Picard groups of moduli space of low degree K3 surfaces
We study the moduli space of quasi-polarized K3 surfaces of degree 6 and 8 via geometric invariant theory. In particular, we verify the Noether-Lefschetz conjecture  in these two cases. The general case is discussed at the end of the paper.
A primitively quasi-polarized K3 surface of degree consists of a K3 surfaces and a semiample line bundle such that is a primitive class and . Let be the moduli space of primitively quasi-polarized surfaces of degree . The Noether-Lefschetz divisors in correspond to surfaces with Picard number at least .
More precisely, for any non-negative integers , we define to be the locus of quasi-polarized K3 surfaces which contains a curve class satisfying
In , Maulik and Pandharipande have conjectured that the Picard group with -coefficients is spanned by those Noether-Lefschetz divisors on .
All the Noether-Lefschetz divisors are irreducible divisors on . When , the Picard group with rational coefficients is spanned by Noether-Lefschetz divisors .
It is well-known that is a connected component of the Shimura variety of orthogonal type by global Torelli theorem. The Noether-Lefschetz conjecture is also closely related to the study of cohomology on such Shimura varieties. The vanishing of the first cohomology of is proved in , and actually we have the following result:
The Picard group is isomorphic to the cohomology group for any .
Outline of the paper
In section 2, we review the Noether-Lefshcetz (NL) divisors on from an arithmetic perspective and show that they are all irreducible divisors. The projective models of low degree K3 surfaces are described in section 3. In theses cases, we give precise geometry description of elements in certain NL divisors. Theorem 1.1 is proved in the section 4 and section 5 via geometric invariant theory (GIT). Roughly speaking, we can construct an open subset of via GIT and the boundary components are NL divisors. In the last section, we prove a more general result on arbitrary Shimura variety of orthogonal type and Theorem 1.2 is deduced as a corollary.
Acknowledgements. The authors are grateful to Brendan Hassett and Radu Laza for useful discussions.
2. Period space and Heegner divisors
2.1. Period domain of K3 surface
Let be a primitively quasi-polarized K3 surface of degree . The middle cohomology is a unimodular even lattice of signature under the intersection form . The orthogonal complement of the first Chern class of
is an even lattice of signature , and it has a unique representation
where , is the hyperbolic plane and is the unimodular, negative definite even lattice of rank .
The period domain associated to can be realized as
The arithmetic group
naturally acts on According to the Global Torelli theorem of K3 surfaces, there is an isomorphism
via the period map. This implies that is a locally Hermitian symmetric variety. Moreover, is -factorial since it only has quotient singularities.
2.2. Heegner divisors
Given an element , there is an associated hyperplane
It is easy to see that the value and the residue class of modulo the lattice are both invariant under the action of . Thus, for each pair of and , one can define the Heegner divisor of by
Using the identification via period map, Maulik and Pandharipande have showed that the Noether-Lefschetz divisors are exactly the Heegner divisors on .
 The group is generated by the element . The Noether-Lefschetz divisor , where
Similarly as in , we prove the following theorem:
(Irreducibility Theorem) All the Heegner divisors (or equivalently, Noether-Lefschetz divisors ) are irreducible.
Let be a vector satisfying and . Denote by , it corresponds to a primitive vector with norm and level and type . Here we say a primitive vector is of level if and it is of type if . Moreover, they satisfy that is an integer.
Obviously, we have . It is easy to see that for each hyperplane , the arithmetic quotient is irreducible. As the Heegner divisor is a union of for all primitive vectors with given norm, level and type, it suffices to prove that the arithmetic group acts transitively on all such primitive vectors in .
Let be the two generators of the hyperbolic plane . Next, we say that two elements in are congruent if they are in the same orbit under the action of . We claim that any primitive vector with norm and level and type as above is congruent to the vector
Let be a rank one lattice of discriminant . Each primitive vector in of norm corresponds to a primitive imbedding . Two primitive vectors are congruent if the corresponding imbedding differ by an automorphism in .
Now we use the Nikulin’s theory  1.15 on imbedding of quadratic forms to classify all the congruent classes of the primitive imbedding. According to , the primitive imbedding is uniquely determined by the data , where
is a subgroup of .
is a subgroup of .
An isomorphism preserving the quadratic forms restricted to these subgroups, with graph .
An even lattice with signature and discriminant form and an isomorphism . Here is the discriminant quadratic form on and the discriminant quadratic form on .
Two imbeddings and are congruent if and only if and .
In our situation, let be the image of the imbedding. Then the level of actually corresponds to the order of which uniquely determines since the discriminant group is a cyclic group of order . The isomorphism corresponds to an automorphism of which is uniquely determined by the type of . Hence the congruent class of the primitive imbedding can be classified by the level and type. Notice that the primitive vector (2.2) is of level and type . And we prove our claim. ∎
2.5. Dimension formula
where denotes the fraction part and is the generalized quadratic Gauss sum:
Let us denote by . After applying the summation formula proved by Gauss in 1811 (cf.  2.2), one can simply get
and is the Jacobi symbol.
