Compactifications of the moduli space of plane quartics and two lines
Abstract
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.
Keywords
K3 surfaces Variations of GIT quotients Period map Quartic curvesMathematics Subject Classification
14L24 14J28 14J17 32G20 14Q101 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 Open image in new window 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 [1, 2, 17, 18, 22, 23, 30]. 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 [12]. The moduli of del Pezzo surfaces of degree 2 with two anticanonical 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 doublecovers of Open image in new window branched at a (possibly singular) quartic curve. Also, a generic global Torelli for certain double covers of these K3 surfaces (namely, minimal resolutions of bidouble covers of Open image in new window along a quartic and four lines, cf. [14, Section 5.4.2]) can be derived using the results in this article and the methods in [28].
Following the general theory of variations of GIT quotients developed by Dolgachev and Hu [9] and independently by Thaddeus [32], we construct GIT compactifications Open image in new window 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 Sect. 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 [13] of GIT quotients of pairs (X, H) formed by a hypersurface X of degree d in Open image in new window and a hyperplane H to tuples \((X, H_1, \ldots , H_k)\) 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 Open image in new window can also be constructed via Hodge theory (cf. Sect. 3). The idea is to consider the K3 surface \(S_{(C, L_1, L_2)}\) obtained by taking the desingularization of the double cover \(\overline{S}_{(C,L_1,L_2)}\) of Open image in new window branched along the sextic curve \(C+L_1+L_2\). Note that generically \(\overline{S}_{(C,L_1,L_2)}\) admits nine ordinary double points (coming from the intersection points \(C \cap L_1\), \(C \cap L_2\) and \(L_1 \cap L_2\)). 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 Open image in new window . Then the K3 surface \(S_{(C,L_1,L_2)}\) is naturally Mpolarized in the sense of Dolgachev [8]. Let Open image in new window 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 Mpolarized K3 surfaces \(S_{(C,L_1,L_2)}\) one obtains a period map Open image in new window from Open image in new window to a certain period domain Open image in new window . We shall prove that Open image in new window is an isomorphism.
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 Open image in new window .
The approach is analoguos to the one used by Laza [17]. 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 \(T = M_{\Lambda _{K3}}^{\perp }\) (see Proposition 3.13), determine the period domain Open image in new window and choose a suitable arithmetic group \(\Gamma \) (cf. Sect. 3.3, N.B. \(\Gamma \) is not the standard arithmetic group \(O^*(T)\) used in [8] but an extension of \(O^*(T)\)). Finally we extend the construction to the nongeneric case (using the methods and some results of [17]) and apply the global Torelli theorem and the surjectivity of the period map for K3 surfaces to prove the theorem (cf. Sects. 3.4 and 3.5).
Note that the period domain Open image in new window is a type IV Hermitian symmetric domain. The arithmetic quotients of Open image in new window admit canonical compactifications called Baily–Borel compactifications. To compare the GIT compactification and the Baily–Borel compactification we consider a slightly different moduli space Open image in new window (constructed by taking a quotient of the GIT quotient Open image in new window ) parameterizing triples \((C,L,L')\) consisting of quartic curves C and unlabeled lines \(L,L'\). In a similar manner, we construct a period map Open image in new window and prove that Open image in new window is an isomorphism between the locus Open image in new window where \(C+L+L'\) has at worst simple singularities and a certain locally symmetric domain Open image in new window (cf. Sect. 3.6). Moreover, we show in Corollary 2.16 that Open image in new window is the union of three points Open image in new window and five rational curves Open image in new window , Open image in new window whose incidence structure is describe in Fig. 1. The quasiprojective variety Open image in new window has codimension higher than 1 and hence the period map Open image in new window extends to the GIT compactification Open image in new window . Note also that Open image in new window preserves the natural polarizations (the polarization of Open image in new window is induced by the polarization of the moduli of plane sextics and the polarization of Open image in new window comes from the polarization of moduli of degree 2 K3 surfaces). A proof similar to [21, Theorem 7.6] shows that the extension of Open image in new window induces an isomorphism between the GIT quotient Open image in new window and the Baily–Borel compactification Open image in new window (see Sect. 3.7). Some computations and remarks on the Baily–Borel boundary components are also included in the paper (cf. Sect. 3.8).
Theorem 3.24
The period map Open image in new window extends to an isomorphism of projective varieties Open image in new window denotes the Baily–Borel compactification of Open image in new window .
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. [17, Definition 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 Open image in new window and Open image in new window ). Motivated by studying deformations of \(N_{16}\) singularities, Laza [17] 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 [10, 11]) and a classifying space for such pairs may be related to our GIT compactification.
