Compactifications of the moduli space of plane quartics and two lines

We study the moduli space of triples $(C, L_1, L_2)$ consisting of quartic curves $C$ and lines $L_1$ and $L_2$. Specifically, we construct and compactify the moduli space in two ways: via geometric invariant theory (GIT) and by using the period map of certain lattice polarized $K3$ surfaces. The GIT construction depends on two parameters $t_1$ and $t_2$ which correspond to the choice of a linearization. For $t_1=t_2=1$ we describe the GIT moduli explicitly and relate it to the construction via $K3$ surfaces.


Introduction
The construction of compact moduli spaces with geometric meanings is an important problem in algebraic geometry. In this article, we discuss the case of the moduli of K3 surfaces of degree 2 obtained as minimal resolutions of double covers of P 2 branched at a quartic C and two lines L 1 , L 2 , for which we give two constructions, one via Geometric Invariant Theory (GIT) for the plane curves (C, L 1 , L 2 ) depending on a choice of two parameters for each of the lines, and one via the period map of K3 surfaces. For a particular choice of parameters, we show that the constructions agree. Similar examples include [Sha80], [LS07], [Loo09], [Laz09,Laz10] and [ACT02,ACT11]. Our interest on this example arose after the first two authors considered studying the variations of GIT quotients for a cubic surface and a hyperplane section [GMG16]. The moduli of del Pezzo surfaces of degree 2 with two anti-canonical sections seems to be closely related to the moduli of K3 surfaces considered in this article, since del Pezzo surfaces of degree 2 with canonical singularities can be obtained as double-covers of P 2 branched at a (possibly singular) quartic curve. Also, a generic global Torelli for certain double covers of these K3 surfaces (namely, minimal resolutions of bi-double covers of P 2 along a quartic and four lines, cf. [Gar16,§5.4.2]) can be derived using the results in this article and the methods in [PZ17].
Following the general theory of variations of GIT quotients developed by Dolgachev and Hu [DH98] and independently by Thaddeus [Tha96], we construct GIT compactifications M(t 1 , t 2 ) for the moduli space of triples (C, L 1 , L 2 ) consisting of a smooth plane quartic curve C and two labeled lines L 1 , L 2 in Section 2. These compactifications depend on parameters t 1 , t 2 which are the ratio polarizations of the parameter spaces of quartic and linear homogeneous forms representing C and L 1 , L 2 . We generalize the study in [GMG18] of GIT quotients of pairs (X, H) formed by a hypersurface X of degree d in P n+1 and a hyperplane H to tuples (X, H 1 , . . . , H k ) with several hyperplanes H i , considering the relation between the moduli spaces of tuples with labeled and unlabeled hyperplanes. We then apply the setting to the case at hand, namely plane quartic curves and two lines. One sees in Lemma 2.9 that the space where the set of stable points is not empty can be precisely described. Furthermore, given a particular tuple, we can bound the set of parameters for which it is semistable (cf. Lemma 2.11).
Next we focus on the case when t 1 = t 2 = 1. The moduli space M(1, 1) can also be constructed via Hodge theory (cf. Section 3). The idea is to consider the K3 surface S (C,L 1 ,L 2 ) obtained by taking the desingularization of the double coverS (C,L 1 ,L 2 ) of P 2 branched along the sextic curve C + L 1 + L 2 . Note that genericallyS (C,L 1 ,L 2 ) admits nine ordinary double points (coming from the intersection points C ∩ L 1 , C ∩ L 2 and L 1 ∩ L 2 ). It follows that the K3 surface S (C,L 1 ,L 2 ) contains nine (−2)-curves which form a certain configuration. Call the saturated sublattice generated by these curves M ⊂ Pic(S (C,L 1 ,L 2 ) ). Then the K3 surface S (C,L 1 ,L 2 ) is naturally M-polarized in the sense of Dolgachev [Dol96]. Let M 0 ⊂ M(1, 1) be the locus where the sextic curves C + L 1 + L 2 have at worst simple singularities (also known as ADE singularities or Du Val singularities). By associating to the triples (C, L 1 , L 2 ) the periods of the M-polarized K3 surfaces S (C,L 1 ,L 2 ) one obtains a period map P from M 0 to a certain period domain D/Γ. We shall prove that P is an isomorphism.
Theorem 1.1 (Theorem 3.23). Consider the triples (C, L 1 , L 2 ) consisting of quartic curves C and lines L 1 , L 2 such that C +L 1 +L 2 has at worst simple singularities. Let S (C,L 1 ,L 2 ) be the K3 surface obtained by taking the minimal resolution of the double plane branched along C + L 1 + L 2 . The map sending (C, L 1 , L 2 ) to the periods of S (C,L 1 ,L 2 ) extends to an isomorphism P : M 0 → D/Γ.
The approach is analogue to the one used by Laza [Laz09]. Roughly speaking, we first consider the generic case where C is smooth and C +L 1 +L 2 has simple normal crossings. Then we compute the (generic) Picard lattice M and the transcendental lattice T = M ⊥ Λ K3 (see Proposition 3.13), determine the period domain D and choose a suitable arithmetic group Γ (cf. Section 3.3, N.B. Γ is not the standard arithmetic group O * (T ) used in [Dol96] but an extension of O * (T )). Finally we extend the construction to the non-generic case (using the methods and some results of [Laz09]) and apply the global Torelli theorem and the surjectivity of the period map for K3 surfaces to prove the theorem (cf. Section 3.4 and Section 3.5).
