefly recall the classification theorem of non-symplectic involutions [14].
The local action of a non-symplectic involution at a fixed point is of type
1 0
A2,0 =
,
0 −1
so that Fix(σ2 ) is the disjoint union of smooth curves and there are no isolated
fixed points. The lattice S(σ2 ) is 2-elementary thus, according to Theorem 2.1, its
isomorphism class is determined by the invariants r, a and δ.
a
•δ=1
∗δ=0
0
11
10
2 *
8
4
6
8
2 10 *
g
0
*
2
4
4
*
6
4
2
6
*
*
*
*
*
* 8
*
*
*
*
6
*
*
10 12 14 16 18 20
8
k−1
r
Figure 1. Order 2
Theorem 4.1 (Theorem 4.2.2, [14]). The fixed locus of a non-symplectic involution
on a K3 surface is
i) empty if r = 10, a = 10 and δ = 0,
ii) the disjoint union of two elliptic curves if r = 10, a = 8 and δ = 0,
iii) the disjoint union of a curve of genus g and k rational curves otherwise, where
g = (22 − r − a)/2,
k = (r − a)/2.
All possible values of the triple (r, a, δ) and the corresponding invariants (g, k)
of the fixed locus are represented in Figure 1. In particular 0 ≤ g ≤ 10, 0 ≤ k ≤ 9
and both can take any value in the interval.
In [25] D.-Q. Zhang classified the surfaces which can arise as quotients of K3
surfaces modulo non-symplectic involutions.
8
M. Artebani, A. Sarti, S. Taki
5. Order 3
Non-symplectic automorphisms of order 3 have been recently classified in [2] and
[24]. In this case the local action at a fixed point is of one of the following
2
ζ3 0
ζ3 0
A3,1 =
,
A
=
,
3,0
0 ζ32
0 1
so that the fixed locus contains both fixed curves and isolated points. Let (r, a) be
the invariants of S(σ).
Theorem 5.1 (Theorem 1.2, [24]; Table 1, [2]). Let σ be a non-symplectic automorphism of order 3 of a K3 surface. Then its fixed locus is the disjoint union of
n points, a smooth curve of genus g and k smooth rational curves, where
n = r/2 − 1,
g = (22 − r − 2a)/4,
k = (2 + r − 2a)/4.
a
All possible values of the triple (r, a) and the corresponding invariants (g, k, n) of
the fixed locus are represented in Figure 2.
0
7
6
0
2
2
4
4
4
2
k
0
5
6
8
6
4
2
0
2
4
6
8
10
g
n
m
Figure 2. Order 3
Remark 5.2. We give here another construction for the surfaces with n = 3, k = 1,
g = 1 and n = 3, k = 0 (cf. [2, Proposition 4.7]). This construction is used again
in the Examples 6.7, 6.9, 7.1, 7.2, 8.1, 8.2. Consider the elliptic K3 surface Xai ,bi
y 2 = x3 + (t6 + a1 t3 + a2 ) + (t12 + b1 t9 + b2 t6 + b3 t3 + b4 )
the discriminant of the fibration is
∆(t) = 4(t6 + a1 t3 + a2 )3 + 27(t12 + b1 t9 + b2 t6 + b3 t3 + b4 )2 .
For generic ai , bi ∈ C the fibrations has 24 fibers of type I1 over the zeros of ∆(t)
(divided in 8 σ-orbits). The authomorphis of order 3
σ(x, y, t) = (x, y, ζ3 t)
acts non trivially on the basis of the fibration and fixes two smooth elliptic curves
over t = 0 and t = ∞. Since an automorphism of order three on an elliptic curve
either leaves the curve pointwise fixed or has three fixed points we see that σ is
non-symplectic and the only possible case is n = 3, k = 1, g = 1. The quotient
Non-symplectic automorphisms on K3 surfaces
9
surface by σ is a rational elliptic surface πai ,bi : Yai ,bi −→ P1 with Weierstrass
equation
y 2 = x3 + (t2 + a1 t + a2 ) + (t4 + b1 t3 + b2 t2 + b3 t + b4 )
this fibration has one singular fiber of type IV over t = ∞ and 8 fibers I1 over the
zeros of ∆(t) = 4(t2 + a1 t + a2 )3 + (t4 + b1 t3 + b2 t2 + b3 t + b4 )2 . Let p : P −→ P1 be a
non-trivial principal homogeneous space of πai ,bi given by an order 3 element in the
(0) ([4, Ch.V, §4]), then p : P −→ P1 is a rational elliptic surface with a
fiber πa−1
i ,bi
multiple smooth elliptic fiber of multiplicity 3 over 0 (for generic ai ’s and bi ’s). Let
Z be the surface obtained by blowing up the intersection point of the three rational
curves of the fiber p−1 (∞) and then blowing down the three strict transforms of
the three rational curves. Then Z is a log Enriques surface of index 3 with three
singular points. Let X be the canonical cover of Z and σ̄ be a generator of the
covering transformation group of the cover. By [27, Theorem 5.1], since Z has three
singular points of type (3, 1) (these are Hirzebruch-Jung singularities, cf. [3, Section
5]) then X is a K3 surface and σ̄ is an order 3 non-symplectic automorphism of X
with only three fixed points.
The relations between the Examples 6.7, 7.1, 8.1 and the Examples 6.9, 7.2, 8.2 is
the same as here.
6. Order 5
An order five non-symplectic automorphism of a K3 surface has three possible
local actions at a fixed point
A5,0 =
ζ5
0
0
1
, A5,2 =
ζ53
0
0
ζ53
, A5,1 =
ζ52
0
0
ζ54
.
Thus the fixed locus can contain both fixed curves and isolated points of two different types. We start providing two families of examples.
Example 6.1. Let A be the family of plane sextic curves defined by
x0 (x0 − x1 )
4
Y
i=1
(x0 − λi x1 ) + x52 x1 = 0,
where λi ∈ C. Observe that the projective transformation
(4)
σ̄(x0 , x1 , x2 ) = (x0 , x1 , ζ5 x2 )
preserves any curve in A. If C ∈ A is smooth, then the double cover X of P2
branched along C is a K3 surface and σ̄ induces a non-symplectic automorphism
σ of order 5 on X. Since Fix(σ̄) = {(0, 0, 1)} ∪ {x2 = 0}, then Fix(σ) is the union
of an isolated fixed point and a smooth curve of genus two.
If the complex numbers λi ’s are distinct, then the corresponding sextic C ∈ A is
smooth and has six fixed points on the line x2 = 0. Otherwise, if two or three of
the λi ’s coincide, then C has a singular point of type A4 or E8 respectively.
10
M. Artebani, A. Sarti, S. Taki
In particular we have the following cases
Q4
a) x20 (x0 − x1 ) i=2 (x0 − λi x1 ) + x52 x1 = 0
b)
c)
d)
e)
A4
x30 (x0 − x1 )(x0 − λ3 x1 )(x0 − λ4 x1 ) + x52 x1 = 0
E8
x30 (x0 − x1 )2 (x0 − λ4 x1 ) + x52 x1 = 0
A4 ⊕ E8
x20 (x0 − x1 )2 (x0 − λ3 x1 )(x0 − λ4 x1 ) + x52 x1 = 0 A24
x30 (x0
3
− x1 ) +
x52 x1
=0
E82 .
Remark 6.2. Observe that the line x1 = 0 intersects any sextic of A in the isolated
fixed point of σ̄ with multiplicity 6. Thus, the inverse image of the line in X is the
union of two smooth rational curves intersecting in one point with multiplicity 3.
