Picard groups on moduli of K3 surfaces with Mukai models
We discuss the Picard group of the moduli space of quasi-polarized K3 surfaces of genus and . In this range, is unirational, and a general element in is a complete intersection with respect to a vector bundle on a homogenous space, by the work of Mukai. In this paper, we find generators for the Picard group using Noether-Lefschetz theory. This verifies the Noether-Lefschetz conjecture on the moduli of K3 surfaces in these cases.
It is well-known that the moduli space of smooth projective curves of genus is a quasi-projective variety with Picard number one for (cf. [Ha83]). The Picard group with rational coefficients is generated by the first Chern class of the Hodge bundle. In the moduli theory of higher dimensional varieties, the primitively polarized K3 surface of genus can be viewed as a two dimensional analogue of genus smooth projective curve, and it is natural for us to study the Picard group of its moduli space .
Unlike in the curve case, the rank of is no longer constant; it has been shown by O’Grady [OG86] that increases to infinity as is increasing (see also [MP07] ). Besides the Hodge line bundle, there are many other natural divisors on coming from geometry. Actually, the Noether-Lefschetz locus of parametrizing K3 surfaces with Picard number greater than one is a union of countably many irreducible divisors by Hodge theory. We call them the Noether-Lefschetz (NL) divisors on . In this paper, we study the Picard group on for () and find its generators in terms of NL-divisors on .
For conveniency, instead of working on , we will study the moduli space of primitively quasi-polarized K3 surfaces of genus , which is more natural from a Hodge theoretic point of view. It is known that is a locally Hermitian symmetric variety, by the Torelli theorem, and the complement is a divisor parametrizing surfaces containing a -exceptional curve. In this setting, we define the Noether-Lefschetz (NL) divisors on as follows: given , let be the locus of those K3 surfaces whose Picard lattice contains a primitive rank two sublattice
where is the primitive quasi-polarization of and . Each NL divisor is irreducible by [O86]. In [MP07], Maulik and Pandharipande conjectured that the divisors span the group with rational coefficients. Our main result is:
The Picard group of with rational coefficients is spanned by NL divisors for and . Moreover, the basis of of is given by:
For , the group is generated by
The rank of is , so there is a linear relation between the generators in (0.2). See Remark LABEL:relation for more details.
When , a general K3 surface in is a double cover of () or a complete intersection in , and the assertion can be found in [O86][Sh80][Sh81][LT13]. In these cases, the proof relies on the explicit construction of the moduli space of the corresponding complete intersections via geometric invariant theory (GIT). If is greater than , the general K3 surface of genus is no longer a complete intersection in , but it can be interpreted as a complete intersection with respect to certain vector bundles on homogenous spaces, for in the range of our theorem, so there ought to be a similar construction.
Acknowledgments We have benefited from conversations with Brendan Hassett and Jun Li. We are very grateful to Arie Peterson for his useful comments and providing us the relation of NL-divisors in Remark LABEL:relation.
1. Geometry of K3 surfaces
In this section, we review Mukai’s work on the projective models of general low genus quasi-polarized K3 surfaces. We give a precise characterization, in terms of the Picard lattice, of the (non-general) K3 surfaces lying outside the locus of Mukai models. We also include a few examples to illustrate the phenomenon. Throughout this paper, we work over complex numbers.
Let be a primitively quasi-polarized K3 surface of genus , i.e. is big and nef with . The middle cohomology is an even unimodular lattice of signature under the intersection form . Let be the orthogonal complement, which is an even lattice of signature . The period domain associated to can be realized as a connected component of
The monodromy group
naturally acts on where is the identity component of . According to the Global Torelli theorem for K3 surfaces, there is an isomorphism
via the period map. Hence, is a locally Hermitian symmetric variety with only finite quotient singularities, and is thus -factorial. Each NL divisor can be considered as the quotient of a codimension one subdomain in .
The NL divisors we defined here are slightly different from the divisors discussed in [MP07]. Indeed, Maulik and Pandharipande define the NL divisors by specifying a curve class as below. Given the data , let be the locus parametrizing the K3 surfaces containing a class with and . Then is a divisor and it is supported on a collection of NL divisors satisfying for some , where is the determinant of the lattice . It is not hard to see that the span of two sets and are the same, and we have .
1.2. Mukai models
Let be a smooth quasi-polarized K3 surface of genus . The linear system defines a map from to . The image of is called a projective model of . If is birational to its image, the is a degree surface in with at worst rational double points.
Suppose is birational. If the K3 surface contains a -exceptional curve, the morphism contracts this exceptional curve and the image has a rational double point. Thus it is easy to see that is singular only if .
