The unirationality of the moduli spaces of 2-elementary K3 surfaces (with an Appendix by Ken-Ichi Yoshikawa)

Abstract

We prove that the moduli spaces of K3 surfaces with non-symplectic
involutions are unirational. As a by-product we describe configuration spaces
of 4<d<9 points in the projective plane as arithmetic quotients of type IV.

Full text

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arXiv:1011.1963v1 [math.AG] 9 Nov 2010
THE UNIRATIONALITY OF THE MODULI SPACES OF
2-ELEMENTARY K3 SURFACES
SHOUHEI MA
with an Appendix by Ken-Ichi Yoshikawa
Abstract. We prove the unirationality of the moduli spaces of K3 surfaces with
non-symplectic involutions. As a by-product, we describe the configuration
space of 5 ≤ d ≤ 8 points in P2as an arithmetic quotient of type IV.
1. Introduction
K3 surfaces with non-symplectic involutions were classified by Nikulin [26],
and Yoshikawa [31]showed that their moduli spaces are Zariski open sets ofcertain
modular varieties of orthogonal type. In this paper we prove the unirationality of
those moduli spaces. This work was inspired by a recent result of Yoshikawa on the
Kodaira dimensions of those spaces, which is presented by him in the Appendix A
of this paper. Let us begin by recalling basic definitions.
Let X be a complex K3 surface endowed with an involution ι. When ι acts non-
trivially on H0(KX), the pair (X,ι) is called a 2-elementary K3 surface. In this
case, the lattice L+= H2(X,Z)ιof ι-invariant cycles is a hyperbolic lattice with 2-
elementary discriminant form DL+. Themain invariant of (X,ι)is the triplet (r,a,δ)
where r is the rank of L+, a is the length of DL+, and δ is the parity of DL+. Nikulin
[26] proved that the deformation type of (X,ι) is determined by the main invariant
(r,a,δ), and he enumerated all main invariants of 2-elementary K3 surfaces.
By the theory of period mapping, 2-elementary K3 surfaces belonging to a fixed
main invariant (r,a,δ) are parametrized by the Hermitian symmetric domain as-
sociated to a certain lattice L− of signature (2,20 − r). Yoshikawa [31], [33]
determined the correct monodromy group as the orthogonal group O(L−) of L−.
Consequently, he constructed the moduli space M(r,a,δ)of those 2-elementary K3
surfaces as a Zariski open set of the modular variety defined by O(L−).
The principal result of the present paper is the following.
Theorem 1.1. For every main invariant (r,a,δ) the moduli space M(r,a,δ)of 2-
elementary K3 surfaces of type (r,a,δ) is unirational.
We recall that the 2-elementary K3 surfaces in M(1,1,1)are double covers of
P2ramified over smooth sextics so that M(1,1,1)is birational to the orbit space
2000 Mathematics Subject Classification. Primary 14J28, Secondary 14G35, 14H50.
Key words and phrases. K3 surface, non-symplectic involution, unirationality of moduli space,
orthogonal modular variety, point set in projective plane.
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|OP2(6)|/PGL3, which is unirational. This well-known fact is a prototype of The-
orem 1.1. Kond¯ o [16] proved the rationality of M(10,2,0)and M(10,10,0), the latter
being isomorphic to the moduli of Enriques surfaces. Shepherd-Barron [29] prac-
tically established the rationality of M(5,5,1)in the course of proving that of the
moduli of genus 6 curves. Matsumoto-Sasaki-Yoshida [21] constructed general
members of M(16,6,1)starting from six lines on P2. A similar idea was used by
Koike-Shiga-Takayama-Tsutsui [15] to obtain general members of M(14,8,1)from
four bidegree (1,1) curves on P1×P1. In particular, M(16,6,1)and M(14,8,1)are also
unirational. Theorem 1.1 is a generalization of these results.
As we noted in the beginning, the present work was inspired by work of
Yoshikawa. By using a criterion of Gritsenko [10] and Borcherds products, he
found that M(r,a,δ)has Kodaira dimension −∞ if either 13 ≤ r ≤ 17 or r + a = 22,
r ≤ 17. After that he suggested to the author to study the birational type of M(r,a,δ)
through an algebro-geometric approach. The present work grew out of this sugges-
tion. After Theorem 1.1 was proved, Yoshikawa and the author decided to write
both approaches in this paper. Yoshikawa’s work is presented in the Appendix A.
His result initiated systematic study of the birational type of M(r,a,δ). Now the Ko-
daira dimensions ofsome ofM(r,a,δ)may becalculated in twomanners: bymodular
forms on the moduli spaces, and by the geometry of 2-elementary K3 surfaces.
We will prove Theorem 1.1 by using certain Galois covers of the moduli spaces
and isogenies between them. The strategy is as follows.
(1) Let?
ety?
a′< a, δ = δ′.
(3) For each fixed 1 ≤ r ≤ 19, choose a large a and find a moduli interpreta-
tion of (an open set of)?
?
unirational in some way.
M(r,a,δ) be the modular variety associated to the group O(L−)0 of
isometries of L−which act trivially on the discriminant form. The vari-
M(r,a,δ)is a Galois cover of M(r,a,δ).
(2) Construct an isogeny?
M(r,a,δ)→?
M(r,a′,δ′)when either a′< a, δ = 1, or
M(r,a,δ). Then prove the unirationality of?
M(r,a,δ)
by using that interpretation. By the step (2) follows the unirationality of
M(r,a′,δ′)for a′< a.
(4) The remaining moduli spaces M(r,a′′,δ′′), a′′> a, are also proved to be
One of the advantages of studying the covers?
ular varieties. Those isogenies admit geometric interpretation in terms of twisted
Fourier-Mukai partners. Along this strategy we will derive the unirationality of
?
singularities, going one step further from the well-known (but not so geometrical)
interpretation of?
plained in [1], 2-elementary K3 surfaces with r+a ≤ 20 are related to log del Pezzo
M(r,a,δ)rather than M(r,a,δ)is that
we can find isogenies between them so that the problem is reduced to fewer mod-
M(r,a,δ)for seventy (r,a,δ) by studying just twenty-two. In the step (3), we often
regard?
M(r,a,δ)as the moduli of lattice-polarized K3 surfaces.
Let us comment on other possible approaches for Theorem 1.1. Firstly, as ex-
M(r,a,δ)as a moduli of certain plane sextics endowed with markings of the