Example 6.3. Let B be the family of plane sextic curves defined by
a1 x60 + a2 x30 x1 x22 + a3 x20 x31 x2 + x0 (a4 x51 + a5 x52 ) + a6 x21 x42 = 0.
The projective transformation
(5)
σ̄(x0 , x1 , x2 ) = (x0 , ζ5 x1 , ζ52 x2 )
preserves any curve in B. If C ∈ B is smooth, then the double cover X of P2
branched along C is a K3 surface and σ̄ induces a non-symplectic automorphism σ
of order 5 on X. In fact it does not leave invariant the holomorphic 2-form (written
in local coordinates)
dx ∧ dy
√
f
where f denotes the equation above in local coordinates. Since Fix(σ̄) = {(1, 0, 0)}∪
{(0, 1, 0)} ∪ {(0, 0, 1)} and all but one of them belong to C then Fix(σ) is the union
of 4 isolated points.
Observe that the line x0 = 0 intersects any sextic of B in two fixed points, with
multiplicities 2 and 4.
Theorem 6.4. Let σ be a non-symplectic automorphism of order 5 of a K3 surface.
The fixed locus of σ is the disjoint union of ni points of type i and kj curves of
genus j, where (ni , kj ) appears in a row of Table 2.
The same table gives the isomorphism class of the corresponding fixed lattice
S(σ) and its orthogonal complement T (σ).
Proof. In what follows we will determine all possible lattices S = S(σ) and T =
T (σ). Let a be the minimal number of generators of AS ∼
= AT . We start excluding
some cases.
i) The case a = 0 can be excluded by [6] (if T is unimodular then the order has
to divide 66, 44, 42, 36, 28 or 12).
ii) If r(T ) = 12, a = 2 then X has an elliptic fibration with 2 fibers of type I5 .
Moreover, by Proposition 3.4, k0 = 1, n1 = 5 and n2 = 2. Since the fixed
points are not enough to fix the two reducible fibers, this case does not appear.
iii) The case r(T ) = 16, a = 1, 3 can be excluded in a similar way.
iv) The case r(T ) = 20, a = 2 can be excluded because X (Fix(σ)) = −1 by
Proposition 3.4.
The list of lattices in Table 2 can be then obtained by using Theorem 2.1 and Theorem 3.1. We now determine the number and genus of the fixed curves for each
Non-symplectic automorphisms on K3 surfaces
n1
1
3
3
5
5
7
9
n2
0
1
1
2
2
3
4
k0
0
0
0
1
1
2
3
k1
0
1
0
1
0
0
0
k2
1
0
0
0
0
0
0
T (σ)
H 5 ⊕ U ⊕ E 8 ⊕ E8
H5 ⊕ U ⊕ E8 ⊕ A4
H5 ⊕ U (5) ⊕ E8 ⊕ A4
U ⊕ H 5 ⊕ E8
U ⊕ H5 ⊕ A24
U ⊕ H5 ⊕ A4
U ⊕ H5
11
S(σ)
H5
H5 ⊕ A4
H5 ⊕ A∗4 (5)
H5 ⊕ E 8
H5 ⊕ A24
H5 ⊕ E8 ⊕ A4
H5 ⊕ E8 ⊕ E8
Table 2. Order 5
lattice S in the table.
S = H5 Let h be the generator of H5 with h2 = 2. It can be seen that h is
ample and without base locus (see Lemma 3.5, [9]). Hence it defines a double cover
branched along a smooth plane sextic B. The action of σ on X induces a projectivity σ̄ of P2 . Since X (Fix(σ)) = −1 by Theorem 3.4, then the fixed locus contains a
curve of genus g > 1. This implies that σ̄ has a curve in the fixed locus, hence for
a suitable choice of coordinates it is of the form 4. So, we are in Example 6.1 and
the fixed locus is the union of a smooth genus two curve and one point.
Since H5 ⊆ S for each lattice S in the table, by the remark above each K3 surface has a projective model as a double cover of the plane ramified along a sextic
B (eventually singular) and σ induces a projective automorphism σ̄ of P2 . In the
following cases rank(S) ≥ 6, hence by Lemma 3.2 the curves in the fixed locus of
σ have genus at most 1. Note that any (−2)-curve is preserved by σ and contains
exactly two fixed points.
S = H5 ⊕ A4 In this case B has a singular point p of type A4 . By the previous
remark σ has 5 fixed points on the exceptional divisor over p. By Theorem 3.4
n = 4 and X (Fix(σ)) = 0, hence k0 = 0 and the fixed locus contains a fixed curve.
As before, this implies that σ̄ is of type (4). This implies that we are in Example
6.1, a). Hence Fix(σ) is the union of a smooth elliptic curve (the proper transform
of the line x2 = 0) and 4 isolated points.
S = H5 ⊕ E8 In this case the curve B has an E8 singularity. Observe that E8
contains a fixed rational curve. As before, this implies that we are in Example 6.1,
b) and Fix(σ) is the union of a smooth rational curve (the proper transform of the
line x2 = 0) and 7 isolated points.
The cases H5 ⊕ A24 , H5 ⊕ A4 ⊕ E8 and H5 ⊕ E82 can be discussed in a similar way
and correspond to Example 6.1 c), d) and e) respectively.
S = H5 ⊕ A∗4 (5) Since A∗4 (5) does not contain (−2)-curves (see the description of
A∗4 in [23]) the sextic B is smooth. Since the fixed locus does not contain curves of
genus > 1 and B is smooth, then the automorphism σ̄ has at most isolated fixed
12
M. Artebani, A. Sarti, S. Taki
points. This implies that, for a suitable choice of coordinates, it is of the form (5).
Hence we are in Example 6.3 and Fix(σ) is the union of 4 isolated points.
Remark 6.5. In [17] K. Oguiso and D-Q. Zhang showed the uniqueness of a K3
surface with a non-symplectic automorphism of order five and fixed locus containing
no curves of genus ≥ 2 and at least 3 rational curves. In their approach they use
log Enriques surfaces.
Remark 6.6. In [7, §3.1, 3.2] and in [8, Remark 6] S. Kondō considers the minimal
resolution X of the double cover of P2 branched along the union of the line x2 = 0
and the plane quintic curve
x52
=
5
Y
i=1
(x0 − λi x1 ),
where the λi ’s are distinct complex numbers. Then X is a K3 surface with an
automorphism σ induced from σ̄ as in (4) with n = 7, k0 = 1, k1 = 0 and T =
U ⊕H5 ⊕A⊕2
4 as computed in [5, Section 12, p. 53]. Since this family has dimension
2, this gives a different model for the family of K3 surfaces in Example 6.1, c).
In [6, §7,(7.6)] appears the following elliptic K3 surface with order 5 automorphism:
y 2 = x3 + t3 x + t7 , σ(x, y, t) = (ζ53 x, ζ52 y, ζ52 t).
Here the fixed locus has the invariants n = 13, k0 = 3 and k1 = 0, hence this gives
a different model for Example 6.1, e).
Example 6.7. Let Xα,β,γ be an elliptic K3 surface with Weierstrass equation
y 2 = x3 + (t5 + α)x + (βt10 + t5 + γ)
the discriminant of the fibration is
∆(t) = 4(t5 + α)3 + 27(βt10 + t5 + γ)2 .
For generic α, β and γ the fibration has a fiber of type IV on t = ∞ and 20 fibers
of type I1 over the zeros of ∆. Observe that the fibration has an automorphism of
order 5
σ(x, y, t) = (x, y, ζ5 t).