As we mentioned in the introduction, when (), general members in are no longer complete inspections in , but can be interpreted as complete intersections with respect to vector bundles on homogenous spaces in the following sense. Let be a rank vector bundle on a smooth projective variety with local frame . A global section
is non-degenerate at if and is a regular sequence. The section is non-degenerate if it is nondegenerate at every point of , where the zero locus of the section .
A subscheme is a complete intersection with respect to if is the zero locus of a non-degenerate global section of . In particular, one can view the complete intersection of -hypersurfaces in as the complete intersection with respect to the vector bundle
Now we review Mukai’s Brill-Noether theory on K3 surfaces, which essentially gives us the classification of the projective models of general K3 surfaces.
A polarized K3 surface of genus is Brill-Noether (BN) general if the inequality holds for any pair of non-trivial line bundles such that .
When is BN general of genus , for any two integers with , Mukai [Mu02] shows that there exists a rigid and stable vector bundle on of rank such that . The higher cohomology of vanishes and . Then there is a map
The BN theory on K3 surfaces is an analogous to the BN theory on curves. Actually, if a smooth curve is BN general, then the polarized K3 surface is BN general (cf. [Ha02]).
For and , Mukai has shown that image of can be described as complete intersections with respect to a vector bundle on some homogenous subspace in . Here we just summarize Mukai’s results in [Mu02]:
A primitively quasi-polarized K3 surface of genus or is Brill-Noether general if and only if it is birational to a complete intersection with respect to a vector bundle in a homogenous space via , where and the images are listed below:
: and , is a complete intersection of a quadric and a codimension three linear section in ;
: is the isotropic Grassmannian and , is a codimension eight linear section of ;
: and , is a codimension six linear section of ;
: is the Langrangian Grassmannian and , is a codimension four linear section of ;
: is the flag variety of dimension five associated with the adjoint representation of imbedded in and ; is a codimension three linear section of ;
: let be a seven dimensional vector space and . Then is BN general if and only if is birational to a hyperplane of , where is a non-degenerate three dimensional subspace and consists of three dimensional subspaces of such that the restriction of to is zero.
Here, the non-degeneracy of the subspace means that there is no decomposable vector in . When is non-degenerate, is a smooth Fano threefold of index 1 and can be considered as a complete intersection with respect to the vector bundle on , where is the dual of the universal subbundle on .
For , the values of chosen by Mukai are
In the case , Mukai chose the rank 5 vector bundle with . The linear system embeds the K3 surface into a Grassmannian , whose image is contained in the isotropic Grassmannian .
The homogenous space is the quotient of a simply connected semisimple Lie group by a maximal parabolic subgroup. Here, we list some of the associated semisimple Lie groups of (), which will be used later:
: the spin group .
: the special linear group .
: the symplectic group .
: the exceptional group of type .
Next, observing that a smooth codimension three linear section of is the unique (up to isomorphism) Fano threefold of degree five and index two, one can certainly consider the general K3 surface in as a quadric hypersurface in . It is known that the Fano threefold is a quasi-homogenous space with automorphism group . The following Lemma describes the locus of such K3 surfaces.
A BN general K3 surface of genus is contained in the smooth Fano threefold if and only if is not contained in .
By Theorem 1.4, we already know that a BN general K3 surface is a complete intersection of a quadric hypersurface and a codimension three linear section . We know that is isomorphic to if is smooth.
If is singular, we claim that must contain a quadric surface . Actually, the codimension three linear sections in have been determined by Todd and Kimura (cf. [Todd],[Ki82]), and it is straightforward to see the existence of the quadric surface. For instance, a codimension three linear section with an singularity (this is the generic case) is defined in as follows:
Then there exists a quadric cone in . The K3 surface is contained in since the intersection is an elliptic curve of degree four. ∎
1.6. Non-BN general K3 surfaces
In this subsection, we classify all non-BN general K3 surfaces for () and interpret them as a union of NL divisors in . This is natural because non-BN general K3 surfaces must contain some special curve, and hence lie in some NL divisor .
The locus of non-BN general K3 surfaces in is a union of the NL divisors satisfying
In particular, a quasi-polarized K3 surface of genus and is non-BN general if and only if it lies in one of the following NL divisors:
, , ,
, , , , ,
First, suppose is not BN general. Then there exist line bundles satisfying and
To compute and , let us recall some results about the linear systems on K3 surfaces. Let be an effective divisor on . Saint-Donat has shown that
When has no fixed component, then
, then is base point free and . The Riemann-Roch Theorem yields
, then for some base point free line bundle satisfying and .
When with the fixed component and has no fixed component, then is either base point free or by (i). For each connected reduced fixed component of F, we have and or if is base point free; or if . The converse is also true.