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surfaces of index ≤ 2 so that one might study M(r,a,δ)via the moduli of such sur-
faces. For this direction the most relevant is Nakayama’s explicit description of log
del Pezzo surfaces of index 2 [24]. Secondly, by using singular curves on P2and
Fnlike the step (3) above, for each of most (r,a,δ) we may actually find a unira-
tional parameter space which dominates M(r,a,δ). In fact, these two approaches are
closely related. In [20] these will be developed further to derive the rationality of
M(r,a,δ)for sixty-three main invariants. Hence one may establish Theorem 1.1 also
by just studying the remaining moduli spaces. However, the proof of rationality is
delicate and ad hoc, so that the whole proof of unirationality would be lengthy if
we do so. Also the first and second approaches are of rather ad hoc character. We
here prefer the proof using the covers?
In our proof of Theorem 1.1, we relate the covers?
arithmetic quotients of type IV. To be more precise, let Ud ⊂ (P2)d(resp. Vd ⊂
(P2)d) be the variety of ordered d points of which no three are collinear (resp. only
the first three are collinear). Let Ud/G and Vd/G denote the quotient varieties for
the diagonal actions of G = PGL3. Let Lnbe the lattice ?2?2⊕ ?−2?n.
Theorem 1.2. Let 5 ≤ d ≤ 8. For each 1 ≤ n ≤ 8 there exists an arithmetic group
Γn⊂ O(Ln) such that one has birational period maps
Ud/G ? F(Γ2d−8),
where F(Γn) is the modular variety associated to Γn. One has Γn = O(Ln)0for
1 ≤ n ≤ 6, and for n = 7,8 one has Γn⊃ O(Ln)0with Γn/O(Ln)0≃ Sn−5where SN
is the symmetric group.
M(r,a,δ)because it is more systematic, short,
and self-contained. The above mentioned approaches are reserved for [20].
M(r,22−r,δ)for r ≥ 12 with
configuration spaces of points in P2. As a by-product we describe those spaces as
Vd/G ? F(Γ2d−9),
When d ≤ 6, we recover some results of Matsumoto-Sasaki-Yoshida [21]. They
constructed a period map for U6, and then obtained other period maps (though not
in the present form) by specializing the one for U6. The novel part of Theorem 1.2
is the description of Ud/G and Vd/G for d = 7,8 as arithmetic quotients of type IV.
Also we modified the period maps for d ≤ 6 from [21] so that the specialization
picture would extend to d ≤ 8. We hope to study the boundary behavior of the
period maps in detail in a future paper. For Theorem 1.2 we give a proof based on
the Torelli theorem which for d ≤ 6 differs from that of [21] and which derives the
arithmetic groups Γneasily.
Kond¯ o, Dolgachev, and van Geemen [18], [8], [19] described the space Ud/G
for 5 ≤ d ≤ 7 as an arithmetic quotient of a complex ball. It is also known [9]
that U7/G can be expressed as a Siegel modular variety. Thus the space Ud/G for
5 ≤ d ≤ 7 admits the structure of an arithmetic quotient in more than one way. In
view of the relation of Ud/G with the moduli of del Pezzo surfaces, it would be
interesting to describe the induced rational action of the Weyl group on F(Γ2d−8).
The rest of the paper is structured as follows. In Section 2 we review the nec-
essary facts concerning lattice, modular variety, and invariant theory. In Section
3 we gather basic results on 2-elementary K3 surfaces with particular attention to