This automorphism acts non trivially on the basis of the fibration and has two fixed
points over t = 0 and t = ∞, hence it fixes the reducible fiber IV and a smooth
elliptic fiber. This implies that the automorphism is non-symplectic. Checking in
Table 2 one can see that this K3 surface is in the family with n1 = 3, n2 = 1, k1 = 1
and in fact it is easy to see that this is an equation of the generic K3 surface in
that family. Moreover observe that
1) If β = 0 then the fibration has a singular fiber of type III ∗ over t = ∞ and 15
fibers I1 . This corresponds to the case n1 = 5, n2 = 2, k0 = 1, k1 = 1 in Table 2.
2) If α3 = −27/4γ 2 then the fibration has a singular fiber of type IV over t = ∞
and of type I5 over t = 0. This corresponds to n1 = 5, n2 = 2, k0 = 1.
3) If α = γ = 0 then we have a fiber of type IV over t = ∞ and a fiber of type II ∗
over t = 0. This corresponds to n1 = 7, n2 = 3, k0 = 2.
4) If α = β = γ = 0 then we have a fiber of type III ∗ over t = ∞ and a fiber of
type II ∗ over t = 0. This corresponds to n1 = 9, n2 = 4, k0 = 3.
Non-symplectic automorphisms on K3 surfaces
13
We recall that a log-Enriques surface is a normal algebraic surface Y having
at most quotient singularities such that h1 (Y, OY ) = 0 and mKY = OY for some
positive integer m. The index of Y is the smallest positive m with this property.
The canonical covering of q : Ỹ −→ Y (i.e. that induced by the relation mKY =
OY ) is a cyclic cover of order m, étale over the smooth points of Y . The minimal
resolution of Ỹ is known to be either a K3 surface or an abelian surface ([27]).
Remark 6.8. The general theory of log Enriques surfaces has been developed in
[26], [27], [28]. Moreover, log Enriques surfaces of index 2 have been studied in [25],
of index 11 in [15], 13, 17, 19 in [16] and 5 in [17].
Example 6.9. Let πa,b,c : Ya,b,c −→ P1 be the rational jacobian elliptic surface
with Weierstrass equation
y 2 = x3 + tx + (at2 + bt + c), a, b, c ∈ C.
For generic a, b, c ∈ C the elliptic fibration has a fiber of Kodaira type IV ∗ at t = ∞
and 4 fibers of type I1 at the zeros of ∆(t) = 4t3 + 27(at2 + bt + c)2 . Let p : P → P1
be a non-trivial principal homogeneous space of πa,b,c given by an order 5 element
−1
(0) ([4, Ch.V, §4]), then p : P −→ P1 is a rational elliptic surface
in the fiber πa,b,c
with a multiple smooth elliptic fiber of multiplicity 5 over 0 (for generic a, b, c). Let
Z be the surface obtained by blowing up the intersection points of the component
of multiplicity 3 of p−1 (∞) with those of multiplicity 2 and then blowing down
the four connected components of the proper transform of p−1 (∞). Then an easy
computation shows that Z is a log Enriques surface of index 5 with 4 singular points
(the images of the components of p−1 (∞)). Let X be the canonical cover of Z and
σ be a generator of the covering transformation group of the cover. By Theorem
5.1 in [27], since Z has three singular points of type (5, 2) and one of type (5, 1),
then X is a K3 surface. Moreover, σ is an order 5 non-symplectic automorphism
of X with only isolated fixed points.
7. Order 7
The local actions of an order 7 non-symplectic automorphism at a fixed point
are of four types
2
3
4
1 0
ζ7 0
ζ7 0
ζ7 0
A0 =
, A1 =
,
A
=
,
A
=
.
2
3
0 ζ7
0 ζ76
0 ζ75
0 ζ74
Example 7.1. Let Xa,b be an elliptic surface with Weierstrass equation
y 2 = x3 + (at7 + b)x + (t7 − 1).
For generic a, b ∈ C the fibration has one fiber of Kodaira type III over t = ∞ and
21 fibers of type I1 over the zeros of
∆(t) = 4(at7 + b)3 + 27(t7 − 1)2 ,
hence it is a K3 surface (see [10, Table (IV.3.1)]). Moreover, X carries the order 7
automorphism
(6)
σ(x, y, t) = (x, y, ζ7 t).
Observe that
1) X0,b has one singular fiber of type II ∗ over t = ∞ and 14 singular fibers of type
I1 ;
14
M. Artebani, A. Sarti, S. Taki
n1
2
2
4
4
6
n2
1
1
3
3
5
n3
0
0
1
1
2
k0
0
0
1
1
2
k1
1
0
1
0
0
T (σ)
U ⊕ U ⊕ E8 ⊕ A6
U (7) ⊕ U ⊕ E8 ⊕ A6
U ⊕ U ⊕ E8
U (7) ⊕ U ⊕ E8
U ⊕ U ⊕ K7
S(σ)
U ⊕ K7
U (7) ⊕ K7
U ⊕ E8
U (7) ⊕ E8
U ⊕ E8 ⊕ A6
Table 3. Order 7
2) if b3
type I7
3) if b3
type I7
= −27/4, then Xa,b has one singular fiber of type III over t = ∞, one of
over t = 0 and 14 fibers of type I1 ;
= −27/4, then X0,b has one singular fiber of type II ∗ over t = ∞, one of
over t = 0 and 7 of type I1 .
Example 7.2. Let πa,b : Ya,b −→ P1 be the rational jacobian elliptic surface with
Weierstrass equation
y 2 = x3 + tx + (at + b), a, b ∈ C.
For generic a, b ∈ C the elliptic fibration has a fiber of Kodaira type III ∗ at t = ∞
and 3 fibers of type I1 at the zeros of ∆(t) = 4t3 +27(at+b)2 . To this rational surface
we can associate a K3 surface with an order seven non-symplectic automorphism
having zero-dimensional fixed locus. The construction is similar to that described in
Example 6.9. Let p : P → P1 be a non-trivial principal homogeneous space of πa,b
−1
given by an order 7 element in the fiber πa,b
(0). The surface Z obtained by blowing
up the intersection points of the component of multiplicity 4 of p−1 (∞) with those
of multiplicity 3 and then blowing down the three connected components of the
proper transform of p−1 (∞) is a log Enriques surface of index 7 with 3 singular
points (the images of the components of p−1 (∞)). By Theorem 5.2 in [27] the
canonical cover of Z is a K3 surface and a generator of the covering transformation
group is an order 7 non-symplectic automorphism with only isolated fixed points.
Let (r, a) be the invariants of S(σ).
Theorem 7.3. Let σ be a non-symplectic automorphism of order 7 of a K3 surface.
The fixed locus of σ is the disjoint union of ni points and kj curves of genus j where
(ni , kj ) appears in the first row of Table 3.
The same table gives the corresponding fixed lattice S(σ) and its orthogonal complement T (σ).
Proof. The lattices in Table 3 can be found by means of Theorem 2.1 and Theorem
3.1. Observe that each lattice S = S(σ) in the table contains a copy of either U or
U (7). This implies that the generic K3 surface with Picard lattice isomorphic to S
has an elliptic fibration. By Lemma 3.2 it follows that a curve in the fixed locus of
σ has genus ≤ 1.
S = U ⊕ K7 The generic K3 surface with Picard lattice S has a jacobian elliptic
fibration with a unique reducible fiber of type Ã1 (since K7 contains a unique
(−2)-curve, as can be checked directly). According to Theorem 3.4, σ has exactly
3 isolated fixed points and X (Fix(σ)) = 3. Hence σ can have only elliptic fixed
Non-symplectic automorphisms on K3 surfaces
15
curves. Moreover, the action of σ on the basis of the elliptic pencil is not trivial,
since otherwise a smooth fiber would have an order 7 automorphism with a fixed
point in the intersection with the section.