Note that the period domain D is a type IV Hermitian symmetric domain. The arithmetic quotients of D admit canonical compactifications called Baily-Borel compactifications. To compare the GIT compactification and the Baily-Borel compactification we consider a slightly different moduli space M * (constructed by taking a quotient of the GIT quotient M(1, 1)) parameterizing triples (C, L, L ′ ) consisting of quartic curves C and unlabeled lines L, L ′ . In a similar manner, we construct a period map P ′ and prove that P ′ is an isomorphism between the locus M * 0 ⊂ M * where C +L+L ′ has at worst simple singularities and a certain locally symmetric domain D/Γ ′ (cf. Section 3.6). Moreover, we show in Corollary 2.16 that M * \M * 0 is the union of three points III(1), III(2a), III(2b) and five rational curves II(1), II(2a1), II(2a2), II(2b), II(3) whose incidence structure is describe in Figure 1. The quasi-projective variety M * 0 ⊂ M * has codimension higher than 1 and hence the period map P ′ extends to the GIT compactification M * . Note also that P ′ preserves the natural polarizations (the polarization of M * 0 is induced by the polarization of the moduli of plane sextics and the polarization of D/Γ ′ comes from the polarization of moduli of degree 2 K3 surfaces). A proof similar to [Loo03,Thm. 7.6] shows that the extension of P ′ induces an isomorphism between the GIT quotient M * and the Baily-Borel compactification (D/Γ ′ ) * (see Section 3.7). Some computations and remarks on the Baily-Borel boundary components are also included in the paper (cf. Section 3.8). We conclude by the following remarks. The moduli space of quartic triples (C, L 1 , L 2 ) is closely related to the moduli space of degree 5 pairs (cf. [Laz09,Def. 2.1]) consisting of a quintic curve and a line (i.e. given a triple (C, L 1 , L 2 ) that we consider, compare it with the pairs (C + L 1 , L 2 ) and (C + L 2 , L 1 )). Motivated by studying deformations of N 16 singularities, Laza [Laz09] has  constructed the moduli space of degree 5 pairs using both the GIT and Hodge theoretic approaches. His work is an important motivation for us and the prototype of what we do here. Also, the study of singularities and incidences lines on quartic curves is a classical topic (see for example the work of Edge [Edg50,Edg45]) and a classifying space for such pairs may be related to our GIT compactification. at the Institute for Computational and Experimental Research in Mathematics (ICERM) during a visit by the authors as part of a Collaborate@ICERM project. The authors would like to thank ICERM for making this visit possible.

Variations of GIT quotients
In [GMG18] the first two authors introduced a computational framework to construct all GIT quotients of pairs (X, H) formed by a hypersurface X of degree d and a hyperplane H in P n+1 . They drew from the general theory of variations of GIT quotients developed by Dolgachev and Hu [DH98] and independently by Thaddeus [Tha96]. The motivation was to construct compact moduli spaces of log pairs (X, D = X ∩ H) where X is Fano or Calabi-Yau. In this article we need to extend this setting to the case of tuples (C, L 1 , L 2 ) where C is a plane quartic curve and L 1 , L 2 are lines. However, extending our work in [GMG18] to two hyperplanes entails the same difficulties as for an arbitrary number of hyperplanes, while the dimension does not play an important role in the setting. Therefore we will consider the most general setting of a hypersurface in projective space and k hyperplane sections.
2.1. Variations of GIT quotients for n-dimensional hypersurfaces of degree d together with k (labelled) hyperplanes. Let R = R n,d,k be the parameter scheme of tuples (F d , l 1 , . . . , l k ), where F d is a polynomial of degree d and l 1 , . . . , l k are linear forms in variables (x 0 , . . . , x n+1 ), modulo scalar multiplication. We have that R n,d,k = P(H 0 (P n+1 , O P n+1 (d))) × P(H 0 (P n+1 , O P n+1 (1))) × · · · × P(H 0 (P n+1 , O P n+1 (1))) where N = n+1+d d −1 and natural projections π 0 : R n,d,k → P N , π i : R n,d,k → P n+1 for i = 1, . . . , k. The natural action of G = SL n+2 in P n+1 extends to each of the factors in R n,d,k and therefore to R n,d,k itself. The set of G-linearizable line bundles Pic G (R) is isomorphic to Z n+1 . Then a line bundle L ∈ Pic G (R), is ample if and only if a > 0, b i > 0 for i = 1, . . . , k, where The latter is a trivial generalization of [GMG18, Lemma 2.1]. Hence, for L ∼ = O(a, b 1 , . . . , b k ), the GIT quotient is defined as: Next, we explain why it is enough to consider the vector t = (t 1 , . . . , t k ) instead of (a; , b 1 , . . . , b k ). Let us introduce some notation.
Given a maximal torus T ∼ = C n+2 ⊂ G, we can choose projective coordinates (x 0 , . . . , x n+1 ) such that T is diagonal in G. Hence, any one-parameter subgroup λ : C * → T is a diagonal matrix with diagonal entries s r i where r i ∈ Z for all i and n+1 i=0 r i = 0. We say that λ is normalized if r 0 · · · r n+1 and λ is not trivial. Any homogeneous polynomial g of degree d can be written as n+1 , λ := n+1 i=0 d i r i , which we use to introduce the Hilbert-Mumford function for homogeneous polynomials: which is piecewise linear on λ for fixed (f, l 1 , . . . , l k ). Since the Hilbert-Mumford function is functorial[MFK94, Definition 2.2, cf. p. 49], we can generalize [GMG18, Lemma 2.2] to show that a tuple (f, l 1 , . . . , l k ) is (semi-)stable with respect to a polarisation L = O(a, b 1 , . . . , b k ) if and only if is negative (respectively, non-positive) for any normalized non-trivial one-parameter subgroup λ of any maximal torus T of G. Hence the stability of a tuple is independent of the scaling of L and as such, we may define: Notice that the stability of a tuple (f, l 1 , . . . , l k ) is completely determined by the support of f and l 1 , . . . , l k . Moreover, notice that the t-stability of a tuple is invariant under the action of G. Hence, we may say that a tuple (X, H 1 , . . . , H k ) formed by a hypersurface X ⊂ P n+1 and hyperplanes H i ⊂ P n+1 is t-stable (respectively, t-semistable) if some (and hence any) tuple of homogeneous polynomials (f, l 1 , . . . , l k ) defining (X, H 1 , . . . , H k ) is t-stable (respectively, t-semistable). A tuple In [GMG18], for fixed torus T in G, we introduced the fundamental set S n,d of one-parameter subgroups -a finite set-and we showed that if k = 1 it was sufficient to consider the oneparameter subgroups in S n,d for each T to determine the t-stability of any (X, H 1 ). Let us recall the definition -slightly simplified from the original [GMG18, Definition 3.1]-and extend the result to any k.
is the unique solution of a consistent linear system given by n equations chosen from the following set: The set S n,d is finite since there are a finite number of monomials of degree d in n + 2 variables. Observe that S n,d is independent of the value of k. The following lemma is a straight forward generalization of [GMG18, Lemma 3.2] which we include here for the convenience of the reader: Proof. Let (R ns T ) t be the non-t-stable loci of R with respect to a maximal torus T , and let (R ns ) t be the non t-stable loci of R.