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the relation with singular sextic curves. The proof of Theorem 1.1 will be devel-
oped from Section 4 to Section 9. Theorem 1.2 will be proved in Sections 7, 8,
and 9. In Section 10 we remark the unirationality of the moduli spaces of Borcea-
Voisin threefolds as a consequence of Theorem 1.1. In the Appendix A written by
Yoshikawa, the approach by modular forms is presented.
Otherwise stated, we work in the category of algebraic varieties over C.
2. Preliminaries
2.1. Lattices. Let L be a lattice, i.e., a free Z-module of finite rank endowed with
a non-degenerate integral symmetric bilinear form (,). The orthogonal group of L
is denoted by O(L). The dual lattice Hom(L,Z) of L is denoted by L∨. Note that L∨
is canonically embedded in the quadratic space L⊗Q and contains L. For an integer
n ? 0, L(n) denotes the scaled lattice (L,n(,)). The Z-modules underlying L and
L(n) are canonically identified. The lattice L is even if (l,l) ∈ 2Z for all l ∈ L, and
odd if it is not even. If L is an even lattice, on the finite Abelian group DL= L∨/L
we have a quadratic form qL: DL→ Q/2Z defined by qL(x + L) = (x, x) + 2Z.
We call (DL,qL) the discriminant form of L. The associated bilinear form bL :
DL× DL→ Q/Z is given by bL(x + L,y + L) = (x,y) + Z. Notice that the bilinear
form (DL,bL) can be defined even when L is odd. Regardless of the parity of L, we
denote by O(L)0⊂ O(L) the discriminant kernel, i.e., the group of isometries of L
which act trivially on DL. When L is even, we denote by rL: O(L) → O(DL,qL)
the natural homomorphism.
Proposition 2.1 ([25]). Let Λ be an even unimodular lattice and L be a primitive
sublattice of Λ with the orthogonal complement M. Then one has a natural isome-
try λ : (DL,qL) ≃ (DM,−qM) defined by the relation x+ λ(x) ∈ Λ, x ∈ DL. For two
isometries γL∈ O(L) and γM∈ O(M), the isometry γL⊕ γMof L ⊕ M extends to
that of Λ if and only if rL(γL) = λ−1◦ rM(γM) ◦ λ.
A lattice L is 2-elementary if DLis a 2-elementary Abelian group, i.e., DL ≃
(Z/2Z)afor some a ≥ 0. The main invariant of an even 2-elementary lattice L is
the quadruplet (r+,r−,a,δ) where (r+,r−) is the signature of L, a is the length of
DL, and δ ∈ {0,1} is defined by δ = 0 if qL(DL) ⊂ Z/2Z and δ = 1 otherwise. By
Nikulin [25], the isometry class of L is uniquely determined by the main invariant
if r± > 0. When L is hyperbolic, we also call the triplet (1 + r−,a,δ) the main
invariant of L. In this paper we often use the following 2-elementary lattice
(2.1)
Mn= ?2? ⊕ ?−2?n−1= Zh ⊕ Ze1⊕ ··· ⊕ Zen−1,
where (h,h) = 2, (ei,ej) = −2δi,j, and (h,ei) = 0. Let
(2.2)
ΛK3= U3⊕ E2
be the even unimodular lattice of signature (3,19) where U is the hyperbolic plane
U = ?e, f?, (e,e) = (f, f) = 0, (e, f) = 1, and E8is the rank 8 even negative-
definitive unimodular lattice. The following assertion is due to Nikulin.
8
Proposition 2.2 ([25], [26]). Let L be an even hyperbolic 2-elementary lattice. If a
primitive embedding L ֒→ ΛK3exists, then it is unique up to the action of O(ΛK3).