It follows that the fibration has exactly two σ-invariant fibers. The first one is
smooth and belongs to the fixed locus. The second one is the reducible fiber and
has Kodaira type III (if it were of type I2 it should contain a fixed rational curve).
Hence k0 = 0 and k1 = 1.
Since the generic such surface has a jacobian elliptic fibration with an order 7
automorphism acting non trivially on the basis, we can write it in Weierstrass form
y 2 = x3 + f (t)x + g(t),
where σ acts as (x, y, t) 7→ (x, y, ζ7 t). The polynomials f (t), g(t) are invariant for
this action and deg(f ) ≤ 8, deg(g) ≤ 12, hence for a proper choice of coordinates
we can assume f (t) = (at7 + b)tm and g(t) = (ct7 − d)tn where m ≤ 1, n ≤ 5. Since
the discriminant divisor is also invariant and there are exactly 21 fibers of type I1
in P1 \{∞}, then ∆(t) = δ(t7 − α)(t7 − β)(t7 − γ). Looking at the coefficients of
the equality ∆(t) = 4f (t)3 + 27g(t)2 we can deduce that m = n = 0, hence the
Weierstrass equation of the K3 surface is of type
y 2 = x3 + (at7 + b)x + (ct7 − d).
After applying a suitable change of coordinates we obtain a surface in the family
of Example 7.1.
Similar remarks give k0 and k1 for S = U ⊕ E8 , U ⊕ K7 ⊕ A6 and U ⊕ E8 ⊕ A6 .
As before, it can be proved that the generic K3 surfaces in each case belong to the
families a), b) and c) respectively in Example 7.1.
S = U (7) ⊕ K7 We start proving that the generic K3 surface X with SX ∼
= S ad1
mits two σ-invariant elliptic fibrations πi : X → P with generic fibers intersecting
in 7 points. Let e, f denote the generators of U (7), then by Riemann-Roch theorem
we can suppose them to be effective. If they are nef then we are done by [19, §3].
Recall that, on a K3 surface, a divisor is nef iff its intersection with any (-2)-curve
is non-negative. If there exists a (-2)-curve p with (e, p) < 0 and (f, p) < 0, then
the image of e, f by the reflection sp (x) = x+(x, p)p both have positive intersection
with p.
Hence we consider the case when (e, p) < 0 and (f, p) ≥ 0. We write the curve p
in the form p = ae + bf + g where g ∈ K7 . Since (e, p) < 0 and (f, p) ≥ 0 we have
b < 0 and a ≥ 0. Now p2 = 14ab + g 2 ≤ −16 since g 2 ≤ −2, which is impossible.
Hence a = 0 but then (f, p) = 0 so sp (f ) = f and sp (e) intersect the curve p
positively. In particular if a (-2)-curve is a fixed component for e then either it is
also a fixed component for f , or f does not meet that curve (the converse is true
also). Then one can apply the proof of 3) in [19, Theorem 1, §6] to obtain elements
γ1 and γ2 in the group generated by reflections in the (-2)-curves, with γ1 (e) and
γ2 (f ) nef and (γ1 (e), γ2 (f )) = 7.
Assume that σ fixes a smooth elliptic curve, then this is one of the fibers of π1
(otherwise there would be a fixed point on any fiber), similarly it should be a fiber
of π2 . This gives a contradiction since the two fibers intersect in 7 points.
Hence it follows from Theorem 3.4 that σ has exactly 3 fixed points, two with a
16
M. Artebani, A. Sarti, S. Taki
local action of type (7, 2) and one with action of type (7, 3). This implies that π1
has at most one reducible fiber of type III.
Observe that σ induces an order 7 automorphism on the base space P1 with two
fixed points i.e. σ preserves two fibers. We can assume them to be a smooth fiber
over 0 and a fiber of type III over ∞. Moreover, π1 has generically 21 fibers of
type I1 , divided in 3 σ-orbits.
The quotient X/(σ) is a log Enriques surface of index 7 (Lemma 1.7 and its
proof, [18]) and π1 induces and elliptic fibration π̄1 : X/(σ) → P1 . Let Y be the
minimal resolution of X/(σ). The proper transform of π̄1−1 (∞) is a −1 curve. After
contracting this curve we get a minimal rational elliptic surface π̃1 : Ȳ → P1 with
one fiber of type 7I0 , one of type III ∗ and three of type I1 . It can be easily proved
that the jacobian fibration of π̃1 is as in Example 7.2.
Remark 7.4. A different model for the surface with SX = U ⊕ E8 ⊕ A6 was given
by S. Kondō in [6, §7, (7.5)]:
y 2 = x3 + t3 x + t8 , σ(x, y, t) = (ζ 3 x, ζy, ζ 2 t).
The singular fibers are of type III∗ , of type IV∗ and 7 of type I1 . Moreover it follows
from [22, §5] that the rank of the Mordell-Weil group is 1.
8. Order 11
Non-symplectic automorphisms of order 11 on K3 surfaces have been studied by
Oguiso and Zhang in [15]. We provide here an alternative veiw of the calssification.
Example 8.1. Let Xa be an elliptic surface with Weierstrass equation
y 2 = x3 + ax + (t11 − 1).
For generic a ∈ C the fibration has one fiber of Kodaira type II over t = ∞ and 22
fibers of type I1 over the zeros of
∆(t) = 4a3 + 27(t11 − 1)2 ,
hence it is a K3 surface (see [10, Table (IV.3.1)]). Moreover, X carries the order 11
automorphism
(7)
σ(x, y, t) = (x, y, ζ11 t).
Observe that if a3 = −27/4 then Xa has one singular fiber of type II over t = ∞,
of type I11 over t = 0 and 11 of type I1 .
Example 8.2. Let πa : Ya −→ P1 be the rational jacobian elliptic surface with
Weierstrass equation
y 2 = x3 + x + (t − a), a ∈ C.
For generic a the family has a fiber of type II ∗ over t = ∞ and two fibers of type
I1 over the zeros of ∆ = 4 + 27(t − a)2 . As in Examples 6.9 and 7.2, we associate
to Ya a K3 surface with non-symplectic automorphism with zero-dimensional fixed
locus. Let p : P → P1 be a non-trivial principal homogeneous space of πa given by
an element of order 11 in the fiber πa−1 (0). The surface Z obtained as the composite
of the blowing up of the intersection point of the components of multiplicity 5 and
6 in p−1 (∞) and the blowing down of the proper transform of the fiber p−1 (∞) is a
log Enriques surface of index 11 with two singular points. By Theorem 5.1 in [27]
Non-symplectic automorphisms on K3 surfaces
n1 = n4
1
1
3
n2 = n3
0
0
2
n5
0
0
1
k0
0
0
1
k1
1
0
0
T (σ)
U ⊕ U ⊕ E 8 ⊕ E8
U ⊕ U (11) ⊕ E8 ⊕ E8
K11 (−1) ⊕ E8
17
S(σ)
U
U (11)
U ⊕ A10
Table 4. Order 11
the canonical cover of Z is a K3 surface and a generator of the Galois group of the
covering is a non-symplectic order 11 automorphism with two isolated fixed points.
Theorem 8.3. Let σ be a non-symplectic automorphism of order 11 of a K3 surface. The fixed lattice of σ is the union of ni points of type i and kj curves of genus
j where (ni , kj ) appears in a row of Table 4.