Given a fixed (f, l 1 , . . . , l k ), the function µ t ((f, l 1 , . . . , l k ), −) : Q n+2 → Q is piecewise linear and its critical points -the points in Q n+2 where µ t ((f, l 1 , . . . , l k ), −) fails to be linear-correspond to those monomials x I , x I ′ ∈ Supp(f ) such that x I , λ = x I ′ , λ , or equivalently, the points λ ∈ Q n+2 ∩ ∆ such that x I−I ′ , λ = 0 for some x I , x I ′ ∈ Supp(f ). These points define a hyperplane in Q n+2 and the intersection of this hyperplane with ∆ is a simplex ∆ x I ,x I ′ of dimension n. As µ t ((f, l 1 , . . . , l k ), −) is linear on the complement of ∆ x I ,x I ′ , the minimum of µ t ((f, l 1 , . . . , l k ), −) is achieved on the boundary, i.e. either on ∂∆ or on ∆ x I ,x I ′ (for some I, I ′ ), all of which are convex polytopes of dimension n. By finite induction, we conclude that the minimum of µ t ((f, l 1 , . . . , l k , −) is achieved at one of the vertices of ∆ or ∆ x I ,x I ′ , which correspond precisely, up to multiplication by a constant, to the finite set of one-parameter subgroups in S n,d . Indeed, observe that if λ = Diag(s r 0 , . . . , s r n+1 ) is one such vertex, then 0 = x I−I ′ , λ = n+1 i=0 δ i γ i for some δ = (δ 0 , . . . , δ n+1 ) = I − I ′ where n+1 i=0 = 0 and −d δ i d. In addition, observe that we can find one such δ so that 0 = n+1 i=0 δ i γ i = γ i − γ i+1 , thus giving the equations determining the maximal facets of ∆, i.e. those where r i = r i+1 . The lemma follows from the observation that Eq(n, d) ⊂ Eq(n, d + 1).
Definition 2.4. The space of GIT stability conditions is The space of GIT stability conditions is bounded, as it can be realized as a hyperplane section of Amp G (R). Since R is a product of vector spaces (and hence a Mori dream space), Stab(n, d, k) is also a rational polyhedron. It is possible to precisely describe it and we will do this later for Stab(1, d, 2). Moreover, there is a finite number of non-isomorphic GIT compactifications M( t ) as t ∈ Stab(n, d, k) in varies. Therefore we have a natural division of Stab(n, d, k) into a finite number of disjoint rational polyhedrons of dimension k called chambers and the intersection of any two-chambers is a (possibly empty) rational polyhedron of smaller dimension which we will call a wall [DH98, Theorem 0.2.3]. The quotient M( t ) is constant as t moves in the interior of a face or chamber. It is possible to find these walls explicitly by means of Lemma 2.3 (see [GMG18, Theorem 1.1]) for given (n, d, k), since all walls of dimension k − 1 should be a subset of the finite set of equations Another interesting feature is that the the t-stability of tuples (X, H 1 , . . . , H k ) is equivalent of the t-stability of reducible GIT hypersurfaces of higher degree. Indeed: In particular, if t 1 , . . . , t k are natural numbers, (X, H 1 , . . . , H k ) is t-(semi)stable if and only if X + t 1 H 1 + · · · + t k H k (semi)stable in the classical GIT sense.
Proof. Let λ be a normalized one-parameter subgroup, m be a positive integer and g = g I x I be a homogeneous polynomial. Let J be such that Let (f, l 1 , . . . , l k ) be the equations of (X, H 1 , . . . , H k ) under some system of coordinates and let λ be a normalized one-parameter subgroup. Using the above observation, the lemma follows from: Corollary 2.6. Let t = (t 1 , . . . , t j , 0, . . . , 0) and t ′ = (t 1 , . . . , t j ), j k.
Proof. From [OS78, Theorem 2.1], any hypersurface X = {f = 0} where f is a homogeneous polynomial of degree d 3 has dim(Aut(f )) = 0. Hence, for any tuple p = (X, H 1 , . . . , H K ) such that X is smooth and X ∩ H i has simple normal crossings, its stabilizer G p satisfies Now let us consider the case of the symmetric polarization of R n,d,k . In order to do so, observe that the group S k acts on R n,d,k by defining the action of h ∈ S k as h : (f, l 1 , . . . , l k ) → (f, l h(1) , . . . , l h(k) ).
Proof. Since (2) holds, all the spaces in the above diagram are non-empty. As π is finite, the pair [(X, H 1 , · · · , H k )] -represented by the classes of tuples in R ′ n,d,k -is (semi)stable with respect to L * if and only if every (X, H 1 , · · · , H k ) in the class [(X, H 1 , · · · , H k )] is t * -(semi)stable, by [MFK94, Theorem 1.1 and p. 48]. By Lemma 2.5, (X, H 1 , · · · , H k ) -represented by tuples in R n,d,k -is t * -(semi)stable if and only if (X + H j 1 + · · · + H j l , H 1 , . . . ,Ĥ j i , . . . , H k ) -represented by tuples in R ′ n,d+l,k−l -is t * -(semi)stable (note that we use the notation t * for vectors with all entries equal 1, whether t * has k or k − l entries). The last statement regarding closed orbits follows from noting that finite morphisms are closed, and hence φ j 1 ,...,j l is closed.
2.2. Symmetric GIT quotient of a quartic curve and two lines. We have seen how to construct GIT quotients M(t 1 , t 2 ) := M(t 1 , t 2 ) 1,d,2 for (t 1 , t 2 ) ∈ Stab(t 1 , t 2 ). In this section we apply our results to the case of quartic plane curves (d = 4), but let us first show that our setting satisfies condition (2) for arbitrary degree. Hence, for the rest of the article, we assume that n = 1 and k = 2.
Lemma 2.9. The space of GIT stability conditions is In particular, (2) holds.