2 Variations of GIT quotients
In [13] 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 Open image in new window . They drew from the general theory of variations of GIT quotients developed by Dolgachev and Hu [9] and independently by Thaddeus [32]. The motivation was to construct compact moduli spaces of log pairs Open image in new window 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 [13] 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 ndimensional hypersurfaces of degree d together with k (labeled) hyperplanes
Definition 2.1
Let \(\vec t\in (\mathbb Q_{\geqslant 0})^k\). The tuple \((f,l_1,\ldots ,l_k)\) is \(\vec t\)stable (respectively \(\vec t\)semistable) if Open image in new window (respectively Open image in new window ) for all nontrivial normalized oneparameter subgroups \(\lambda \) of G. A tuple \((f,l_1,\ldots ,l_k)\) is \(\vec t\)unstable if it is not \(\vec t\)semistable. A tuple \((f,l_1,\ldots ,l_k)\) is strictly\(\vec t\)semistable if it is \(\vec t\)semistable but not \(\vec t\)stable.
Notice that the stability of a tuple \((f,l_1,\ldots ,l_k)\) is completely determined by the support of f and \(l_1,\ldots ,l_k\). Moreover, notice that the \(\vec t\)stability of a tuple is invariant under the action of G. Hence, we may say that a tuple \((X,H_1,\ldots , H_k)\) formed by a hypersurface Open image in new window and hyperplanes Open image in new window is \(\vec t\)stable (respectively, \(\vec t\)semistable) if some (and hence any) tuple of homogeneous polynomials \((f,l_1,\ldots ,l_k)\) defining \((X,H_1,\ldots ,H_k)\) is \(\vec t\)stable (respectively, \(\vec t\)semistable). A tuple \((X,H_1,\ldots , H_k)\) is \(\vec t\)unstable if it is not \(\vec t\)semistable.
In [13], for fixed torus T in G, we introduced the fundamental set\(S_{n,d}\)of oneparameter 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 \(\vec t\)stability of any \((X,H_1)\). Let us recall the definition—slightly simplified from the original [13, Definition 3.1]—and extend the result to any k.
Definition 2.2

Open image in new window such that Open image in new window for all \(i=0,\ldots ,n+1\) and Open image in new window .
 \((\gamma _0,\ldots ,\gamma _{n+1})\) 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 [13, Lemma 3.2] which we include here for the convenience of the reader:
Lemma 2.3
Proof
Let Open image in new window be the non\({\vec t}\)stable loci of Open image in new window with respect to a maximal torus T, and let Open image in new window be the non\(\vec t\)stable loci of Open image in new window .
By [7, p. 137], Open image in new window . Let \((f,l_1,\ldots ,l_k)\) be the equations in some coordinate system—inducing a maximal torus \(T\subset G\)—of a non\(\vec t\)stable tuple \((X,H_1,\ldots , H_k)\). Then, Open image in new window for some \(\rho \in T'\) in a maximal torus \(T'\) which may be different from T. All the maximal tori are conjugate to each other in G, and by [7, Exercise 9.2 (i)], we have Open image in new window for all \(g\in G\). Hence, there is Open image in new window such that Open image in new window is normalized and Open image in new window satisfies Open image in new window . Normalized oneparameter subgroups in the coordinate system induced by T are the intersection of \(\sum r_i=0\) and the convex hull of \(r_ir_{i+1}\geqslant 0\), where \(i=0,\ldots , n\). The restriction of the \(n+1\) linearly independent inequalities in \(n+1\) variables to \(\sum r_i=0\) gives a closed convex polyhedral subset \(\Delta \) of dimension \(n+1\) (in fact, a simplex) in the \(\mathbb Q\)lattice of characters of T—isomorphic to the lattice of monomials (in variables \(x_0,\ldots , x_{n+1}\)) tensored by \(\mathbb Q\), which in turn is isomorphic to \(\mathbb Q^{n+2}\).
