Moduli Space of Quasi-Polarized K3 Surfaces of Degree 6 and 8*

Zhiyuan LI Zhiyu TIAN

Abstract In this paper， the authors study the moduli space of quasi-polarized complex K3 surfaces of degree 6 and 8 via geometric invariant theory.The general members in such moduli spaces are complete intersections in projective spaces and they have natural GIT constructions for the corresponding moduli spaces and they show that the K3 surfaces with at worst ADE singularities are GIT stable.They give a concrete description of boundary of the compactification of the degree 6 case via the Hilbert-Mumford criterion.They compute the Picard group via Noether-Lefschetz theory and discuss the connection to the Looijenga’s compactifications from arithmetic perspective.One of the main ingredients is the study of the projective models of K3 surfaces in terms of Noether-Lefschetz divisors.

Keywords K3 surfaces，GIT，Noether-Lefschetz divisor，Looijenga compactification

1 Introduction

A primitively quasi-polarized K3 surface (S，L) of degree 2? over C consists of a complex K3 surfaces， a big and nef line bundle L such that c1(L) ∈H2(S，Z) is a primitive class and L2= 2?.Let F2?be the moduli space of primitively quasi-polarized complex K3 surfaces of degree 2?.It is well-known that the period map behaves very well on F2?.Namely，if we denote by D the period domain of K3 surfaces and Γ2?the monodromy group， global Torelli theorem tells us that F2?is isomorphic to Γ2?D via the period map.

Besides the Hodge theoretical construction， there is also explicit algebraic construction of F2?via geometric invariant theory (GIT for short) for low degree K3 surfaces， where such a general K3 surface is a complete intersection in the projective space.For instance， the GIT construction of F2and F4has been worked out by Shah (see [20–21]).When 2? = 6 or 8， a general element (S，L) ∈F2?is a complete intersection of a smooth quadric and a cubic in P4or a complete intersection of three quadrics in P5respectively.

In this paper，we describe the GIT construction of moduli space of these complete intersects and characterize the image of period map for such complete intersections as a complement of certain Noether-Lefschetz (NL for short)divisors in Γ2?D.The latter one has a natural arithmetic compactification constructed by Looijenga(see[14])， and we will compare this arithmetic compactification with the natural GIT compactification.

More precisely， for any non-negative integers d，g， the NL-divisoris defined to be the locus of quasi-polarized complex K3 surfaces (S，L) ∈F2?such that Pic(S)contains a rank two primitive sublattice of the following form:

for some β ∈Pic(S).For simplicity of notations， we identifyas divisors on Γ2?D via period map.One of our main result is in the following.

Theorem 1.1For? = 3，4， the complete intersections inP?+1of degree2?with at worst simple singularities(i.e.， isolatedADEsingularities)areGIT-stable.LetM2?be the moduli space of such complete intersections with at worst simple singularities.Then the period map extends toM2?and its image inΓ2?Dis the complement of，and.

Furthermore， the naturalGITcompactificationis not isomorphic to Looijenga’s compactification of the complement

Remark 1.1We refer the readers to [6] for the analysis of GIT stability for such complete intersections with semi log canonical singularities.

Secondly， we have classified the boundary components of M6in its GIT compactification.The main result is as follows.

Theorem 1.2The boundaryconsists of9irreducible components whose general memberXis described as follows:

α)(Semitable)Xhas two corank3singularities， but not a union of a quadric surface and a quadric cone with multiplicity two.

γ)(Semistable)Xhas two simple elliptic singularities of type， whose projective tangent cone intersectXalong lines， and not the union of three quadric cones.

δ)(Stable)Xhas an isolatedsingularity.

∈)(Stable)Xhas an isolatedsingularity， whose projective tangent cone meetsXat a point.

ζ)(Stable)Xis singular along a line.

η)(Stable)Xis singular along a conic.

θ)(Stable)Xis singular along a twisted cubic.

φ)(Stable)Xis singular along a rational normal curve of degree4.

The stratumαis6-dimensional，is7-dimensional(it contains a2-dimensional semistable lociβ)， γandφare2-dimensional，δandζare11-dimensional，∈is8-dimensional，ηis7-dimensional，θis3-dimensional.

According to the work of[22]，the Baily-Borel compactification(Γ6D)*of the Shimura variety Γ6D consists of 10 irreducible components.The extended period map induces a birational map

According to the spirit of Hassett-Keel-Looijenga program raised by[12]，it is expected that the map (1.2) can factor through a sequence of elementary birational transformations of Shimura type， i.e.， the exceptional loci comes from Shimura subvarieties.This problem will be solved in a forthcoming paper (see [8]).

In [16]， Maulik and Pandharipande have conjectured that the Picard group of F2?with Q-coefficients is spanned by the NL-divisorsThis conjecture has been verified in [3] via automorphic representation theory and finding a geometric approach remains highly interesting.Here， using the main theorem， we can compute the Picard group of F6and F8from the GIT construction.

Corollary 1.1When2? = 6or8， the Picard groupPicQ(F2?)with rational coefficients isspanned byNL-divisors，d=1，2，3，4.Moreover，

for2?=6or8.

The first part of this result has been also obtained by O’grady in [18] using a slightly different method.We just point out that the similar approach has been applied to K3 surfaces with Mukai models (i.e.， 10 ≤2? ≤18 or 2?=22) in [9].

2 Noether-Lefschetz Divisors for K3 Surfaces

Let us recall the Noether-Lefschetz theory on K3 surfaces.