Proof. Let t = (t 1 , t 2 ) be a vector and (C, L 1 , L 2 ) be a t-semistable tuple. By a choosing an appropriate change of coordinates, we may assume Let λ = Diag(s 2 , s −1 , s −1 ). Then, as t 1 0, t 2 0, we have Similarly, by taking a change of coordinates such that Recall that the space of GIT stability conditions is convex [DH98, 0.2.1]. Hence it is enough to show that all the vertices of the right hand side in (4) have a semistable tuple (C, L 1 , L 2 ) (and hence, they belong to Stab(1, d, 2)). These vertices correspond to the points (0, 0), ( d 2 , 0), (0, d 2 ) and (d, d). By Corollary 2.6, a tuple (C, . Hence, we only need to exhibit a tuple (C, is semistable in the usual GIT sense. The latter follows from the centroid criterion [GMG18, Lemma 1.5]. There are two natural problems regarding the subdivision of Stab(n, d, k) into chambers and walls. One of them is to determine the walls and the solution is usually rather heavy computationally and geometrically speaking (see [GMG18,GMG16] for the case (n, d, k) = (2, 3, 1) and for a partial answer when k = 1 and (n, d) are arbitrary). Given a tuple (X, H 1 , · · · , H k ) the second problem consists on determining for which chambers and walls this tuple is (semi)stable. This problem may be easier to solve, especially when the answer to the first problem is known. The problem is simpler when k = 1, as then Stab(n, d, 1) is one-dimensional has a natural order. Nevertheless, we can give a partial answer when n = 1, k = 2 and d is arbitrary.
Lemma 2.11. Suppose that C is a plane curve of degree d whose only singular point p ∈ C is a linearly semi-quasihomogeneous singularity [Laz09, Def. 2.21] with respect to the weights w = (w 1 , w 2 ), w 1 w 2 > 0. Suppose further that C + L 1 + L 2 have simple normal crossings in C \ {p}. Let f be the localization of the equation of f at p and w(f ) be its weighted degree with respect to w.
For the second statement, the lines L 1 and L 2 have equation l 1 (x 0 , x 1 ) and l 2 (x 0 , x 1 ) + x 2 , respectively, where l 1 , l 2 are linear forms. If t 2 − t 1 For the rest of the paper we consider tuples (C, L 1 , L 2 ) formed by a plane quartic C and two lines L 1 , L 2 ⊂ P 2 . The following result will come useful: Lemma 2.12 (Shah [Sha80, Section 2], cf. [Laz16, Theorem 1.3]). Let Z be a plane sextic, and X the double cover of P 2 branched along Z. Then X has semi-log canonical singularities if and only if Z is semistable and the closure of the orbit of Z does not contain the orbit of the triple conic. In particular, a sextic plane curve with simple singularities is stable.
Lemma 2.13. Let t = (1, 1) and (C, L 1 , L 2 ) be a tuple such that the sextic C + L 1 + L 2 is reduced. Then, (C, L 1 , L 2 ) is t-(semi)stable if and only if the double cover X of P 2 branched at C + L 1 + L 2 has at worst simple singularities (respectively simple elliptic or cuspidal singularities).
Proof. The sextic Z := C + L 1 + L 2 = {f · l 1 · l 2 = 0} (where f is a quartic curve and l 1 , l 2 are distinct linear forms not in the support of f ) cannot degenerate to a triple conic and it is reduced by hypothesis. By Lemma 2.12, Z is a GIT-semistable sextic curve if and only if X has semi-log canonical (slc) singularities. The surface X is normal, as Z is reduced [CD89, Proposition 0.1.1]. In particular X := {w 2 = f · l 1 · l 2 } ⊂ P(1, 1, 1, 3) has hypersurface log canonical singularities away from the singular point (0 : 0 : 0 : 1) ∈ X, and by the classification of such singularities in [LR12, Table 1], they can only consist of either simple, simple elliptic or cuspidal singularities. If Z has only simple singularities then Z is GIT-stable by Lemma 2.12. Now suppose Z is GIT-stable and reduced. By [Laz16, Theorem 1.3 and Remark 1.4] a GIT-semistable plane sextic curve has either simple singularities or it is in the open orbit of a sextic containing a double conic or a triple conic in its support, contradicting the fact that Z is reduced. Hence Z has only simple singularities. The proof follows from Lemma 2.5.
Remark 2.14. Although, we will not discuss other polarizations. It is worth to notice that for t = (ǫ, ǫ) the stability is very similar to the one of plane quartics. In particular, if C is a semistable quartic and L 1 , L 2 are lines in general position. Then, the triple (C, L 1 , L 2 ) is stable.
Let R s t and R ss t be the set of t-stable and t-semistable tuples (f, l 1 , l 2 ), respectively. Let R 0 := {(f, l 1 , l 2 ) | the sextic {f · l 1 · l 2 = 0} is reduced and has at worst simple singularities} . 1). We are interested in describing the compactification of M 0 by M(1, 1). We use the notation in [Laz16].
Lemma 2.15. The quotient M(1, 1) is the compactification of M 0 by three points and six rational curves. The three points correspond to the closed orbit of tuples (C, L 1 , L 2 ) defined up to projective equivalence by the following tuples: The six rational curves correspond to the closed orbit of tuples (C, L 1 , L 2 ) defined up to projective equivalence by the following cases: where a = 0, 1, ∞.
Lemma 2.15, together with Proposition 2.8 gives us the following compactification which will be of interest for the next section:   3. Moduli of quartic plane curves and two lines via K3 surfaces 3.1. On K3 surfaces and lattices. By a lattice we mean a finite dimensional free Z-module L together with a symmetric bilinear form (−, −). The basic invariants of a lattice are its rank and signature. A lattice is even if (x, x) ∈ 2Z for every x ∈ L. The direct sum L 1 ⊕ L 2 of two lattices L 1 and L 2 is always assumed to be orthogonal, which will be denoted by L 1 ⊥ L 2 . For a lattice M ⊂ L, M ⊥ L denotes the orthogonal complement of M in L. Given two lattices L and L ′ and a lattice embedding L ֒→ L ′ , we call it a primitive embedding if L ′ /L is torsion free.
We shall use the following lattices: the (negative definite) root lattices A n (n 1), D m (m 4), E r (r = 6, 7, 8) and the hyperbolic plane U. Given a lattice L, L(n) denotes the lattice with the same underlying Z-module as L but with the bilinear form multiplied by n.