In particular, we have , when .
3. Projective models of K3 surfaces
Let be a smooth K3 surface with a primitive quasi-polarization satisfying and for every curve . The linear system defines a map from to . The image of is called a projective model of .
In , Saint-Donat gives a precise description of all projective models of when is not a birational morphism.
 Let be the primitive quasi-polarization of degree on and let be the map defined by . Then there are following possibilities:
is birational to a degree surface in . In particular, is a closed embedding when is ample.
is a generically map and is a smooth rational normal scroll of degree , or a cone over a rational normal curve of degree .
has a fixed component , which is a smooth rational curve. Moreover, is a rational normal curve of degree in .
We call K3 surfaces of type , , nonhyperelliptic, unigonal, and digonal K3 surfaces accordingly. When , the projective model of a general quasi-polarized K3 surface is a complete intersection in the projective space .
Assume that is a birational morphism. Then one can easily see that is not ample if and only if there exists an exceptional curve . The morphism will factor through a contraction where is a singular K3 surface with A-D-E singularities.
Recalling that the Noether-Lefschetz divisor parametrize all K3 surfaces of degree with exceptional curves. Therefore, the projective model of a general member in is a surface in of degree with A-D-E singularities.
In this paper, we mainly consider the case and , where the above classification can be easily read off from the Picard lattice of .
Let be a smooth quasi-polarized K3 surface of degree (). Then
if and only if is digonal except
and , where is a rational curve, is an irreducible elliptic curve and is irreducible of genus two with and . The image is contained in a cone over cubic surface in .
if and only if is unigonal.
if and only if is one of the following:
when , is birational to the complete intersection of a singular quadric and a cubic in via .
when , is either birational to a bidegree hypersurface of the Serge variety via or is in case .
Now we suppose that a quasi-polarized K3 surface is neither unigonal or diagonal. Then is a birational map to a complete intersection of a quadric and a cubic. Our first statement of (3) comes from the fact any quadric threefold containing a plane cubic must be singular. If , the assertion follows from  Proposition 7.15 and Example 7.19. ∎
4. Complete intersection of a quadric and a cubic
In this section, we construct the moduli space of the complete intersection of a smooth quadric and a cubic in via geometric invariant theory.
4.1. Terminology and Notations
In the rest of this paper, we will use the following terminology. Let be an analytic function in whose leading term defines an isolated singularity at the origin. We have the following types of singularities:
Simple singualrities: isolated , , singularities.
Simple elliptic singularities :
We will use the notation as linear, quadratic and cubic polynomials of .
4.2. Cubic sections on quadric threefolds
Let be the smooth quadric threefold in defined by the equation
Since every nonsingular quadric hypersurface in is projectively equivalent to , a complete intersection of a smooth quadric and a cubic can be identified with an element in .
The automorphism group of is the reductive Lie group which is isomorphic to . Then we can naturally describe the moduli space of the complete intersection of a smooth quadric and a cubic as the GIT quotient of the linear system , where is a -dimensional vector space defined by the exact sequence
Let us take the set of monomials
to be a basis of . Sometimes, we may change the basis for simpler computations.
4.3. Numerical criterion
Now we classify stability of the points in under the action of by applying the Hilbert-Mumford numerical criterion .
As is customary, a one parameter subgroup (1-PS) of can be diagonalized as
for some . We call such a normalized 1-PS of if .
Let be a normalized 1-PS of . Then the weight of a monomial with respect to is
If we denote by (resp. ) the set of monomials of degree which have non-positive (resp. negative) weight with respect to , one can easily compute the maximal subsets (resp. ), as listed in Table 1 (resp. Table 2) .
According to the Hilbert-Mumford criterion, an element is not properly stable (resp. unstable) if and only if the weight of some monomial in is non-positive (resp. negative). Thus we obtain:
Let be the surface defined by an element in . Then is not properly stable if and only if for some cubic hypersurface defined by a cubic polynomial in one of following cases:
For not properly stable, using the destabilizing 1-PS , the limit exists and it is invariant with respect to . The invariant part of polynomials of type are the followings:
Similarly, we get
With the notation above, is not semistable if and only if for some cubic hypersurface defined by one of the following equations:
, and has no term.
4.6. Geometric interpretation of stability
Let be a hypersurface singularity given by an equation . The corank of is minus the rank of the Hessian of at .