2.1 Noether-Lefschetz divisors

Let(S，L)be a primitively quasi-polarized K3 surface of degree 2?.The middle cohomology Λ:=H2(S，Z) is a unimodular even lattice of signature (3，19) under the intersection form 〈，〉.Let h2?=c1(L)， then the orthogonal complement Λ2?:=?Λ is an even lattice of signature(2，19)， which has a unique representation:

where 〈ω，ω〉=-2?， U is the hyperbolic plane and E8(-1) is the unimodular， negative definite even lattice of rank 8.

The monodromy group

naturally acts on D， where Aut(Λ2?)+is the identity component of Aut(Λ2?).According to the global Torelli theorem of K3 surfaces， there is an isomorphism

via the period map.Then F2?is a locally Hermitian symmetric variety with only quotient singularities， and hence Q-factorial.

where M runs for all rank two primitive sublattice of Λ2?of the form (1.1).In the language of Heegner divisors， the right-hand side of (2.2) is called the arithmetic quotient of hyperplane arrangement in D.As known in [18， Proposition 1.3]， we have the irreducibility theorem.

Theorem 2.1All theNL-divisors∈PicQ(F2?)are irreducible.

Remark 2.1The definition of NL-divisors we used here is slightly different from the one in[16].Maulik and Pandharipande define the NL-divisors without the assumption of primitivity of the sublattice M in(2.2).But the span of these divisors are the same as ours(see[16，§0.2]).

2.2 Dimension formula

Let us denote by PicQ(Γ2?D)NLthe subgroup of PicQ(Γ2?D) generated by NL-divisors with Q-coefficients.By [5， 16]， we know that the dimension ρ2?of the span of Heegner divisors on Γ2?D can be explicitly computed by the following formula:

where {，} denotes the fraction part and G(a，b) is the generalized quadratic Gauss sum:

Lemma 2.1

where

andis the Jacobi symbol.

As shown in[16]，the span of NL-divisors are the same as the span of non irreducible divisors on Γ2?D.

2.3 Projective models of K3 surfaces

Let (S，L) be a smooth K3 surface with a primitive quasi-polarization L of degree 2?.The linear system |L| defines a map ψLfrom S to P?+1.The image of ψLis called a projective model of S.

In [19]， Saint-Donat gives a precise description of all projective models of(S，L) when ψLis not a birational morphism.

Proposition 2.1(see [19])LetLbe the primitive quasi-polarization of degree2?onSand letψLbe the map defined by|L|.Then there are the following possibilities:

1.ψLis birational to a degree2?surface inP?+1.In particular，ψLis a closed embedding whenLis ample.

2.ψLis a generically2 : 1map andψL(S)is a smooth rational normal scroll of degree?，or a cone over a rational normal curve of degree?.

3.|L|has a fixed componentD， which is a smooth rational curve.Moreover，ψL(S)is a rational normal curve of degree?+1inP?+1.

We call K3 surfaces of type(1)，(2)，(3) nonhyperelliptic，unigonal，and digonal K3 surfaces accordingly.When ? = 2，3，4， the projective model of a general quasi-polarized K3 surface(S，L) is a complete intersection in the projective space P?+1.

Remark 2.2Assume that ψLis a birational morphism.Then one can easily see that L is not ample if and only if there exists an exceptional (-2) curve D ?S.The morphism ψLwill factor through a contraction π : S →， whereis a singular K3 surface with simple singularities.

Recalling that the NL-divisorparametrizes all K3 surfaces (S，L) of degree 2? with exceptional (-2) curves.Therefore， the projective model of a general member inis a surface in Pl+1of degree 2? with simple singularities.

In this paper，we mainly consider the case 2?=6 and 8，where the classification of projective models of S can be read offfrom the Picard lattice of S.

Lemma 2.2Let(S，L)be a smooth quasi-polarizedK3surface of degree2? (2? = 6or8).Then

(*) L2= 8andL = L′+E +C， whereCis a rational curve，Eis an irreducible elliptic curve andL′is irreducible of genus two withL′·C = E ·C = 1andL′·E = 2.The imageψL(S)is contained in a cone over cubic surface inP4.

·When?=3，Sis birational to the complete intersection of a singular quadric and a cubic inP4viaψL.

·When? = 4，Sis either birational to a bidegree(2，3)hypersurface of the Serge varietyP1×P2→P5viaψLor in case(*).

ProofThe proof of (1) and (2) are straightforward from Proposition 2.1.See also [19， §2，§5 ] for more detailed discussion.

Now we suppose that a quasi-polarized K3 surface (S，L) ∈is neither unigonal nor diagonal.Then ψLis a birational map to a complete intersection of a quadric and a cubic.Our first statement of (3) comes from the fact that any quadric threefold containing a plane cubic must be singular.If (S，L) ∈， the assertion follows from [19， Proposition 7.15 and Example 7.19].

Remark 2.3We would like to refer the readers to [9–10] for a detailed description of projective models of low degree (2? ≤22) K3 surfaces.

3 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 P4via geometric invariant theory.

3.1 Terminology and notations

In the rest of this paper， we will use the following terminology.Let f(u，v，w)be an analytic function in C[[u，v，w]]whose leading term defines an isolated singularity at the origin.We have the following types of singularities:

· Simple singualrities: Isolated An， Dk， Ersingularities.

· Simple elliptic singularities:

We will use the notation l(x)，q(x) and c(x) as linear， quadratic and cubic polynomials of x=(x0，··· ，xn).