Notation 3.1. Given any even lattice L, we define: In our context, a polarization for a K3 surface is the class of a nef and big divisor H (and not the most restrictive notion of ample divisor, we follow the terminology in [Laz09]) and H 2 is its degree. More generally there is a notion of lattice polarization. We shall consider the period map for (lattice) polarized K3 surfaces and use the standard facts on K3 surfaces: the global Torelli theorem and the surjectivity of the period map. We also need the following theorem (see [Mor88,p. 40]   3.2. The K3 surfaces associated to a generic triple. We first consider the K3 surfaces arising as a double cover of P 2 branched at a smooth quartic curve C and two different lines L 1 and L 2 such that C + L 1 + L 2 has simple normal crossings. We shall show that these K3 surfaces are naturally polarized by a certain lattice.
Denote byS (C,L 1 ,L 2 ) the double cover of P 2 branched along C + L 1 + L 2 . Let S (C,L 1 ,L 2 ) be the K3 surface obtained as the minimal resolution of the 9 singular points ofS (C,L 1 ,L 2 ) . Let π : S (C,L 1 ,L 2 ) → P 2 be the natural morphism. Note that π : S (C,L 1 ,L 2 ) → P 2 also factors as the composition of the blow-up of P 2 at the singularities of C + L 1 + L 2 and the double cover of the blow-up branched along the strict transforms of C, L 1 and L 2 (see [BHPVdV04,§III.7
Corollary 3.7. Let C be a smooth plane quartic curve and L 1 , L 2 two distinct lines such that C + L 1 + L 2 has simple normal crossings and let  : M ֒→ Pic(S (C,L 1 ,L 2 ) ) be the lattice embedding given in Notation 3.3. Then  is a primitive embedding.
The proof of Proposition 3.6 can easily be adapted to proof the following Lemma.
Lemma 3.8. Let S be a K3 surface and  : M ֒→ Pic(S) be a lattice embedding. If none of (ξ), is divisible by 2 in Pic(S), then the embedding  is primitive.
Proposition 3.9. Assume that S is a K3 surface such that Pic(S) is isomorphic to the lattice M. Then S is the double cover of P 2 branched over a reducible curve C + L 1 + L 2 where C is a smooth plane quartic, L 1 , L 2 are lines and C + L 1 + L 2 has simple normal crossings.
As h is nef and (h, h) = 2 > 0, h is a polarization of degree 2. We will show that h is base point free by reductio ad absurdum. By Theorem 3.2, there exists a divisor D such that (D, D) = 0 and (h, D) = 1. Note that this is a numerical condition. Write D as a linear combination of γ, l ′ 1 , α 1 , . . . , α 3 , l ′ 2 , β 1 , . . . , β 3 and ξ, with coefficients c 1 , . . . , c 10 . Let S (Q,L,L ′ ) be the K3 surface associated to a smooth quartic curve Q and two lines L, L ′ such that Q + L + L ′ has simple normal crossings. Find the curve classes corresponding to γ, l ′ 1 , α 1 , . . . , α 3 , l ′ 2 , β 1 , . . . , β 3 , ξ (as what we did at the beginning of this subsection) and consider their linear combination D ′ with coefficients c 1 , . . . , c 10 , the same values as in the expression for D. Then, both D and D ′ satisfy the same numerical conditions in S (respectively S (Q,L,L ′ ) ) with respect to the divisor class h = γ + ξ. Again, by Theorem 3.2 the pull-back h of ∼ O P 2 (1) in S (Q,L,L ′ ) has base points, which gives a contradiction. So the linear system of h defines a degree two map π : S → P 2 . Since S is a K3 surface of degree 2, the branching locus must be a sextic curve B.
As M ֒→ (Pic(S (C,L 1 ,L 2 ) ) and they have the same rank and discriminant, then M ∼ = (Pic(S (C,L 1 ,L 2 ) ). Now let us consider the case when C has at worst simple singularities not contained in L 1 + L 2 and C + L 1 + L 2 has simple normal crossings away from the singularities of C. We still use S (C,L 1 ,L 2 ) to denote the K3 surface obtained as a minimal resolution of the double cover of P 2 along C + L 1 + L 2 . The rank 10 lattice M is the same as in Notation 3.3.
Lemma 3.11. If C has at worst simple singularities not contained in L 1 + L 2 and C + L 1 + L 2 has simple normal crossings away from the singularities of C, then there exists a primitive embedding  : M ֒→ Pic(S (C,L 1 ,L 2 ) ) such that (h) is a base point free degree two polarization.
Proof. Thanks to the transversal intersection, we define the embedding  as in the generic case. In particular, the morphism π : S (C,L 1 ,L 2 ) → P 2 is defined by (h). The embedding  is primitive by Proposition 3.6.
3.3. M-polarized K3 surfaces and the period map. In this subsection let us compute the (generic) Picard lattice M and the transcendental lattice T . Then we shall determine the period domain D and define a period map for generic triples (C, L 1 , L 2 ) via the periods of M-polarized K3 surfaces S (C,L 1 ,L 2 ) .
Definition 3.12. Let M be the lattice defined in Notation 3.3. An M-polarized K3 surface is a pair (S, ) such that  : M ֒→ Pic(S) is a primitive lattice embedding. The embedding  is called the M-polarization of S. We will simply say that S is an M-polarized K3 surface when no confusion about  is likely.
We now determine the lattice M and show that it admits a unique primitive embedding into the K3 lattice Λ K3 .
Proposition 3.13. Let M be the lattice defined in Notation 3.3. Then M is isomorphic to the lattice U(2) ⊥ A 2 1 ⊥ D 6 and admits a unique primitive embedding (up to isometry) M ֒→ Λ K3 into the K3 lattice Λ K3 . The orthogonal complement T := M ⊥ Λ K3 with respect to the embedding is isometric to U ⊥ U(2) ⊥ A 2 1 ⊥ D 6 . Proof. By [Nik79, Corollary 1.13.3] the lattice M is uniquely determined by its invariants which can be easily computed from the Gram matrix G M (see also Lemma 3.4 and Remark 3.5).