A complete intersection is not properly stable if and only if satisfies one of the following conditions:
has a hypersurface singularity of corank .
is singular along a line and there exists a plane such that and is contained in the projective tangent cone for any point .
has a singularity of corank at least 2 and the restriction of the projective cone to contains a line passing through with multiplicity at least .
As a consequence of Lemma 4.4, it suffices to find the geometric characterizations of the complete intersections of type . Here we do it case by case.
(i). If is of type , then can be considered as the intersection of and a cubic cone with the vertex . It is easy to see that is a corank of 3 singularity of .
Conversely, we write the equation of as
If we choose the affine coordinate
then the affine equation near is
It has a corank singularity at the origin if and only if the quadric is .
(ii). If is of type , then the equation of is given by
and therefore is singular along the line .
Moreover, for any point , the projective tangent cone at is defined as
which contains the plane for each and .
Conversely, since the intersection of and is a double line , we may certainly assume that the plane is defined by
after some coordinate transform persevering the quadric form . Then the line is given by .
Because is singular along , the equation of can be written as:
Then the projective tangent cone
contains the plane for each point only if the quadrics have no term.
(iii). For of type , a similar discussion is as follows: if is defined by
then is singular at . After choosing the affine coordinates as (4.3), the affine equation near is
for some polynomials with . Therefore, is a hypersurface singularity of corank and its projective tangent cone is a double plane . The remaining part is straightforward.
Conversely, we take to be the singular point as before. Then the equation of can be written as
Then the quadric is of the form for some linear polynomial because is singular of corank at least .
After we make a coordinate change preserving and , the equation of can be written as either
The projective tangent cone at is a double plane
The line contained in the restriction of to has to be defined by . It follows that the last case (4.10) can not happen since contains with multiplicity at least .
Finally, the multiplicity condition implies that the quadric does not have terms. ∎
In the case of (N3), if we set and , then we see from (4.8) that the local analytic function near is equivalent to
in . So a general member of type will have an isolated simple elliptic singularity of type .
A complete intersection is unstable if and only if satisfies one of the following conditions:
is reducible, where is a cone over a conic with vertex and is singular at ;
is singular along a line satisfying the condition: there exist a plane such that for any point .
It suffices to check the complete intersections of type case by case.
(). Suppose is a union of two surfaces satisfying the desired conditions. We can also assume that the vertex of is and is defined by
for a suitable change of coordinates preserving . Therefore, the equation of has the form
Since the other component is singular at , there is no terms in the quadric . The converse is obvious.
(). To simplify the proof, we choose another monomial basis of as below:
Then the polynomial of type has the form
At this time, is singular along the line and satisfies the condition described in ().
On the other hand, the line on can be written as
for a suitable change of coordinates preserving . Then the equation of has the form
where does not contain term.
Moreover, for any point , the projective tangent cone is given by
They have a common plane with multiplicity if and only if is defined by and does not contain the terms. ∎
A complete intersection is semistable (reps. stable) if has at worst isolated singularities (reps. simple singularities).
By Theorem 4.9, the singular locus of is at least one dimensional if it is unstable. Then has to be semistable if it has at worst isolated singularities.
Now it makes sense to talk about the moduli space of complete intersections of a smooth quadric and a cubic with simple singularities. Let be the open subset of parameterizing such complete intersections in . Then we have .
There is a natural open immersion , and the complement of the image in is the union of three Noether-Lefschetz divisors and . In particular, the Picard group is spanned by .
For the first statement, one only need the fact that the complete intersections with simple singularities correspond to degree 6 quasi-polarized K3 surfaces containing a curve. Therefore, we obtain an open immersion . By Lemma 3.3, we know that the boundary divisors of the image is the union of and .
Next, we claim that the dimension of is at most one. Observe that is constructed via the GIT quotient , and has rank one since the boundary of in has codimension at least two. Let be the set of -linearized line bundles on . There is an injection
by  Proposition 4.2. Our assertion follows from the fact the forgetful map is an injection.
Since the complement of in is the union of three irreducible divisors and , it follows that is spanned by the set of Noether-Lefschetz divisors by dimension considerations. ∎
There is another natural GIT construction of moduli space of complete intersections in projective space, see . There exists a projective bundle parameterizing all complete intersections of a quadric and a cubic in . Then one can consider the GIT quotient
for the line bundle .
We want to point out that is isomorphic to our GIT quotient when . This can be obtained via a similar argument as in . It will be interesting to study the variation of GIT on .
4.13. Minimal orbits
In this subsection, we give a description of the boundary components of the GIT compactification. The boundary of the GIT compactification consists of strictly semistable points with minimal orbits. From , it suffices to discuss the points of type . As in , our approach is to use Luna’s criterion:
(Luna’s criterion) Let be a reductive group acting on an affine variety . If is a reductive subgroup of and is stabilized by , then the orbit is closed if and only if is closed.
To start with, we first observe that Type , and