The same table gives the isomorphism class of the corresponding fixed lattice
S(σ) and its orthogonal complement T (σ).
Proof. Since 10 divides the rank of T , then rank S(σ) is either 12 or 2. As before,
the lattices in the table can be found by means of Theorem 2.1 and Theorem 3.1.
Note that also in this case any K3 surfaces admits an elliptic fibration, hence by
Lemma 3.2 the fixed locus contains curves of genus at most 1.
S = U ⊕ A10 In this case n = 11 and α = 1 by Theorem 3.4. Since the genus of
a curve in Fix(σ) is at most 1, this implies that k0 = 1, i.e. the fixed locus contains
exactly one smooth rational curve R. Note that X in this case has a jacobian fibration with one fiber of type I11 and R is necessarily a component of this reducible
fiber. Hence the fiber I11 contains 9 fixed points. The remaining two fixed points
belong to an irreducible fiber of type II. In particular k1 = 0.
S = U The surface X has a jacobian fibration with no reducible fibers. The order
11 automorphism acts on the basis of the fibration with two fixed points i.e. it
preserves two fibers. It follows that there is one singular fiber (of type II) with two
fixed points and a smooth fiber, which is pointwise fixed. This implies that k0 = 0
and k1 = 1.
S = U (11) In this case X carries two elliptic fibrations intersecting in 11 points.
Infact since U (11) does not contain (-2)–classes the generators e and f of U (11)
are nef. Hence by [19, §3] there are two elliptic fibrations with fibers intersecting
in 11 points.
By the same idea used in the proof of Theorem 7.3, we conclude that there are no
curves in the fixed locus. By the Theorem 3.4 it follows that σ has two isolated
fixed points, one with local action of type (11, 2) and the other of type (11, 5).
Hence π : X −→ P1 , which denotes one of the two stable elliptic fibrations, has
at most one reducible fiber of type II. Since σ induces an order 11 automorphism
of P1 there are two fibers which are preserved. These are a fiber of type II and a
smooth elliptic fiber I0 , which we can assume to be over ∞ and 0. For the generic
K3 the other singular fibers are 22 of type I1 , divided in two orbits of length 11.
The quotient X/(σ) is a log Enriques surface of index 11 (Lemma 1.7 and its proof,
[18]) with two singular points (the images of the fixed points on X) and an elliptic
18
M. Artebani, A. Sarti, S. Taki
n1 = n2
3
n3
2
n4
1
n5 = n6
0
k
1
T (σ)
U ⊕ H13 ⊕ E8
S(σ)
H13 ⊕ E8
Table 5. Order 13
n1 = n2 = n3 = n4
0
n5 = n8
1
n6
2
n7
3
k
0
T (σ)
U ⊕ U ⊕ E8 ⊕ L17
S(σ)
U ⊕ L17
Table 6. Order 17
fibration π̄ : X/(σ) → P1 . We can now consider the minimal resolution of X/(σ),
where the proper transform of π̄ −1 (∞) is a (−1) curve. After contracting this (−1)
curve we obtain a rational elliptic smooth surface Y → P1 with one fiber of type
11I0 , one fiber of type II ∗ and two fibers of type I1 . This is Example 8.2.
Example 8.4 ([6]). Let X be the K3 surface
5
2
2
y 2 = x3 + t5 x + t2 , σ(x, y, t) = (ζ11
x, ζ11
y, ζ11
t).
The elliptic fibration has one singular fiber of type IV , one of type III ∗ and 11
of type I1 . This is the only K3 surface with order 11 automorphism such that
rank T (σ) = 10. Note that the fiber of type III ∗ contains 7 fixed points and a
rational fixed curve, while the fiber of type IV contains 4 fixed points.
9. Order 13, 17, 19
K3 surfaces with non-symplectic automorphisms of order 13, 17 and 19 are well
known and studied in [6] and [16], here we give an alternative point of view. The
following examples are due to Kondō.
Example 9.1. Let X be the K3 surface with Weierstrass model
5
2
y 2 = x3 + t5 x + t4 , (x, y, t) 7→ (ζ13
x, ζ13 y, ζ13
t).
The elliptic fibration has one singular fiber of type II, one of type III ∗ and 13
fibers of type I1 . Note that the fiber of type III ∗ contains 7 fixed points and a
rational fixed curve, while the fiber of type II contains 2 fixed points.
Example 9.2. Let X be the K3 surface with Weierstrass model
7
2
2
y 2 = x3 + t7 x + t2 , (x, y, t) 7→ (ζ17
x, ζ17
y, ζ17
t).
The elliptic fibration has one singular fiber of type IV , one of type III and 17
fibers of type I1 . Note that the fiber of type IV contains 4 fixed points, while the
fiber of type III contains 3 fixed points.
Example 9.3. Let X be the K3 surface with Weierstrass model
7
2
y 2 = x3 + t7 x + t, (x, y, t) 7→ (ζ19
x, ζ19 y, ζ19
t).
The elliptic fibration has one singular fiber of type II, one of type III and 19 fibers
of type I1 . Note that the fiber of type II contains 2 fixed points, while the fiber of
type III contains 3 fixed points.
Non-symplectic automorphisms on K3 surfaces
n1 = n2 = n3 = n8 = n9
0
n4 = n6 = n7
1
n5
2
k
0
T (σ)
K19 (−1) ⊕ E8 ⊕ E8
19
S(σ)
U ⊕ K19
Table 7. Order 19
Theorem 9.4. A K3 surface with a non-symplectic automorphism of order p ∈
{13, 17, 19} is isomorphic to the surface in Example 9.1, 9.2 or 9.3 respectively.
The fixed locus of such automorphism is the union of ni points of type i and k
smooth rational curves, as described in Tables 5, 6 and 7 respectively. The same
table gives the corresponding fixed lattices S(σ) and their orthogonal complements.
Proof. The lattices in the table can be found by means of Theorem 2.1, Theorem
3.1 and [6]. Since for p = 13, 17, 19 the rank of S(σ) is ≥ 5, then by Lemma 3.2 the
fixed locus of σ contains curves of genus at most 1.
If p = 13 then S(σ) ∼
= H13 ⊕ E8 . Let e1 , e2 be the generators of H13 with
intersection matrix as in section 1 and f ∈ E8 . The vectors e1 − e2 + f, e2 + f and
a basis of f ⊥ ∼
= E7 ⊂ E8 generate a primitive sublattice S of S(σ) isometric to
U ⊕ E7 such that S ⊥ contains no (−2)-curves. It follows that X admits a jacobian
elliptic fibration f13 with a unique reducible fiber F of type III ∗ . Observe that σ
induces a non-trivially action on the basis of f13 , since otherwise the general fiber
of f13 would have an order 13 automorphism with a fixed point (the intersection
with a section of f ). Thus σ preserves exactly two fibers of f . Since X (X) = 24
and X (F ) = 9, then f13 has also a σ-orbit of 13 singular fibers of type I1 and a
σ-invariant fiber of type II. Working as in the proof of Theorem 7.3 it can be
proved that there is only one jacobian fibration with this property (see also [16,
§4]). Thus X is isomorphic to the surface in Example 9.1.
The proofs for p = 17, 19 are similar. In these cases S(σ) contains a primitive
sublattice S ∼
= U ⊕ A1 respectively such that S ⊥ contains no
= U ⊕ A2 ⊕ A1 and ∼
(−2)-curves. Thus the surface admits a jacobian fibration with reducible fibers of
types Ã2 ⊕ Ã1 and Ã1 respectively. This implies, together with the fact that σ acts
non-trivially on the basis of the fibration and a computation of X (X), that X is
isomorphic to either the surface in Example 9.2 or 9.3.