Remark 3.14. Note that both M and T are even indefinite 2-elementary lattices (a lattice L is 2-elementary if L * /L ∼ = (Z/2Z) k for some k). One could also invoke Nikulin's classification [Nik79, Theorem 3.6.2] of such lattices to prove the previous proposition. Moreover, M and T are orthogonal to each other in a unimodular lattice and hence ( The moduli space of M-polarized K3 surfaces is a quotient D/Γ for a certain Hermitian symmetric domain D of type IV and some arithmetic group Γ (see [Dol96]). Fix the (unique) embedding M ֒→ Λ K3 and define to be one of the two connected components. Note that D can also be identified with To specify the moduli of M-polarized K3 surfaces, one also needs to determine the arithmetic group Γ. In the standard situation considered in [Dol96] it is required that the M-polarization is pointwise fixed by the arithmetic group and one takes Γ to be O * (T ). In our geometric context the choice is different. Specifically, the permutations among α 1 , . . . , α 4 and among β 1 , . . . , β 4 are allowed. Observe that at the moment we view the lines L 1 and L 2 as labeled lines, distinguishing the tuples (C, L 1 , L 2 ) and (C, L 2 , L 1 ) and we do not consider the isometry of M induced by flipping the two lines.
Let L be an even lattice. Recall that any g ∈ O(L) naturally induces g * ∈ O(L * ) by g * ϕ : v → ϕ(g −1 v) (which further defines an automorphism of A L preserving q L , therefore giving a natural homomorphism r L : O(L) → O(q L )). Lemma 3.16. Let g M (respectively g T ) be an automorphism of M (respectively T ). If r M (g M ) = r T (g T ), then g M can be lifted to g ∈ O(Λ K3 ) with g| T = g T . The same statement holds for g T .
Proof. First let us describe the automorphisms of A M induced by the transpositions in Σ α and Σ β . We consider Σ α and the case of Σ β is analogous. The image of the transposition ( (1 j 3) and ξ * invariant. The automorphism of A M induced by the transposition (α 1 α 4 ) between α 1 and α 4 is given by , and γ * , α * 1 , β * j (1 j 3) are invariant by this action. The case of transpositions (α 2 α 4 ) and (α 3 α 4 ) is analogous. As it is well-known, the transpositions generate Σ α and Σ β . It is easy to compute the image of Σ α × Σ β in O(q M ) using the previous descriptions.
Recall that M 0 ⊂ M(1, 1) is the moduli space of triples (C, L 1 , L 2 ) consisting of a quartic curve C and (labeled) lines L 1 , L 2 such that C + L 1 + L 2 has at worst simple singularities.
Proposition 3.18. The period map P that associates to a (generic) triple (C, L 1 , L 2 ) the periods of the K3 surface S (C,L 1 ,L 2 ) defines a birational map P : M 0 D/Γ.
Proof. Let U be an open subset of M 0 parameterizing triples (C, L 1 , L 2 ) with C smooth quartic curves and L 1 , L 2 two (labeled) lines such that C + L 1 + L 2 has simple normal crossings. Set Σ 4 to be the permutation group of 4 elements and note that Σ α ∼ = Σ β ∼ = Σ 4 . Let U be the (Σ 4 ×Σ 4 )-cover of U that parametrizes quintuples (C, L 1 , L 2 , σ 1 , σ 2 ) where σ k : {1, 2, 3, 4} → C ∩ L k (k = 1, 2) is a labeling of the intersection points of C ∩ L k . Note that the monodromy group acts as the permutation group Σ 4 on the four points of intersection C ∩ L k . By Corollary 3.7, σ 1 and σ 2 determine an M-polarization  of the K3 surface S (C,L 1 ,L 2 ) . Therefore there is a well-defined map By the global Torelli theorem and the surjectivity of the period map for K3 surfaces (see also Proposition 3.9), the map P is a birational morphism. The group Σ 4 × Σ 4 acts naturally on both U and D/O * (T ) as Γ is an extension of Σ 4 × Σ 4 and O * (T ). Essentially, the actions are induced by the permutation of the labeling of the intersection points C ∩ L k (k = 0, 1). It follows that P is (Σ 4 × Σ 4 )-equivariant and descends to the birational map P : M 0 D/Γ (see also Lemma 3.16 and Lemma 3.17).
3.4. M-polarization for non-generic intersections. We will show in this section that the birational map P : M 0 D/Γ in Proposition 3.18 extends to a birational morphism P : M 0 → D/Γ. To do this, we need to extend the construction of M-polarization  : M ֒→ Pic(S (C,L 1 ,L 2 ) ) to the non-generic triples (C, L 1 , L 2 ) and show that the construction fits in families. The idea is to use the normalized lattice polarization (cf. [Laz09,Definition 4.24]) for degree 5 pairs constructed in [Laz09, §4.2.3]. A degree d pair (D, L) consists of a degree d plane curve D and a line L ⊂ P 2 (see [Laz09, Definition 2.1]). Given a triple (C, L 1 , L 2 ) of a quartic curve C and two different lines L 1 and L 2 , one can construct a degree 5 pair in two ways: (C + L 2 , L 1 ) or (C + L 1 , L 2 ). We follow the notation of the previous subsections, especially Notation 3.3. We will determine the images of γ, l ′ 1 , α 1 , . . . , α 4 (respectively γ, l ′ 2 , β 1 , . . . , β 4 ) using the degree 5 pair (C + L 2 , L 1 ) (respectively (C + L 1 , L 2 )).
Let us briefly review the construction of normalized lattice polarization for degree 5 pairs. See The construction of a normalized lattice polarization for degree 5 pairs is a modification of the process of canonical resolution. We may choose a labeling of the intersection points of D and L, which means a surjective map σ : {0, 1, 2, 3, 4} → D ∩ L satisfying |σ −1 (p)| = mult p (D ∩ L) for every p ∈ D ∩ L. As argued in [Laz09,Proposition 4.25], L is blown-up exactly 5 times in the desingularization process described above. The blown-up points can be chosen as the first five steps of the sequence of blow-ups: S ′ → . . . S 0 → P 2 , and the labeling determines the order of these first 5 blow-ups. Let {p k } 4 k=0 be the centers of these blow-ups and E k be the exceptional divisors. Note that p k ∈ S k−1 (the image of p k under the contraction to P 2 is a point of intersection D ∩ L) and E k is a divisor on S k for k = 0, . . . , 4. We define the following divisors L) is Artin's fundamental cycle (see for example [BHPVdV04,p. 76]) associated to the simple singularity of the curve B k−1 at the point p k .