3.2 Cubic sections on quadric threefolds

Let Q be the smooth quadric threefold in P4defined by the equation

Since every nonsingular quadric hypersurface in P4is projectively equivalent to Q， a complete intersection of a smooth quadric and a cubic can be identified with an element in |OQ(3)|.

The automorphism group of Q is the reductive Lie group SO(Q)(C) which is isomorphic to SO(5)(C).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 |OQ(3)| = P(V)， where V is a 30-dimensional vector space defined by the exact sequence

Let us take the set of monomials

to be a basis of V.Sometimes， we may change the basis for simpler computations.

3.3 Numerical criterion

Now we classify stability of the points in P(V) under the action of SO(Q)(C) by applying the Hilbert-Mumford numerical criterion (see [17]).

As is customary， a one parameter subgroup (1-PS) of SO(Q)(C) can be diagonalized as

for some u，v ∈Z.We call such λu，v: C*→SO(Q)(C) a normalized 1-PS of SO(Q)(C) if u ≥v ≥0.

Let λu，vbe a normalized 1-PS of SO(Q)(C).Then the weight of a monomialB with respect to λu，vis

If we denote by M≤0(λu，v) (resp.M＜0(λu，v)) the set of monomials of degree 3 which have non-positive (resp.negative) weight with respect to λu，v， one can easily compute the maximal subsets M≤0(λu，v) (resp.M＜0(λu，v))， as listed in Table 1 (resp.Table 2).

Table 1 Maximal subsets M≤0(λ)

Table 2 Maximal subsets M＜0(λ)

According to the Hilbert-Mumford criterion， an element f(x0，··· ，x4)∈P(V) is not properly stable (resp.unstable) if and only if the weight of all monomial in f is non-positive (resp.negative) for some 1-PS.Thus we obtain the following lemma.

Lemma 3.1LetXbe the surface defined by an element inP(V).ThenXis not properly stable if and only ifX =Q∩Yfor some cubic hypersurfaceY ?P4defined by a cubic polynomial in one of following cases:

· c(x1，x2，x3，x4);

· x0x3l(x2，x3)+x1x2l1(x3，x4)+x1q(x3，x4)+c(x2，x3，x4);

· x0+x1x3l1(x2，x3)+x1x4l2(x1，x2，x3，x4)+c(x2，x3，x4).

For f ∈P(V)not properly stable，using the destabilizing 1-PS λ， the limitexists and it is invariant with respect to λ.The invariant part of polynomials of type (N1)–(N3) are the followings:

(α) c(x1，x2，x3)=0;

(β) λ1+λ2x1x2x3+λ3x0x2x3+λ4x1x2x4=0， λi∈C;

(γ) λ1+λ2x1x2x3+λ3x0x23+λ4x21x4=0， λi∈C.

Similarly， we get the following lemma.

Lemma 3.2With the notation above，Xis not semistable if and only ifX = Q ∩Yfor some cubic hypersurfaceYdefined by one of the following equations:

· x0+x1q(x3，x4)+c(x2，x3，x4)， andc(x2，x3，x4)has noterm;

· x4q1(x1，x2，x3，x4)+x3q2(x2，x3)+λx1.

3.4 Geometric interpretation of stability

We use the terminology of the corank of the hypersurface singularities as in [1， 12].

Definition 3.1Let0 ∈Cnbe a hypersurface singularity given by the equationf(z1，··· ，zn)=0.The corank of0isnminus the rank of the Hessian off(z1，··· ，zn)at0.

Theorem 3.1A complete intersectionX = Q ∩Yis not properly stable if and only ifXsatisfies one of the following conditions:

(i) Xhas a hypersurface singularity of corank3.

(ii) Xis singular along a lineLand there exists a planePsuch thatP ∩Q=2LandPis contained in the projective tangent coneP(CTp(X))for any pointp ∈L.

(iii) Xhas a singularitypwhich deforms to a singularity ofclass， and the restriction of the projective coneP(CTp(X))toXcontains a lineLpassing throughpwith multiplicity at least6.

ProofAs a consequence of Lemma 3.1， it suffices to find the geometric characterizations of the complete intersections of type (N1)–(N3).Here we do it case by case.

(i) If X is of type (N1)， then X can be considered as the intersection of Q and a cubic cone Y with the vertex p0=[1，0，0，0，0]∈Q.It is easy to see that p0is a corank of 3 singularity of X.

Conversely， we write the equation of Y as

If we choose the affine coordinate

then the affine equation near p0is

in C3.It has a corank 3 singularity at the origin if and only if the quadric q is 0.

(ii) If X is of type (N2)， then the equation of Y is given by

and therefore X is singular along the line L:x2=x3=x4=0.

Moreover，for any point p=[z0，z1，0，0，0]∈L， the projective tangent cone P(CTp(X)) at p is defined as

which contains the plane P :x3=x4=0 for each p ∈L and P ∩Q=2L.

Conversely， since the intersection of P and Q is a double line L， we may certainly assume that the plane P is defined by

after some coordinate transform persevering the quadric form Q.Then the line L = P ∩Q is given by x2=x3=x4=0.

Because X is singular along L， the equation of Y can be written as:

Then the projective tangent cone

contains the plane P for each point p = [z0，z1，0，0，0] ∈L only if the quadrics qihave noterm.

(iii) For X of type (N3)， a similar discussion is as follows: If Y is defined by

then X = Q ∩Y is singular at p0.After choosing the affine coordinates as (3.3)， the affine equation near p0is

for some polynomials ?，f，g with ? linear，deg(f)≥1，deg(g)≥3.Therefore，p0is a hypersurface singularity of corank 2 and its projective tangent cone is a double plane 2P :=x4=0.The remaining part is straightforward.