10. Moduli spaces
Let ρ ∈ O(LK3 ) be an isometry of prime order p with fixed lattice
S(ρ) = {x ∈ LK3 : ρ(x) = x}
of signature (1, r − 1) and [ρ] be its conjugacy class in O(LK3 ). A [ρ]-polarized K3
surface is an algebraic K3 surface carrying a non-symplectic automorphism σ such
that
σ ∗ (ωX ) = ζp ωX
σ ∗ = φ ◦ ρ ◦ φ−1
for some isometry φ : LK3 → H 2 (X, Z).
A moduli space for such polarized surfaces can be constructed as follows (see
[5]). Let S(ρ)⊥ = T (ρ) and V ρ = {x ∈ LK3 ⊗ C : ρ(x) = ζp x} ⊂ T (ρ) ⊗ C be an
eigenspace of the natural extension ρC to LK3 ⊗ C. Consider the space
Dρ = {w ∈ P(V ρ ) : (w, w̄) > 0, (w, w) = 0}.
20
M. Artebani, A. Sarti, S. Taki
This is a type IV Hermitian symmetric space of dimension r(T (ρ)) − 2 if p = 2 and
it is isomorphic to a complex ball of dimension r(T (ρ))/(p − 1) − 1 if p > 2 (note
that if ζp ∈
/ R then the condition (w, w) = 0 is automatically true).
Furthermore consider the divisor
[
∆ρ =
Dρ ∩ δ ⊥
δ∈T (ρ),δ 2 =−2
and the discrete group
Γρ = {γ ∈ O(LK3 ) : γ ◦ ρ = ρ ◦ γ}.
Theorem 10.1. The orbit space Γρ \(Dρ \∆ρ ) parametrizes isomorphism classes of
[ρ]-polarized K3 surfaces.
Proof. In Theorem 11.3, [5] it is proved that Γρ0 \(Dρ \∆ρ ), where Γρ0 = {γ ∈ Γρ :
γ|S(ρ) = id}, parametrizes isomorphisms classes of ρ-polarized K3 surfaces with the
extra data of a primitive embedding j : S(ρ) → SX . It follows easily from the proof
that, forgetting the polarization j, is equivalent to remove the condition γ|S(ρ) = id
(see also Remark 11.4 in [5]).
Proof. We follow the proofs of [5, Theorems 11.2, 11.3]. Let l ∈ Dρ \∆ρ this is a
line in V ρ fixed under ρ.
Let φ : LK3 −→ H 2 (X, Z) be a marking of a K3 surface X, such that φ(l) =
H 2,0 (X). A necessary condition for X to have a non-symplectic automorphism is
that X is algebraic [12, Theorem 3.1], hence there exists an ample line bundle on
X and we can assume that φ ◦ ρ ◦ φ−1 acts identically on L.
By [3, Corollary 3.11] φ ◦ ρ ◦ φ−1 is an effective Hodge isometry so there exists
σ ∈ Aut(X) with σ ∗ = φ ◦ ρ ◦ φ−1 . Now the group Γρ preserves the space V ρ , hence
we obtain the statement.
It follows from the Torelli type theorem that a moduli space for K3 surfaces
carrying a non-symplectic automorphism of order p is the quotient
MpK3 = ∪ρ (Dρ \∆ρ )/ O(LK3 )
where ρ ∈ O(LK3 ) has order p and has hyperbolic fixed lattice. If we just take
the union over all isometries in the conjugacy class of a fixed isometry ρ̄, we get a
subspace
M[ρ̄] = ∪ρ∈[ρ̄] (Dρ \∆ρ )/ O(LK3 )
which is clearly isomorphic to Γρ̄ \(Dρ̄ \∆ρ̄ ) and parametrizes [ρ̄]-polarized K3 surfaces.
Theorem 10.2. The following table gives the number # of irreducible components
of the moduli space MpK3 , their dimensions and the Picard lattice of the generic K3
surface in each component, for any prime p.
Proof. Since MpK3 = ∪M[ρ] and the spaces M[ρ] are irreducible, then we only need
to understand the mutual inclusions between their closures in order to determine
the irreducible components of MpK3 . The moduli space M[ρ] is in the closure of
M[ρ2 ] if and only if there is ρ1 ∈ [ρ] such that Vρ1 ⊂ Vρ2 . This is equivalent to say
that T (ρ1 ) ⊂ T (ρ2 ) and ρ2 = ρ1 on T (ρ1 ).
Non-symplectic automorphisms on K3 surfaces
p
2
3
5
7
11
13
17
19
# dim
2 19, 18
3 9, 9, 6
2
4, 3
2
2, 2
2
1, 1
1
0
1
0
1
0
21
S(σ)
(2), U (2)
U, U (3), U (3) ⊕ E6∗ (3)
H5 , H5 ⊕ A∗4 (5)
U ⊕ K7 , U (7) ⊕ K7
U, U (11)
H13 ⊕ E8
U ⊕ L17
U ⊕ K19
Table 8. Irreducible components of MpK3
If p = 2, then ρi = −id on T (ρi ), hence M[ρ] ⊂ M[ρ2 ] if and only if T (ρ1 ) ⊂
T (ρ2 ), or equivalently S(ρ1 ) ⊃ S(ρ2 ). As a consequence of Theorem 1, any fixed
lattice S(σ) contains a primitive sublattice which is isometric to either (2) or U (2).
The lattice (2) clearly is not a sublattice of U (2). Thus M2K3 has two irreducible
components of dimensions 19 and 18 respectively. The case p = 3 is Theorem 5.6
in [2].
In Theorem 6.4 we proved that a K3 surface with an order 5 non-symplectic
automorphism either belongs to the family in Example 6.1 or to the one in Example 6.3. These two families are clearly irreducible and of dimensions 4 and 3
respectively. Thus we only need to prove that the second component is not contained in the first one. Assume that the generic K3 surface (X, σ1 ) in the first
family (we assume σ1∗ (ωX ) = ζ5 ωX ), with SX = S(σ1 ) ∼
= H5 ⊕ A∗4 (5), also belongs
to the second family. Since the orthogonal complement of H5 in SX contains no
−2 curves, then by Theorem 10.1, X carries an automorphisms σ2 of order 5 such
that σ2∗ (ωX ) = ζ5 ωX and S(σ2 ) ∼
= H5 . Let h ∈ S(σ1 ) be the class with h2 = 2 as
in the proof of Theorem 6.4. The morphism associated to the class h is a double
cover of P2 branched along a smooth sextic curve C (since A∗4 (5) contains no −2
curves). Since h is fixed by σi∗ , i = 1, 2, then σi induces a projectivity σ̄i of P2
which preserves C. The automorphism group of C is finite, thus σ̄14 ◦ σ̄2 has finite
order and σ14 ◦ σ2 is a symplectic automorphism of finite order of X. By [12] this
would imply that the Picard lattice of X has rank > 8, giving a contradiction.
If p = 7, then in Theorem 7.3 we proved that a K3 surface with an order 7
non-symplectic automorphism either belongs to the family in Example 7.1 or to
the one in Example 7.2. Both are clearly irreducible of dimension 2, thus they are
the irreducible components of M7K3 .
If p = 11, then in Theorem 8.3 we proved that a K3 surface with an order 11
non-symplectic automorphism either belongs to the family in Example 8.1 or to the
one in Example 8.2. Both are clearly irreducible and 1-dimensional, thus they are
the irreducible components of M11
K3 .