The procedure described above produces 5 divisors: D 0 , . . . , D 4 . One can also consider the strict transform of L in S ′ and take its preimage in S (D,L) . This is a smooth rational curve on the K3 surface and we will denote its corresponding class by L ′ . We summarize the properties of these 6 divisors L ′ , D 0 , . . . , D 4 in the following result. Given families of curves (C, L), we can carry out a simultaneous resolution in families. As a result the construction above fits well in families, see [Laz09, p. 2141].
We also require that (h) is the class of the base point free polarization π * O P 2 (1).
(1) (h) is the class of the base point free polarization π * O P 2 (1).
Proof. This follows from a case by case analysis. Specifically, we check the conditions of Lemma 3.8 as in the proof of Proposition 3.6.
Proposition 3.21. The birational map in Proposition 3.18 extends to a morphism P : M 0 → D/Γ. Moreover, the map P is injective.
Proof. Given a triple (C, L 1 , L 2 ) consisting of a quartic curve C and lines L 1 , L 2 such that C +L 1 + L 2 has at worst simple singularities, we consider the M-polarized K3 surface (S (C,L 1 ,L 2 ) , ) where S (C,L 1 ,L 2 ) is the K3 surface obtained by taking the minimal resolution of the double plane branched along C + L 1 + L 2 and  is the lattice polarization constructed above. By [Dol96], the M-polarized K3 surface (S (C,L 1 ,L 2 ) , ) corresponds to a point in D/O * (T ). The polarization  depends only on (C, L 1 , L 2 ), the ordering of C ∩ L 1 and the ordering of C ∩ L 2 , and it is compatible with the action of Γ/O * (T ) = Σ 4 × Σ 4 . Consequently, we can associate to every triple (C, L 1 , L 2 ) a point in D/Γ. In other words, we have a well-defined morphism P : M 0 → D/Γ extending the birational map in Proposition 3.18.
Choose a point ω ∈ D/Γ (more precisely, ω is a Γ-orbit) which corresponds to an M-polarization K3 surface S (C,L 1 ,L 2 ) . Lemma 3.16 allows us to extend an element of Γ to an isometry of the K3 lattice Λ K3 . The global Torelli theorem for K3 surfaces implies that the period ω determines the isomorphism class of the K3 surface S (C+L 1 +L 2 ) . By our construction the classes h, l ′ 1 and l ′ 2 are fixed by Γ. It follows that the period point ω uniquely expresses the K3 surface as a double cover of P 2 and determines two line components of the branched locus. Now we conclude that ω determines uniquely the triple (C, L 1 , L 2 ).
3.5. Surjectivity of the period map. We will show in this section that the period map P : M 0 → D/Γ is surjective. Given the general result of surjectivity of period maps for (lattice) polarized K3, one has to establish that any K3 surface carrying an M-polarization is of type S (C,L 1 ,L 2 ) .
Proposition 3.22. Let S be a K3 surface such that there exists a primitive embedding  : M ֒→ Pic(S). Then there exists a plane quartic curve C and two different lines L 1 , L 2 ⊂ P 2 such that S ∼ = S (C,L 1 ,L 2 ) and C + L 1 + L 2 has at worst simple singularities.
Theorem 3.23. Consider the triples (C, L 1 , L 2 ) consisting of a quartic curve C and lines L 1 , L 2 such that C + L 1 + L 2 has at worst simple singularities. Let S (C,L 1 ,L 2 ) be the K3 surface obtained by taking the minimal resolution of the double plane branched along C + L 1 + L 2 . The birational map sending (C, L 1 , L 2 ) to the periods of S (C,L 1 ,L 2 ) in Proposition 3.18 extends to an isomorphism P : M 0 → D/Γ.
Proof. It suffices to prove that P is surjective. Let ω ∈ D/Γ be a period point. By the surjectivity of the period map of lattice-polarized K3 surfaces (see [Dol96,  The embeddings | M 1 and | M 2 are unique up to permutation of the classes α 1 , . . . , α 4 (β 1 , . . . , β 4 , respectively) thanks to [Laz09,Lemma 4.29]. It follows that the polarization  is unique up to action of Σ 4 × Σ 4 and coincides with our construction in Subsection 3.4. By Proposition 3.18 and Proposition 3.21 the period map P : M 0 → D/Γ is a bijective birational morphism between normal varieties. As a result, P is an isomorphism, by Zariski's Main Theorem.
3.6. The period map for unlabeled triples. Consider the compact space M * and M * 0 ⊂ M * , consisting on the subset of triples (C, L, L ′ ) formed by a quartic curve C and unlabeled lines L, L ′ such that the sextic curve C + L + L ′ is reduced and has at worst simple singularities, as constructed in Corollary 2.16. In this subsection we define the period map P ′ : M * 0 → D/Γ ′ -for an appropriately chosen arithmetic group Γ ′ -and show that P ′ is an isomorphism. We use the same approach taken to define P : M 0 → D/Γ (Proposition 3.18 and Proposition 3.21) and to prove Theorem 3.23. The modification one needs to do is to choose a different arithmetic group Γ ′ . We follow the same notation as in the previous subsections, especially regarding the description of the subgroup Σ α × Σ β in Lemma 3.17. Consider the subgroup Σ α × Σ β ⋊ Z/2Z ⊂ O(M) where the factor Z/2Z corresponds to the swap of α i 's and β i 's for 0 i 4 (the induced action on A M exchanges α * i with β * i for 1 i 3 and fixes γ * and ξ * ). As in the proof of Lemma 3.17, we can verify that the composition Σ α × Σ β ⋊ Z/2Z ֒→ O(M) → O(q M ) is injective. Now we define Γ ′ to be the following extension: For (C, L, L ′ ) ∈ M * 0 (such that C + L + L ′ is reduced and has at worst simple singularities) we consider the period of the K3 surface S (C,L,L ′ ) which is the minimal resolution of the double cover of P 2 branched along C +L+L ′ . Because S (C,L,L ′ ) is polarized by the lattice M, the period corresponds to a point in D. The lattice polarization depends on the labeling of L and L ′ , the ordering of C ∩ L and the ordering of C ∩ L ′ , and thus is compatible with the action of Γ ′ /O * (T ) = Σ 4 × Σ 4 ⋊ Z/2Z. Therefore we have a well-defined period map P ′ : M * 0 → D/Γ ′ (see also Proposition 3.18 and Proposition 3.21). Moreover, the same argument in Proposition 3.21 and Theorem 3.23 allows us to prove that the period map P ′ : M * 0 → D/Γ ′ is an isomorphism.