A connected component of is said to be of Type I if it is reduced with (resp. ) and of Type II otherwise. The first cohomology of vanish if only has Type I fixed component and is base point free.
Coming back to the proof, we discuss two cases:
Case 1. If contains a fixed component, then is an elliptic K3 surface with a section which lies in (See also 1.7). The assertion holds.
Case 2. If has no fixed component which forces , we claim that we can find divisors such that and . Admitting this, we set and , then and . Without loss of generality, we assume . Thus we obtain
Now we prove the claim. Note that the inequality still holds if we replace by its base point free part. So we can assume that is base point free.
Denote by () the sum of all Type I (Type II) fixed components of . Then we can write and has no fixed component. Then the line bundles
and has zero first cohomology. This is because is nef, which implies and have no Type II fixed component by a simple intersection computation. Thus we have proven the claim.
Moreover, our assertion follows easily from the fact that the union of is the same as the union of for satisfying the condition (1.3).
Conversely, if lies in one of the NL divisors listed in the statement, then contains an element such that and satisfying the given condition. We have the inequality
A similar computation can be found in [JK04] Chapter 5, where Johnsen and Knutsen classify all non-Clifford general K3 surfaces of genus .
1.8. Projective model of non-BN general K3 surfaces
In this subsection, we describe the projective models of non-BN general K3 surfaces, which will be used later. Indeed, there is a projective model for the general K3 surface in each of NL divisors of Lemma 1.6. Let us start with Saint-Donat’s result:
[Sa74] If is not a birational to its image, then
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. The image of is a rational normal curve of degree in .
The first (resp. second) case happens if and only if admits an elliptic fibration with a section (resp. bisection), which means that contains an elliptic curve of degree one (resp. two). Moreover, we have
For and , the K3 surface lies in if and only if is not birational to its image. Moreover, when , and .
By the discussion above, is not birational if and only if it contains an elliptic curve of degree one or two. The locus of those containing an elliptic curve of degree one is the irreducible divisor because the corresponding rank two sublattice
is always primitive.
Similarly, for and , the locus of those containing an elliptic curve of degree two is the irreducible divisor , except in the case . When , this locus is the union of the two irreducible divisors and , because there are two primitive lattices
which contain an element satisfying and . ∎
For the case is non-BN general and is birational, the image is contained in some rational normal scrolls , i.e. projective bundles over . Here we say a rational number scroll is of type if
for . The image of the natural morphism may be singular if for some , and we say that is a singular rational normal scroll.
For the low genera appearing in this paper, Johnsen, Knutsen [JK04] and Hana [Ha02] have classified all the non-BN general K3 surfaces whose projective models lie in various rational normal scrolls. Using their results, one can find the projective models of the K3 surfaces lying in each irreducible component of the non-BN general locus. For instance, when , we have
if is the hypersurface of a rational normal scroll of type ;
if is the complete intersection of a singular rational normal scroll of type .
We refer the readers to [JK04] Chapter and [Ha02] Chapter for the complete list of all possible projective models.
2. Birational models of
This section is devoted to the description of a birational model of via geometric invariant theory (GIT). More precisely, we can interpret the moduli space of BN general K3 surfaces as a moduli space of smooth complete intersections, using Theorem 1.4. The latter can be constructed via GIT and this allows us to compute the Picard group of directly.
2.1. GIT construction
Recall that the general K3 surface in among the range of Mukai models is a complete intersection with respect to some vector bundle on a homogenous space , by Theorem 1.4. We obtain the GIT model of the moduli space as follows:
(I). For , let , the complete intersections are parametrized by Grassmannians with a natural group action of coming from the action on :
, is the quotient of by its center;
We denote by
the natural rational map to , where is the semistable locus of . As proved by Mukai (cf. [Mu88] Theorem 0.2), is birational, and thus the image of contains an open subset of . Our goal of this section is to describe the image of smooth complete intersections in via .
(II). The case of is slightly different. The BN general K3 surface is a hyperplane section of the smooth Fano threefold for some non-degenerate . This is a complete intersection with respect to the vector bundle on . Since , we have a natural parameter space of non-degenerate which is birational to the GIT quotient
The moduli space is birational to the -bundle , where the fiber over an element is the projective space . In the rest of this section, we will discuss the GIT stability of the smooth elements in . We start with the study of discriminant loci of complete intersections, which plays an important role in the proof of the main theorem.
2.2. Discriminant loci of complete intersections
First, we discuss general discriminant loci in moduli of complete intersections with respect to vector bundles, and then we restrict to our cases.
Let be a smooth projective variety and a globally generated line bundle on . The discriminant locus of is defined to the subset of parameterizing singular elements in . When is very ample, we set and let be the dual projective space. The discriminant locus is isomorphic to the dual variety , which is an irreducible subvariety in . Moreover, one can easily get:
The dual variety is an irreducible hypersurface of if there is an element such that has only isolated singularities.