Conversely， we take p0to be the isolated singular point which deforms to a singularity ofclass.As it has corank at least 2， the equation of Y can be written as

Then the quadric q1(x1，x2，x3)is of the form l(x1，x2，x3)2for some linear polynomial l because p0is singular of corank at least 2.

After we make a coordinate change preserving Q and p0， the defining equation of Y has two possibilities:

1.x0+x1q(x1，x2，x3，x4)+c(x2，x3，x4)=0，

2.x0+x1q(x1，x2，x3，x4)+c(x2，x3，x4)=0.

The projective tangent cone at P(CTp0(X)) is a double plane

The line L contained in the restriction of 2P to X has to be defined by x2= x3= x4= 0.It follows that the first case can not happen since P ∩X contains L with multiplicity at least 3.

In the second case，the multiplicity condition further implies that the quadric q(x1，x2，x3，x4)does not haveterms.To see that there is no x1x3term， note that the affine local equationnear p0can be written as

where a，b ∈C， g，c are polynomials with deg(c) ≥3.Since p0deforms to， we know that a has to be 0.

Theorem 3.2A complete intersectionX =Q∩Yis unstable if and only ifXsatisfies one of the following conditions:

(i′) Xis singular along a lineLsatisfying the condition:There exists a planePsuch thatP(CTp(X))=2Pfor any pointp ∈L;

(ii′)there exists a planePwhose restriction toXis a lineLwith multiplicity6andXhas a corank3singularityponL.Moreover， the projective tangent coneP(CTp(X))atpis the union of the planePand a quadric surface and they meet atLwith multiplicity two.

ProofWe check the complete intersections of type (U1)–(U2) case by case.

(i′) To simplify the proof， we choose another monomial basis of V as below:

Then the polynomial of type (U1) has the form

At this time， X is singular along the line L : x2= x3= x4= 0 and satisfies the condition described in (i′).

On the other hand， the line L on Q can be written as

for a suitable change of coordinates preserving Q.Then the equation of Y has the form

where qidoes not containterm.

Moreover， for any point p = [z0，z1，0，0，0] ∈L， the projective tangent cone P(CTp(X)) is given by

They have a common plane P with multiplicity 2 if and only if P is defined by x3= x4= 0 and qi(x2，x3，x4) does not contain the x2x3，x2x4terms.(ii′) When Y has the equation

one observe that X contains the line L : x2= x3= x4= 0 which is contained in the plane P := x3= x4= 0.It is easy to see that P intersect with X is the line L with multiplicity 6.Moreover，X is singular at p0=[1，0，0，0，0] and the projective cone at p0is given by

which is the union of the plane X1: x3= x4= 0 and the quadratic surface X2: x4=q2(x2，x3)+λx1x3= 0 satisfying the desired conditions.The proof of the converse is quite similar as the previous cases and we omit the details here.

Corollary 3.1A complete intersectionX =Q ∩Yis semistable(resp.stable)ifXhas at worst isolated singularities(resp.simple singularities).

ProofBy Theorem 3.2，the singular locus of X is at least one dimensional if it is unstable.Then X has to be semistable if it has at worst isolated singularities.

Next， from Theorem 3.1， we know that if X is not properly stable， then either X is singular along a curve or it contains at least an isolated simple elliptic singularity.It follows that X with simple singularities is stable.

Now it makes sense to talk about the moduli space M6of complete intersections of a smooth quadric and a cubic with simple singularities.Let U6be the open subset of P(V)sparameterizing such complete intersections in P4.Then we have M6=U6/SO(5)(C).

Theorem 3.3There is an open immersionP6: M6→F6via the period map and the image ofP6inF6is the complement of threeNL-divisors，and.The Picard groupPicQ(F6)is spanned by{， 1 ≤d ≤4}.

ProofFor 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(-2)curve.Therefore， we obtain an open immersion P6:M6→F6from Torelli theorem.By Lemma 2.2，we know that the boundary divisors of the image P6(F6) is the union of，and.

Next， the moduli space M6is isomorphic to the quotient U6/SO(5)(C).Observing that Pic(U6) ～= Pic(P(V)) has rank one since the boundary of U6in P(V) has codimension at least two， we claim that the dimension of PicQ(M6) is at most one.Denote by Pic(U6)SO(5)(C)the set of SO(5)(C)-linearized line bundles on U6.There is an injection

by[11，Proposition 4.2]for the reductive group SO(5)(C).Our claim then follows from the fact that the forgetful map Pic(U6)SO(5)(C)→Pic(U6) is an injection.Actually， one can easily see that PicQ(M6) is spanned by the descent of the tautological line bundle OU6(1) on U6to the quotient U6/SO(5)(C)， and we denote it by OM6(1).

Since the complement of P6(M6) in F6is the union of three irreducible divisors and dimQ(Pic(F6)) ≥4， it follows that PicQ(F6) is spanned by NL-divisors {，1 ≤d ≤4}by the dimension consideration.

Remark 3.1There is another natural GIT construction of moduli space of complete intersections in projective spaces， see [2， 14].There exists a projective bundle π : PE →P(H0(P5，OP5(2))) ～= P14parameterizing all complete intersections of a quadric and a cubic in P5.Then one can consider the GIT quotient

for the line bundle Ht= π*OP14(1)+tOPE(1).We want to point out that P(E)//HtSL5(C)is isomorphic to our GIT quotient P(V)//SO(5)(C) when t ＜.This will be discussed in the upcoming paper [8].