If p = 13, 17 or 19 then by Theorem 9.4 the moduli space MpK3 is irreducible
and 0-dimensional.
Remark 10.3. The general members of the two irreducible components of M2K3
are the double cover of the plane branched along a smooth sextic curve and the
double cover of a quadric along a smooth curve of bidegree (4, 4).
22
M. Artebani, A. Sarti, S. Taki
Projective models for the general members of M3K3 are described in [2] and [24]. Observe that for p = 3, 5, 7, 11, 17, 19 one irreducible component of MpK3 parametrizes
K3 surfaces carrying a non-symplectic automorphism of order p having only isolated
fixed points.
In [12] V.V. Nikulin proved that the action of a symplectic automorphism on
the K3 lattice (up to conjugacy) only depends on its order or, equivalently, on
the number of its fixed points. We give a similar statement for non-symplectic
automorphisms of prime order.
Corollary 10.4. Let X1 , X2 be algebraic K3 surfaces and σi ∈ Aut(Xi ), i = 1, 2 be
non-symplectic automorphisms of prime order p > 2 such that Fix(σ1 ) and Fix(σ2 )
are homeomorphic. Then there exist isometries φi : LK3 → H 2 (Xi , Z) such that
−1
∗
∗
φ−1
1 ◦ σ 1 ◦ φ1 = φ2 ◦ σ 2 ◦ φ2 .
Proof. For p = 3 the theorem is [2, Corollary 5.3]. For p > 3 the result follows
from Theorems 6.4, 7.3, 8.3, 9.4 and their proofs. These theorems imply that, if
Fix(σ1 ) and Fix(σ2 ) have the same topology, then S(σ1 ) ∼
= S(σ2 ). Moreover, it
follows from their proofs that any pair (X, σ), with S(σ) ∼
= S(σi ) belongs to an
irreducible family in MpK3 . Thus, if (Xi , σi ) ∈ M[ρi ] , then M[ρ1 ] = M[ρ2 ] , which
is a just a reformulation of the statement.
References
[1] V. Alexeev, V.V. Nikulin. Del Pezzo and K3 surfaces. MSJ Memoirs, Vol. 15, Mathematical
Society of Japan, Tokyo, 2006.
[2] M. Artebani, A. Sarti. Non symplectic automorphisms of order 3 on K3 surfaces. Math.
Annalen (2008), 342, 903–921.
[3] W. Barth, C. Peters, A. van de Ven Compact complex surfaces. Springer, Berlin, Heidelberg,
New York (1984) Math. Annalen (2008), 342, 903–921.
[4] F. Cossec, I. Dolgachev. Enriques Surfaces I. Progress in Mathematics, 76. Birkhuser Boston,
Inc., Boston, MA, 1989.
[5] I. Dolgachev, S. Kondō. Moduli spaces of K3 surfaces and complex ball quotients. Arithmetic
and geometry around hypergeometric functions, 43–100, Progr. Math., 260, Birkhuser, Basel,
2007.
[6] S. Kondō. Automorphisms of algebraic K3 surfaces which act trivially on Picard groups. J.
Math. Soc. Japan, 44 (1992), 75–98.
[7] S. Kondō. The moduli space of 5 points on P1 and K3 surfaces. Progress in Mathematics,
260 (2007), 189–206.
[8] S. Kondō. The moduli space of curves of genus 4 and Deligne-Mostow’s complex reflection
groups. Advanced Studies in Pure Math., 36, Algebraic Geometry 2000, Azumino, 383–400.
[9] N. Machida and K. Oguiso. On K3 surfaces admitting finite non-symplectic group actions.
J. Math. Sci. Univ. Tokyo, 5, n.2 (1998), 273–297.
[10] R. Miranda. The basic theory of elliptic surfaces. Dottorato di ricerca in Matematica, ETS
Editrice, Pisa (1989).
[11] J. Milnor. On simply connected 4-manifolds. International symposium in algebraic topology,
(1958), Universidad Nacional Autonoma de México and UNESCO, Mexico city, 122–128.
[12] V.V. Nikulin. Finite groups of automorphisms of Kählerian surfaces of type K3. Moscow
Math. Soc., 38 (1980), 71–137.
[13] V.V. Nikulin. Integral symmetric bilinear forms and some of their applications. Math. USSR
Izv., 14 (1980), 103–167.
[14] V.V. Nikulin Factor groups of groups of the automorphisms of hyperbolic forms with respect
to subgroups generated by 2-reflections. J. Soviet Math., 22 (1983), 1401–1475.
[15] K. Oguiso and D-Q. Zhang K3 surfaces with order 11 automorphisms. preprint
[16] K. Oguiso and D-Q. Zhang. On Vorontsov’s theorem on K3 surfaces with non-symplectic
group actions Proc. Amer. Math. Soc. 128 (2000), no. 6, 1571-1580.
Non-symplectic automorphisms on K3 surfaces
23
[17] K. Oguiso and D-Q. Zhang. K3 surfaces with order five automorphisms J. Math. Kyoto Univ.
(1998) 419-438.
[18] K. Oguiso and D-Q. Zhang. On extremal log Enriques surfaces, II Tohoku Math. J. 50 (1998),
419-436.
[19] I.I. Pjateckii-Shapiro and I.R. Shafarevich. A Torelli Theorem for algebraic surfaces of type
K3 Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971) 530–572. Math. USSR Izvestija Vol. 5 (1971)
547–588
[20] A.N. Rudakov and I. Shafarevich. Surfaces of type K3 over fields of finite characteristic. In:
I. Shafarevich, Collected mathematical papers, Springer, Berlin (1989), 657–714.
[21] J.-P. Serre. A course in arithmetic. Graduate Texts in Mathematics, No.7, Springer-Verlag,
New York-Heidelberg, 1973.
[22] T. Shioda. An explicit algorithm for computing the Picard number of certain algebraic K3
surfaces. American Journal of Mathematics, 108 (1986), 415–432.
[23] N.J.A. Sloane, AT&T Labs-Research and G. Nebe. Catalogue of Lattices. University of Ulm,
http://www.research.att.com/ njas/lattices.
[24] S. Taki. Classification of non-symplectic automorphisms of order 3 on K3 surfaces. Preprint,
2008.
[25] D.-Q. Zhang Quotients of K3 surfaces modulo involutions. Preprint, 1999.
[26] D.-Q. Zhang Normal Algebraic Surfaces with trivial tricanonical divisors. Publ. RIMS, Kyoto
Univ. 33 (1997), 427–442
[27] D.-Q. Zhang Logarithmic Enriques surfaces. J. Math. Kyoto Univ. 31–2 (1991), 419–466
[28] D.-Q. Zhang Normal Logarithmic Enriques surfaces, II. J. Math. Kyoto Univ. 33–2 (1993),
357–397
Michela Artebani, Departamento de Matemática, Universidad de Concepción, Casilla 160C, Concepción, Chile. e-mail: [email protected].
Alessandra Sarti, Université de Poitiers, Laboratoire de Mathématiques et Applications,
Téléport 2 Boulevard Marie et Pierre Curie BP 30179, 86962 Futuroscope Chasseneuil
Cedex, France. e-mail [email protected],
URL http://www.mathematik.uni-mainz.de/∼sarti.
Shingo Taki, Graduate School of Mathematics, Nagoya University, Chikusa-ku Nagoya
464-8602 Japan. e-mail [email protected].
24
M. Artebani, A. Sarti, S. Taki
Appendix: On Naruki’s K3 surface
Shigeyuki Kondō1
As a moduli space of some K3 surfaces with a non-symplectic automorphism of
order 5, the quintic del Pezzo surface appears ([K2], see the following Remark 1.5
for more details). In this appendix we shall give a similar example in case of K3
surfaces with a non-symplectic automorphism of order 7.