3.7. Comparison of the GIT and the Baily-Borel compactifications. Consider the moduli space M * 0 ⊂ M * of triples (C, L, L ′ ) formed by a quartic curve C and unlabeled lines L, L ′ such that the sextic curve C + L + L ′ has at worst simple singularities. We have constructed a period map P ′ : M * 0 → D/Γ ′ in Subsection 3.6 and have shown that it is an isomorphism. There are two natural ways to compactify M * 0 as the GIT quotient M * := M * 1,4,2 defined in (3) and described in Corollary 2.16, or as the Baily-Borel compactification [BB66]. We compare these two compactifications by applying some general results of Looijenga [Loo03]. See also [Laz09, Theorem 4.2]. and their polarizations agree by restriction of the isomorphic polarizations for the GIT of sextic curves and for the compact moduli of K3 surfaces of degree 2 (see [Loo03,§8]). Hence, by [Loo03,Theorem 7.6] P ′ : M * 0 → D/Γ ′ is an isomorphism for polarized varieties. Question 3.25. Does the period map for labeled triples P : M 0 → D/Γ (cf. Theorem 3.23) preserve the natural polarizations?
A positive answer to this question would imply that the period map P : M 0 → D/Γ can be extended to an isomorphism P : M(1, 1) → (D/Γ) * . We strongly believe that the answer is yes (by pulling back the polarizations for sextic curves and degree 2 K3 surfaces via the double covers M(1, 1) → M * and D/Γ → D/Γ ′ ).
3.8. The Baily-Borel compactification. The locally symmetric space D/Γ ′ admits a canonical minimal compactification, the Baily-Borel compactification (D/Γ ′ ) * (cf. [BB66]). The boundary components of (D/Γ ′ ) * are either 0-dimensional (Type III components) or 1-dimensional (Type II components), and they correspond to the primitive rank 1, respectively, rank 2 isotropic sublattices of T up to Γ ′ -equivalence. Following the approach of [Sca87], [Ste91] and [Laz09], we determine the number of the Type III boundary components of (D/Γ ′ ) * and compute certain invariants for the Type II boundary components. Notice that by Theorem Theorem 3.24, the number of these boundary components and some of their invariants (such as the dimension) can be worked out from the boundary components of the GIT quotient M * described in Corollary 2.16.
We determine the 0-dimensional components of (D/Γ ′ ) * using [Sca87, Proposition 4.1.3]. The 0dimensional boundary components are in one-to-one correspondence with the Γ ′ -orbits of primitive isotropic rank 1 sublattices of T . By Proposition 3.13 and [Nik79, Theorem 3.6.2] we have T ∼ = U ⊥ U(2) ⊥ A 2 1 ⊥ D 6 ∼ = U ⊥ U ⊥ A 4 1 ⊥ D 4 . (In particular, T contains two hyperbolic planes.) Write Γ * = Σ α × Σ β ⋊ Z/2Z ⊂ O(q T ) (see Subsection 3.6). Note that for v ∈ T one can associate a vectorv ∈ A T = T * /T defined byv ≡ v div(v) mod T (where div(v) is the divisor of v which is a positive integer such that (v, T ) = div(v)Z). If v is a primitive isotropic vector thenv is an isotropic element in A T . By [Sca87, Proposition 4.1.3] the map Zv →v induces a bijection between the equivalence classes of primitive isotropic rank 1 sublattices of T and Γ * -orbits of isotropic elements of A T . Because T is the orthogonal complement of M in Λ K3 , one has (A M , q M ) ∼ = (A T , −q T ). We have computed the discriminant quadratic form q M in Lemma 3.4. In particular, there are 20 isotropic elements in A M ∼ = A T : 0, γ * , α * i + β * j and α * i + β * j + γ * (1 i, j 3). The action of Γ * has been described in the proof of Lemma 3.17 and Subsection 3.6. It is easy to see that α * i + β * j and α * i + β * j + γ * (1 i, j 3) form one Γ * -orbit. As a result, the Baily-Borel compactification (D/Γ ′ ) * consists of three 0-dimensional boundary components.   (1, 9)) 2-elementary lattice. By a direct computation we get the following (see also [Nik79, Theorem 3.6.2]).
To determine the 1-dimensional components of (D/Γ ′ ) * , one needs to compute the equivalence classes of primitive isotropic rank 2 sublattices of T . We use the algorithm for classifying isotropic vectors in hyperbolic lattices due to Vinberg [Vin75]. Specifically, for each of the equivalence classes of primitive isotropic rank 1 sublattices Zv of T we apply Vinberg's algorithm to the hyperbolic lattice v ⊥ /Zv (with respect to the action by the stabilizer Γ ′ v of v). Now we briefly recall Vinberg's algorithm [Vin75] (see also [Ste91,§4.3]). Let N be a hyperbolic lattice of signature (1, n). (In our case we take N = v ⊥ /Zv.) The algorithm starts by fixing an element h ∈ N of positive square. Then one needs to inductively choose roots δ 1 , δ 2 , . . . such that the distance function (h,δ) 2 |(δ,δ)| is minimized. The algorithm stops with the choice of δ N if every connected parabolic subdiagram (i.e. the extended Dynkin diagram of a root system) of the Dynkin diagram Σ associated to the roots δ 1 , δ 2 , . . . , δ N is a connected component of some parabolic subdiagram of rank n − 1. If the algorithm stops then the W (N)-orbits of the isotropic lines in N correspond to the parabolic subdiagrams of rank n − 1 of Σ (N.B. the isomorphism classes of E ⊥ /E, where E is an isotropic rank 2 sublattice of T containing v, are determined by the Dynkin diagrams of the parabolic subdiagrams). To determine the equivalence classes of the isotropic vectors by a larger group which contains the Weyl group W (N) as a subgroup of finite index, one should take certain symmetries of Σ into consideration.
In our case, a straightforward application of Vinberg's algorithm allows us to compute the isomorphism classes of E ⊥ /E.
Remark 3.28. Using the Clemens-Schmid exact sequence and the incidence relation of the GIT boundary components (see also [Laz09,Theorem 4