3.5 Minimal orbits

In this section， we give a description of the semistable boundary components of the GIT compactification.It consists of strictly semistable points with minimal orbits.From Subsection 3.2， it suffices to discuss the points of type (α)–(γ).As in [12]， our approach is to use Luna’s criterion.

Lemma 3.3(Luna’s criterion (see [15]))LetGbe a reductive group acting on an affine varietyV.IfHis a reductive subgroup ofGandx ∈Vis stabilized byH， then the orbitG·xis closed if and only ifCG(H)·xis closed.

To start with， we first observe that Type (α)， (β) and (γ) have a common specialization，which we denote by Type (ξ):

Lemma 3.4IfXis of Type(ξ)， it is strictly semistable with closed orbits.

ProofThe stabilizer of Type (ξ) contains a 1-PS:

of distinct weights.So the center

is a maximal torus.It acts onIt is straightforward to see any element of Type (ξ) is semistable with closed orbit in VHunder the action.Then the statement follows from Luna’s criterion.

Proposition 3.1LetXbe a surface of Type(α).Then it has two corank3singularities.Moreover， we have

1.Xis unstable if it is union of a quadric surface and a quadric cone with multiplicity two.

2.The orbit ofXis not closed ifXis singular along two lines.It degenerates to type(ξ).Otherwise，Xis semistable with closed orbit.

ProofThe stabilizer of Type (α) contains a 1-PS:

The center CG(H1) ～= SO(Q1)(C)×SO(Q2)(C)， where Q1= x0x4and Q2= x1x3+.The group SO(Q1)(C) ～= SO(2;C) acts linearly on variable x0，x4， while SO(Q2)(C) ～= SO(3)(C)acts linearly on the variables x1，x2and x3.

The action of CG(H1) onis equivalent to the action of SO(Q2)(C) on the set of cubic polynomials in three variables x1，x2，x3preserving the quadratic form Q2.By Luna’s criterion， we can reduce our problem to a simpler GIT question VH1//SO(3)(C).Any 1-PS λ:C*→SO(Q2)(C)of SO(Q2)(C)can be diagonalized in the form

The remaining cases can be shown in a similar way.Here we omit the proof.

Proposition 3.2LetXbe a surface of type(β).Then it is a union of a quadric surface and a complete intersection of two quadrics.Moreover， we have

(i) Xis unstable ifXconsists of two quadric cones and a quadric surface intersecting at a line.

(ii)The orbit ofXis not closed if its equation can be written asλ1+λ2x1x2x3+λ3x1x2x4up to a coordinate transform preservingQ.It degenerates to type(ξ).Otherwise，Xis semistable with closed orbit.

Proposition 3.3A general memberXof type(γ)has two simple elliptic singularities of type.Moreover， we have

(i) Xis unstable ifXconsists of three quadric cones.

(ii)The orbit ofXis not closed if its equation has the formλ1+λ2x1x2x3+λ3x4up to a coordinate change preservingQ.

Otherwise，Xis semistable with closed orbit.

4 Stable Singular Complete Intersection

In this section， we will discuss the stable loci of singular complete intersections of a smooth quadric and a cubic hypersurface.As a result， we prove Theorem 1.2.

4.1 Stable complete intersection with isolated singularity

Let X = Q ∩Y be a complete intersection of the smooth quadric Q and a cubic threefold Y.A first observation is in the following.

Proposition 4.1IfX = Q ∩Yhas only isolated singularity， thenXis stable if and only if the non-ADEsingularities can only be one of the following situations:

Up to a coordinate change， the general equation ofYforY ∩Qwith ansingularity is

Similarly， the general equation ofYforY ∩Qwith ansingularity of typeii)is

where?is linear andcis a cubic polynomial inx1，x3，x4.

ProofSuppose X has only at worst isolated singularities of type i) or ii).This means that X does not have a corank 3 singularity orsingularity whose projective tangent cone meets X along a line.By Theorem 3.1， we know that X is stable.Conversely， suppose X is stable and it has a non-ADE isolated singularity at p = [1，0，0，0，0].If p is not of type i) or ii)， by Theorem 3.1(i)， p has to be atype singularity.Let us analysis the local equation of p.As in the proof in Theorem 3.1， up to a change of coordinates preserving the quadric Q， the defining equation of Y has two possibilities:

1.x0+?(x1，x2，x3，x4)+x1q(x2，x3，x4)+c(x2，x3，x4)=0，

2.x0+?(x1，x2，x3，x4)+x1q(x2，x3，x4)+c(x2，x3，x4)=0.

In the first case， the affine local equation near p can be written as

where fdis a homogenous polynomial of degree d.Note that there is no termandin the fourth and fifth jet.One can easily see that p can not be antype singularity.

In the second case， we know that the projective tangent cone P(CTp(X)) meets X along either a line L:x2=x3=x4=0 or the point p(with multiplicity).If the intersection is a line，X can not be stable by the proof in Theorem 3.1.The only possibility is that the intersection is a point.In this situation， the third jet contains the termand the weights on variables x1，x2，x3areThe equation of Y is of the form

where ? is linear and c is a cubic in x1，x3，x4.

At the end， let us give the general equations for X with ansingularity.Without loss of generality， we assume the singularity is at p = [1，0，0，0，0].As it is corank 2， the defining equation of Y can be written as

or

The weights on(x1，x2，x3)are eitherIn either case，the direct computation shows that all monomials in c(x1，x3，x4) and c′(x1，x2，x4) must have x4term.The assertions follows.