The proof is very standard, so we give only a sketch. Consider the incidence variety
with two projections and .
By our assumption, the image of the first projection is and is generically finite. Also, the second projection is surjective and each fiber of is a projective space , where . Thus is irreducible of codimension one. ∎
The irreducibility of the dual variety actually holds for any irreducible variety . It is usually of codimension one in . When has codimension for some , then is -ruled (cf. [GKZ94] Chapter 1.1).
If is not very ample, we denote by the morphism given by the linear system and let be the image of . The pullback induces an isomorphism
Let be the union of the singular locus of the morphism and the singular locus of . Suppose that there exists an element whose zero locus has only isolated singularities lying outside . Then we have
the discriminant locus of is irreducible of codimension one.
the discriminant locus contains an open subset of the image , and hence contains as an irreducible component.
It follows that
Suppose that there exists an element whose zero locus has only isolated singularities lying outside . There is an irreducible component , which is isomorphic to and has codimension one in . In particular, if the morphism is smooth.
Without the assumption on the singularities of , one can still obtain an irreducible subvariety of isomorphic to , but it may not be an irreducible component of . See Example 1.1.0 in [LM08].
Now, we extend the results above to the case of a globally generated vector bundle on . This will allow us to deal with complete intersections. Let us define
Then we have:
Let be the associated projective bundle on , and its relatively ample line bundle. There is a natural isomorphism
The zero locus of a section in is smooth if and only if the section is non-degenerate and the zero locus of is smooth in . In other words, is isomorphic to .
Moreover, with the same assumption as in Proposition 2.4 for the line bundle , contains an irreducible hypersurface in .
See [Mu92] Proposition 1.9 for the first assertion. The last assertion follows directly from Proposition 2.4. ∎
If the vector bundle is a direct sum of -copies of a globally generated vector bundle on , we parametrize the complete intersections with respect to by the Grassmannian and define
which is called the discriminant locus of in .
The same assertion of Proposition 2.5 holds for in .
One can use a similar argument as in Lemma 2.3, where the fiber of the second projection will be an irreducible Schubert variety, or just note that is the quotient . We omit the details here. ∎
For our purpose, we actually only deal with the case where the vector bundle is very ample, so the discriminant locus will be irreducible. One may observe that the dual of universal subbundle on is not ample, but we have another construction for that case. Proposition 2.5 and Lemma 2.6 are motivated by the study of K3 surfaces of genus and (cf. [Mu92]).
2.7. GIT Stability
Now we discuss GIT stability of the smooth complete intersections in 2.1. We say that a non-degenerate threefold is non-special if it does not belong to Prokhorov’s class of genus 12 Fano threefolds defined in [Pr90]. Then we start with a useful lemma:
Let be either a K3 surface or a non-special smooth non-degenerate threefold . Let be the natural polarization on . The group of automorphisms that preserve the polarization , i.e. , is finite.
See [Hu] Chapter 5, Proposition 3.3 for K3 surfaces and [Pr90] for automorphism groups of Fano threefolds. ∎
By [Pr90], there are only three types of Fano threefolds of genus with infinite automorphism group. One is the Mukai-Umemura manifold constructed in [MU83] with automorphism group , and the other two have automorphism group and . The moduli space for each of these types is at most one dimensional.
For , if is one of the smooth complete intersections described above, then is GIT stable in . Moreover, the smooth non-degenerate complete intersection is GIT stable in if it is non-special.
Let denote the discriminant locus, in the sense of (2.4), for . By Lemma 2.6, is an irreducible hypersurface in , since there are complete intersections with only one rational double point. Indeed, the K3 surface with a primitive Picard lattice
will be such a singular complete intersection.
The discriminant locus is thus cut out by a single homogeneous equation , named the discriminant form. Moreover, it is easy to see that the discriminant form is -invariant because the property of singularity is preserved under changes of coordinates. Hence is a -invariant function. The semistability of the smooth surface then follows from the fact that does not vanish at . It follows that is GIT stable because only has finite stabilizer by Lemma 2.8.
For the case , we employ a similar idea, using a discriminant form. Let be the locus of non-degenerate elements in and the complement of . As any smooth non-degnerate has only finitely many automorphisms if it is non-special, it suffices to show that is an irreducible -invariant divisor in . To prove this, let us consider the incident variety:
with the first projection . The second projection is surjective, and the fiber for any can be described as follows:
Set , and the quotient vector space. Then is isomorphic to the -dimensional irreducible variety
where is the space of decomposable vectors in , and acts simultaneously by scaling on and .
Next, we define the correspondence