√
1.1. Naruki’s K3 surface. Let ζ = e2π −1/7 . We introduce a hermitian form of
signature (1, 2) with variables z = (z1 , z2 , z3 ) by setting
H(z) = (ζ + ζ̄)z1 z̄1 − z2 z̄2 − z3 z̄3 .
We denote by SU (1, 2) the group of (3, 3)-matrices of determinant 1 which are
unitary with respect to H. The group SU (1, 2) naturally acts on the complex ball
of dimension 2
D = {(z1 , z2 , z3 ) ∈ P2 : H(z) > 0}.
We denote by Γ the subgroup of SU (1, 2) consisting of elements whose entries are
integers in Q(ζ). It is known that Γ acts on D properly discontinuously and the
quotient D/Γ is compact. We further denote by Γ′ the subgroup of Γ consisting of
matrices which are congruent to the identity matrix modulo the principal ideal P
generated by 1 − ζ. Naruki [N] showed that the quotient D/Γ′ is isomorphic to a
K3 surface X.
1.2. K3 surfaces with a non-symplectic automorphism of order 7. In the
following we shall show that the Naruki’s K3 surface X is the moduli space of pairs
of a K3 surface and a non-symplectic automorphism of order 7.
Let S = U (7) ⊕ K7 and its orthogonal complement T = U ⊕ U (7) ⊕ E8 ⊕ A6 in
LK3 (see Table 3).
Remark-Definition 1.1. By Theorem 2.1 in [?] it follows that T is isomorphic to
T ′ = U ⊕ U ⊕ K7 ⊕ A26 . An order 7 isometry without non-zero fixed vectors on T
can be thus explicitely described as follows.
Let U1 , U2 be two copies of the hyperbolic plane U and let ei , fi be a basis of Ui ,
(i = 1, 2) satisfying e2i = fi2 = 0, (ei , fi ) = 1. Let x, y be a basis of K7 satisfying
x2 = −2, y 2 = −4, (x, y) = 1. Let ρ0 be the isometry of U1 ⊕ U2 ⊕ K7 defined by
ρ0 (e1 ) = e1 + f1 + e2 + f2 − y
ρ0 (f1 ) = 2e1 + e2 + 2f2 − y
ρ0 (e2 ) = −f1 + e2 + f2 + x
ρ0 (f2 ) = −f1 + e2
ρ0 (x) = e1 + 2f1 − e2 + f2 − x − y, ρ0 (y) = 3e1 − f1 + 4e2 + 3f2 + x − 2y.
It is easy to see that ρ0 has order 7 and acts trivially on the discriminant group of
U1 ⊕ U2 ⊕ K7 .
An easy calculation shows that
v = (−1 + ζ 2 + ζ 4 − ζ 5 )e1 + (ζ 3 − 1)f1 + (ζ − ζ 5 )e2 + (ζ 2 − ζ 5 )f2 + x + (1 + ζ 5 )y
is an eigenvector of ρ0 with the eigenvalue ζ and
(v, v̄) = 7(ζ + ζ 6 ).
1
Research of the author is partially supported by Grant-in-Aidfor Scientific Research A18204001 and Houga-20654001, Japan
Non-symplectic automorphisms on K3 surfaces
25
On the other hand, let r1 , . . . , r6 be a basis of A6 such that ri2 = −2, (ri , ri+1 ) = 1
and the other ri ’s and rj ’s are orthogonal. Consider the isometry of A6 defined by
ρ6 (ri ) = ri+1 , (1 ≤ i ≤ 5),
ρ6 (r6 ) = −(r1 + r2 + r3 + r4 + r5 + r6 ).
It is easy to see that ρ6 acts trivially on the discriminant group of A6 and that
w = r1 + (ζ 6 + 1)r2 + (1 + ζ 5 + ζ 6 )r3 − (ζ + ζ 2 + ζ 3 )r4 − (ζ + ζ 2 )r5 − ζr6
is an eigenvector of ρ6 with the eigenvalue ζ and
(w, w̄) = −7.
Combining ρ0 and ρ6 , we define an isometry ρ of T of order 7 and without nonzero
fixed vectors. Moreover the action of ρ on the discriminant group T ∗ /T is trivial.
In Definition-Remark 1.1 we explicitely described an order 7 isometry ρ on T
without nonzero fixed vectors and acting trivially on the discriminant group. Hence
ρ can be extended to an isometry ρ (we use the same symbol) of LK3 acting trivially
on S.
Now we consider a K3 surface Y with SY ∼
= S. Then the transcendental lattice
TY of Y is isomorphic to T . We identify LK3 and H 2 (Y, Z) so that S = SY and
T = TY . If the period ωY ∈ T ⊗ C is an eigenvector of ρ, then it follows from the
Torelli type theorem for K3 surfaces that ρ can be realized as an automorphism g
of Y of order 7: g ∗ = ρ. Now consider the eigenspace decomposition of ρ:
T ⊗ C = ⊕6i=1 Vζ i
where Vζ i is the eigenspace corresponding to the eigenvalue ζ i . The period domain
of the pair (Y, g) is given by
D′ = {ω ∈ P(Vζ ) : hω, ω̄i > 0}.
Then the above calculations show that the hermitian form on Vζ defined by hω, ω̄i
is given by 7H(ξ). We define an arithmetic subgroup Γ̃ by
Γ̃ = {ϕ ∈ O(T ) : ϕ ◦ ρ = ρ ◦ ϕ}
and a subgroup Γ̃′ by
Γ̃′ = Γ̃ ∩ Ker{O(T ) → O(qT )}.
S
Let ∆ = δ ⊥ ∩ D′ where δ moves over all (−2)-vectors in T . Then (D′ \ ∆)/Γ̃′ is
the moduli of the pair (Y, g). Note that ρ has discriminant −1 and is contained in
Γ̃′ . Moreover ρ acts trivially on B. By using the same method as in [K1], we have
1.3. Theorem. X ∼
= D′ /Γ̃′ .
= D/Γ′ ∼
1.4. Remark. Naruki [N] showed that X has an elliptic fibration with three singular fibers of type I7 in the sense of Kodaira and with 7 sections. Thus X contains
28 smooth rational curves. In particular X has the Picard number 20. We can see
that ∆/Γ′ consists of 28 curves corresponding to 28 smooth rational curves on X.
We omit the proof of this fact here.
26
M. Artebani, A. Sarti, S. Taki
1.5. Remark. Let Z be a K3 surface with the Picard lattice
2 1
∼
SZ =
⊕ A4 ⊕ A4
1 −2
and with a non-symplectic automorphism σ of order 5 acting trivially on SZ . The
author [K2] showed that the moduli space of ordered 5-points on P1 is isomorphic
to the moduli space of the pairs of such (Z, σ). Moreover these moduli spaces can
be written birationally as an arithmetic quotient of a 2-dimensional complex ball
which is isomorphic to the quintic del Pezzo surface.
References
[K1]
[K2]
[N]
S. Kondō, A complex hyperbolic structure of the moduli space of curves of genus three, J.
reine angew. Math., 525 (2000), 219–232.
S. Kondō, The moduli space of 5 points on P1 and K3 surfaces, Progress in Mathematics,
vol. 260 (2007), 189–206.
I. Naruki, On a K3 surface which is a ball quotient, Max Planck Institute Preprint Series.
Shigeyuki Kondō, Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602,
Japan. e-mail [email protected]