From the proof， we can see that if X has only isolated singularities oftype， then X will be stable if the projective tangent cone P(CTp(X)) of the singularity meets X at a point and X is strictly semistable if P(CTp(X)) meets X along a line.

4.2 Stable loci of complete intersection with non-isolated singularity

Let us consider the non-normal case.With the notations as above， we denote by Sing(X)the singular loci of X.Then we have the following theorem.

Theorem 4.1LetXbe a complete intersection of a smooth quadricQand a cubic hypersurfaceYwith non-isolated singularities.Then one of the following holds:

i) Sing(X)contains a line.The general equations of suchXare of the form

whereq1，q2are quadrics andcis a cubic.

ii) Sing(X)contains a conic.The general equations of suchXare of the form

where?iare linear polynomials inx0，··· ，x4.

iii) Sing(X)contains a twisted cubic.The general equations ofXare of the form

whereai，bi∈C，?′is a linear polynomial inx0，··· ，x4and?represents a linear polynomial in four variables.

iv) Sing(X)contains an elliptic curve of degree four.The general equations of suchXare of the form

where?is a linear polynomial andqis a quadric polynomial.

v) Sing(X)contains a rational normal curve of degree four.The general equations of suchXare of the form

where?iare linear polynomials in(4.6)andΔiare quadric polynomials defined in(4.4).Moreover， the general members of each type is stable.

ProofLet C ?Sing(X) be an irreducible curve.If X = X1∪X2is reducible， then deg(Xi) = 2 or 4 as Xiis contained in a smooth quadric threefold Q.The only possibility is X =Q ∩Y with Y a union of a P3and a quadric threefold.This is exactly type ().

If X is irreducible， taking a general hyperplane H， then H ∩X is an irreducible curve singular along H ∩C.Note that the arithmetic genus of H is at most 4， H ∩C has at most 4 points.It follows that the degree of C is at most 4.Hence C can be a line， a conic， a plane cubic， a twisted cubic or a rational normal curve of degree 4.If C is a plane cubic， then C is contained in the intersection P2∩Q， which is a conic.This is clearly impossible.Let us now describe their equations of X case by case.

i) Take the quadric threefold Q : x0x4+x1x3+= 0 and we can assume X is singular along the line C :x0=x1=x2=0.The equation of Y is of the form

where ?iare linear， qiare quadric and c is a cubic polynomial in x0，x1and x2.Then the Jacobian of X on the line C given by

has rank one.The only possibility is that all ?i=0.This gives the equation (ξ).

ii) Take the quadric as above and we assume that X is singular along a smooth conic C : x0= x4= x1x3+= 0.The equation of a cubic hypersurface Y containing C is of the form

for some quadric polynomials q1and q2.Similarly as i)， one can compute that the equation of Y is of type (η).

iii)Take Q as above.If C is an elliptic curve of degree 4， the span of C is a three dimension linear subspace，denoted by H.Note that H ∩X can not be a curve as C is contained in H ∩X with multiplicity at least 2.This means H ∩X is a surface and thus X is reducible.Such X is of type ().

iv) If C is a twisted cubic， the defining equations of C can be written as

Then the equations of a complete intersection X containing C can be written as

for some linear polynomials ?i，?，?′and a quadric polynomial q.If X is singular along C， then via computing the Jacobian of (4.3) ， we get the equation (θ).

vi) Without loss of generality， we can assume that the curve C is defined by equations

and Q is defined by the equation 2Δ1-Δ6= 0.As Y contains C， we may assume that the equation of Y is given by

for some linear polynomial ?i.As X is singular along C， the Jacobian matrix of X along C is

Then via a computation， one can get the linear functions ?iare of the form

There are nine parameters ai，bifor i = 0，1，2 and cjfor j = 1，2，3.We left the details to readers.

Finally， the assertion of stability follows directly from Theorem 3.1.

Proof of Theorem 1.2It basically follows from the combination of Propositions 3.1–3.3， 4.1 and Theorem 4.1.The strata α，β，γ are strictly semistable， which are described in Propositions 3.1，3.2 and 3.3 respectively.The dimension of these components can be computed via Luna’s slice theorem as below.

With the notations as in Propositions 3.1–3.3， we have

1.For ζ， note that the NL-divisorparametrizing X containing a line has dimension 18.The general member inis of the form (4.1).Thus one can see that ζ has codimension 7 inand it follows dim ζ =11.

2.Similarly， the general equations of elements inare given in (4.2)，(4.3) and (4.5) respectively.Then one can directly see that η has codimension 7 in， while θ has codimension 15 inand φ has codimension 16 in.

4.For δ， it can be viewed as the quotient space P(V)/G1， where V is the vector space spanned by monomials in the equation (δ) and G1is the subgroup of SO(5)(C) fixing the singular point p0and the hyperplane x2=0.As dim V =17 and dim G1=5，we get dim δ =11.

5.Similar as above， ∈is the quotient space P(V′)/G2with dim V′=14 and dim G2=5.It follows that dim ∈=8.

5 Complete Intersection of Three Quadrics in P5

Let W = H0(P5，OP5(2)) be the space of global sections of OP5(2).Since every complete intersection X is determined by a net of quadrics Q1，Q2，Q3， the complete intersection of three quadrics are parametrized by the Grassmannian Gr(3，W).The moduli space of complete intersections can be constructed as the GIT quotient Gr(3，W)ss//SL6(C).In this situation，the complete GIT strata is very complicated.For example，see[7]for the GIT stability of a net of quadrics in P4.However， we are satisfied with the following result.