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2.2. Orthogonal modular varieties. Let L be a lattice of signature (2,r−) and let
Γ ⊂ O(L) be a finite-index subgroup. The group Γ acts properly discontinuously
on the complex manifold
ΩL= { Cω ∈ P(L ⊗ C) | (ω,ω) = 0, (ω, ¯ ω) > 0 }.
The domain ΩLhas two connected components, say Ω+
the group of those isometries in Γ which preserve Ω+
Land Ω−
L. The quotient space
L. We denote by Γ+
(2.3)
FL(Γ+) = Γ+\Ω+
L
is a normal quasi-projective variety of dimension r−([2]) and called the modular
variety associated to Γ+. When the lattice L is understood from the context, we
abbreviate FL(Γ+) as F(Γ+).
Proposition 2.3. Let L be a finite-index sublattice of a lattice M of signature
(2,r−). Then there exists a finite surjective morphism F(O(L)+
Proof. We have the sequence L ⊂ M ⊂ M∨⊂ L∨of inclusions. If we regard the
finite groups G1= M/L and G2= M∨/L as subgroups of DL, then we have G2=
{x ∈ DL,bL(x,G1) ≡ 0} and the induced bilinear form (G2/G1,bL) is canonically
isometric to (DM,bM). Since the isometries in O(L)0act trivially on both G1and
G2, they preserve the overlattice M of L, and as isometries of M act trivially on
DM. Thus we have a finite-index embedding O(L)0֒→ O(M)0of groups. Via the
natural identification ΩL= ΩM⊂ P(L ⊗ C) = P(M ⊗ C), this embedding induces a
finite morphism F(O(L)+
The following proposition was used by Kond¯ o [16] to prove the rationality of
the moduli space of Enriques surfaces.
0) → F(O(M)+
0).
0) → F(O(M)+
0).
?
Proposition 2.4 (cf. [16]). Let L be an even 2-elementary lattice of signature
(2,r−). Then the lattice M = L∨(2) is 2-elementary and we have an isomorphism
F(O(L)+) ≃ F(O(M)+).
Proof. Since L(2) ⊂ M ⊂1
identification L ⊗ Q = L∨⊗ Q we have the coincidence O(L) = O(L∨) ⊂ O(L ⊗ Q)
because of the double dual relation L∨∨= L. Thus we have the isomorphisms
FL(O(L)+) ≃ FL∨(O(L∨)+) ≃ FM(O(M)+).
2L(2) = M∨, we see that M is 2-elementary. Under the
?
2.3. Geometric Invariant Theory. We review some facts from Geometric Invari-
ant Theory. Throughout this section let X be a variety acted on by a reductive
algebraic group G. A G-invariant morphism π : X → Y to a variety Y is a geo-
metric quotient of X by G if (i) π is surjective, (ii) OY ≃ (π∗OX)G, (iii) a subset
U ⊂ Y is open if π−1(U) ⊂ X is open, and (iv) the fibers of π are the G-orbits. We
sometimes denote Y = X/G and omit π. A geometric quotient π : X → Y enjoys
the following universality: for every G-invariant morphism f : X → Z there exists
a unique morphism g : Y → Z with g ◦ π = f. In particular, a geometric quotient,
if it exists, is unique up to isomorphism.