Theorem 5.1LetXbe a complete intersection of three quadrics inP5.IfXhas at worst simple singularities， thenXisGITstable.

5.1 Set up

We first make some notations.Given a net of quadrics{Q1，Q2，Q3}，the Pl¨ucker coordinates of {Q1，Q2，Q3} in P(∧3W) can be represented by

for three distinct pairs (ik，jk).

Let λ:C*→SL6(C)be a normalized one-parameter subgroup，i.e.，λ(t)=diag(ta0，ta1，··· ，ta5)satisfying a0≥a1≥···≥a5andWe denote by

the weight of the monomial xixjwith respect to λ.The weight of a Pl¨ucker coordinate xi1xj1∧xi2xj2∧xi3xj3with respect to λ is simply

5.2 Numerical criterion for nets

By the Hilbert-Mumford numerical criterion， a net of quadrics{Q1，Q2，Q3} is not properly stable if and only if for a suitable choice of coordinates， there exists a normalized 1-PS λ :t →diag(ta0，ta1，··· ，ta5) such that the weight of all Pl¨ucker coordinates of {Q1，Q2，Q3} with respect to λ is not positive.We say that {Q1，Q2，Q3} is not properly stable with respect to λ.

Given a normalized 1-PS λ:C*→SL6(C)， we can define two complete orders on quadratic monomials:

2.“＞λ”:xixj＞λxkxlif either wλ(xixj)＞wλ(xkxl) or wλ(xixj)=wλ(xkxl) for a given normalized 1-PS λ and xixj＞xkxl.

Since the 1-PS λ:C*→SL6(C) is normalized， xixj＞λxkxlimplies max{i，j}＞min{k，l}.

We denote by mithe leading term of Qiwith respect to the order “ ＞λ” and we say that a monomial xkxl/∈Qiif the quadratic polynomial Qidoes not contain xkxlterm.Moreover，we can always set

up to replacing Q1，Q2，Q3with a linear combination of the three polynomials.Then the term m1∧m2∧m3appears in the Pl¨ucker coordinates of Q1∧Q2∧Q3and has the largest weight with respect to λ.Hence the net {Q1，Q2，Q3} is not properly stable with respect to λ if and only if wλ(m1∧m2∧m3)≤0.

Lemma 5.1With the notation above， letXbe the complete intersectionQ1∩Q2∩Q3.ThenXhas a singularity with multiplicity greater than two if one of the following conditions does not hold:

(1) m1≥λx0x4;

(2) m2≥λx1x5ifm1=， andm2≥λx0x5otherwise;

(3) m3≥λifm1＜λx0x3.Moreover，Xis singular along a curve if one of the following conditions does not hold:

(1′) m1≥λifm3＜λx1x5;orm1≥λmax{x1x3，}ifm2＜λx1x4;

(2′)m2≥λifm3＜λx2x5;m2≥λmax{x1x4，}ifm1＜λ;andm2≥λmax{x2x4，}otherwise;

(3′) m3≥λmax{x3x5，}.

ProofLet p0be the point [1，0，0，0，0，0] in P5.For (1) and (2)， if either m1＜λx0x4or m2＜λx0x5and m1＜the surface X contains the point p0and two quadrics Q2，Q3are both singular at p0.It follows that multiplicity of p0is greater than 2.

If m1=and m2＜λx1x5， then X is singular along the two points

with multiplicity greater than 2.Similarly， one can easily check our assertion for (3).

For(1′)， (2′)and(3′)，we will only list the singular locus of X and leave the proof to readers:

· X is singular along the line L:x2=x3=x4=x5=0 if condition (1′) is invalid.

· X is either reducible or singular along L or C1: x3=x4=x5= Q1=0 if condition (2′)is invalid.

· X is either reducible or singular along the curve C2:x4=x5=Q1=Q2=0 if condition(3′) is invalid.

As before，we need to know the maximal set M≤0(λ)of triples of distinct quadratic monomials {q1，q2，q3}， whose sum of their weights with respect to λ is non-positive.Instead of looking at all maximal subsets，we are interested in the maximal subset(λ)which contains a triple{m1，m2，m3} satisfying the conditions (1)–(3)and (1′)–(3′) in Lemma 5.1.It is not difficult to compute that there are four such maximal subset.See Table 3 below.

Table 3 Maximal set (λ)

Table 3 Maximal set (λ)

a) Maximal triples {q1，q2，q3}Cases λ=(a0，··· ，5 q1 q2 q3(N1′) (2，1，0，0，-1，-2) x0x2，x21 x0x5，x1x4，x22 x2x5，x24(N2′) (3，1，1，-1，-1，-3) x0x3，x21 x0x5，x1x3 x1x5，x23(N3′) (4，1，1，-2，-2，-2) x0x3，x21 x0x3，x21 x23(N4′) (5，3，1，-1，-3，-5) x0x4，x1x3，x22 x0x5，x1x4，x2x3 x1x5，x2x4，x23

The lemma below gives a geometric description of X of type (N1′)–(N4′).

Lemma 5.2LetXbe a general element of type(N1′)–(N4′).ThenXhas an isolated simple elliptic singularity.

ProofObviously， X is singular at p0= [1，0，0，0，0，0].Moreover， p0is an isolated hypersurface singularity when X is general.To show that it is simple elliptic， let us compute the analytic type of p0case by case.

If X is a general element of type (N1′)， then the equations of Qican be written as

up to a linear change of the coordinates.Let us take the local coordinates near p0:

From the first two quadratic equations， one can get

for some formal power series f1∈C[[y1，y3，y4]]≥2，f2∈C[[y1，y3，y4]]≥4and some constants b，b′∈C.Therefore， the local equation of p0is

for some complex number αi.According to Subsection 3.1， the singularity p0is simple elliptic of type.

If X is a general element of type (N2′)， we write the equations as

Still， we take the affine coordinate (5.2) near p0and then we have

for some f ∈C[[y1，y2，y4]]≥2.Thus the local equation around p0is

where g ∈C[[y1，y2]]≥4，g′∈C[[y1，y2，y4]]≥2and α ∈C is a constant.Hence p0is simple elliptic of typeby Subsection 3.1.

One can similarly prove that X has a simple elliptic singularity p0of typewhen it is general of type (N3′)， and of typewhen it is general of type (N4′).

5.3 Image of the period map

Let U8?Gr(3，W) be the open subset consisting of all complete intersections with at worst simple singularities.By Lemma 5.2， we know that U8is contained in the stable locus of Gr(3，W).This proves Theorem 5.1.Moreover， similarly as Theorem 3.3， we can get the following result.

Theorem 5.2LetM8= U8//SL6(C)be the moduli space of the complete intersection of three quadrics inP5with simplest simple singularities.Then

(i)the boundary ofM8inhas codimension≥2;

(ii)there is an open immersionP8:M8→F8as the extended period map and the complement ofP8(M8)inF8is the union of threeNL-divisorsThe Picard groupPicQ(F8)is spanned by{， 1 ≤d ≤4}.

ProofFor (i)， let Δ ?Gr(3，W) be the discriminant divisor which parameterizes singular complete intersections.Then Δ is SL6(C)-invariant and irreducible(see[9]).Consider the GIT quotient Δ//SL6(C).By Theorem 5.1， the general members in Δ is stable， so the boundarylies in the boundary of Δ//SL6(C) as a proper closed subset.It follows thathas codimension two in.

For (ii)， this follows from the same arguments as in Theorem 3.3.

6 Arithmetic Compactification of Locally Hermitian Symmetric Varieties

Baily and Borel compactify the arithmetic quotient Γ2?D to a normal projective varietyby adding finitely many modular curves and singletons，which correspond to the classes of Q-isotropic subspaces ofof dimensions 2 and 1.In [14]， Looijgenga gives an arithmetic compactification of the completment of hyperplane arrangements in Γ2?D in the spirit of Satake-Baily-Borel theory.In our situation，the hyperplane arrangement which we are interested in will be the union of three NL-divisorsfor d=1，2，3.

6.1 A review of Looijenga’s work

Let E be a collection of elements in Λ.The orthogonal complement of β ∈E and h2?in Λ defines a hyperplane Hβ?P().Set DHβ= D ∩Hβto be the hyperplane arrangement and define

to be the complement of all subdomains obtained from E.The quotientis the complement of Heenger divisors.

Looijenga constructed the compactificationfrom the strata of decomposition of rational cones.It can also be viewed as the natural blowdown of certain minimal normal blowup of the Baily-Borel compactificationThe structure of the birational map is explicitly provided that how the hyperplanes Hβintersect inside the period domain D.To make it precise， we fix our temporary notation as follows:

· PO(E): The collection of subspaces M ?which are intersection of the hyperplane arrangements from E.Denote by

the natural projection.The projection also defines a natural subdomain(see [14，Section 7]).

· I(E): The collection of the common intersection of I⊥and hyperplane arrangements from E containing I， where I is a Q-isotropic subspace of.

by [14， Corollary 7.5].

6.2 Application to moduli problem via GIT

Let us discuss the possible geometric interpretation ofFor many known geometric examples，such as Enrique surface，K3 surfaces of degree 2 and cubic fourfolds，the natural GIT compactification is precisely Looijenga’s compactification (see [13–14]).So it is interesting to investigate the relations between the compactifiations from GIT and arithmetic for K3 surfaces.For K3 surface with Mukai models， Laza has first found that the two compactifications do not necessarily coincide.This fails for quartic surfaces in P3.We can show that this actually happens quite often.

In our case，let E2?be the collection of elements β ∈Λ satisfying β2=0 and β·h2?=1，2 or 3.Thenis the completement of three NL-diviosrsfor d= 1，2，3.The following lemma gives a rough description of the dimension of the boundary strata of

Lemma 6.1When2?=6and8， the boundary ofhas codimension1.

ProofTo understand the dimension of boundary， it suffices to consider the intersection of hyperplane arrangements from E2?.Let M be an even lattice of signature (1，17) spanned by h2?and elements e1，e2，··· ，e17satisfying that= 0，eiej= 1 and h2?ei= 3 for i /= j.It is easy to check that this lattice has signature(1，17)and thus can be embedded into Λ.Then the lattice M can represent the intersection of 17 hyperplane arrangements from E2?.This proves the assertion by (6.2).

Corollary 6.1For2? = 6and8， theGITquoitentis not isomorphic to Looijenga’s compactification.

ProofSince M2?is isomorphic tothis is obtained by comparing the dimension of the boundary of

In general，we believe that natural GIT compactifications of K3 surfaces with Mukai models constructed in [9] will not be the same as Looijenga’s compactification.This can be achieved by a similar method.

Another interesting problem is to study the birational maps betweenand M2?.In a sequel to this paper， the authors together with Greer and Laza will study the birational geometry of F6via the variation of GIT and Looijenga’s arithmetic approach.

AcknowledgementWe are grateful to O’Grady and Laza for many useful comments.