Given a fixed \((f,l_1,\ldots ,l_k)\), the function Open image in new window is piecewise linear and its critical points—the points in \(\mathbb Q^{n+2}\) where Open image in new window fails to be linear—correspond to those monomials Open image in new window such that Open image in new window , or equivalently, the points Open image in new window such that Open image in new window for some Open image in new window . These points define a hyperplane in \(\mathbb Q^{n+2}\) and the intersection of this hyperplane with \(\Delta \) is a simplex Open image in new window of dimension n. As Open image in new window is linear on the complement of Open image in new window , the minimum of Open image in new window is achieved on the boundary, i.e. either on \(\partial \Delta \) or on Open image in new window (for some \(I,I'\)), all of which are convex polytopes of dimension n. By finite induction, we conclude that the minimum of Open image in new window is achieved at one of the vertices of \(\Delta \) or Open image in new window , which correspond precisely, up to multiplication by a constant, to the finite set of oneparameter subgroups in \(S_{n,d}\). Indeed, observe that if Open image in new window is one such vertex, then Open image in new window for some Open image in new window where Open image in new window and Open image in new window . In addition, observe that we can find one such \(\delta \) so that Open image in new window , thus giving the equations determining the maximal facets of \(\Delta \), i.e. those where \(r_i=r_{i+1}\). The lemma follows from the observation that Open image in new window .\(\square \)
Definition 2.4
Lemma 2.5
In particular, if \(t_1,\ldots , t_k\) are natural numbers, \((X,H_1,\ldots ,H_k)\) is \(\vec t\)(semi)stable if and only if \(X+{t_1}H_1+\cdots +{t_k}H_{k}\) (semi)stable in the classical GIT sense.
Proof
Corollary 2.6
Let Open image in new window and Open image in new window , \(j\leqslant k\). Then a tuple \((X,H_1,\ldots , H_k)\) is \(\vec t\)semistable if and only if Open image in new window is \(\vec {t'}\)semistable.
Lemma 2.7
Proof
Proposition 2.8
Proof
Since (2) holds, all the spaces in the above diagram are nonempty. As \(\pi \) is finite, the pair \([(X,H_1,\dots ,H_k)]\)—represented by the classes of tuples in Open image in new window —is (semi)stable with respect to Open image in new window if and only if every \((X,H_1, \dots ,H_k)\) in the class \([(X,H_1,\cdots ,H_k)]\) is \(\vec t_*\)(semi)stable, by [25, Theorem 1.1 and p. 48]. By Lemma 2.5, \((X,H_1, \dots ,H_k)\)—represented by tuples in Open image in new window —is \( \vec t_*\)(semi)stable if and only if Open image in new window —represented by tuples in Open image in new window —is \(\vec t_{*}\)(semi)stable (note that we use the notation \(\vec t_*\) for vectors with all entries equal 1, whether \(\vec t_*\) has k or \(kl\) entries). The last statement regarding closed orbits follows from noting that finite morphisms are closed, and hence Open image in new window is closed.\(\square \)
2.2 Symmetric GIT quotient of a quartic curve and two lines
We have seen how to construct GIT quotients Open image in new window for Open image in new window . 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
Proof
Recall that the space of GIT stability conditions is convex [9, 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 Open image in new window ). These vertices correspond to the points (0, 0), (d / 2, 0), (0, d / 2) and (d, d). By Corollary 2.6, a tuple \((C, L_1,L_2)\) is (d / 2, 0)semistable if and only if \((C, L_1)\) is (d / 2)semistable, but the space of GIT tstability conditions for plane curves and one hyperplane is Open image in new window [13, Theorem 1.1]. A mirrored argument applies for the stability point (0, d / 2).
Hence, we only need to exhibit a tuple \((C, L_1,L_2)\) which is (d, d)semistable. Let Open image in new window . By Lemma 2.5, such a pair is tsemistable if and only if the reducible curve \(C+dL_1+dL_2\) (defined by the equation \(x_0^dx_1^dx_2^d=0\)) of degree 3d is semistable in the usual GIT sense. The latter follows from the centroid criterion [13, Lemma 1.5].\(\square \)
There are two natural problems regarding the subdivision of Open image in new window into chambers and walls. One of them is to determine the walls and the solution is usually rather heavy computationally and geometrically speaking (see [12, 13] 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 \((X, H_1,\cdots , H_k)\) 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 Open image in new window is onedimensional has a natural order. Nevertheless, we can give a partial answer when \(n=1\), \(k=2\) and d is arbitrary.
Definition 2.10
Let Open image in new window be the loci such that Open image in new window if and only \((C,L_1,L_2)\) is tsemistable.
Lemma 2.11
 (a)Suppose that \(p\not \in L_1\cup L_2\). Then
 (b)
Proof
We may choose a coordinate system such that Open image in new window is the singular point of C. We consider the oneparameter subgroup \(\lambda = (2w_1w_2, 2w_2w_1,w_1w_2)\) which is normalized, as \(w_1 \geqslant w_2\).
(a) The first statement is equivalent to show that if Open image in new window and Open image in new window then the triple is Open image in new window unstable.
For the rest of the paper we consider tuples \((C, L_1, L_2)\) formed by a plane quartic C and two lines Open image in new window . The following result will come useful:
Lemma 2.12
(Shah [30, Section 2], cf. [19, Theorem 1.3]) Let Z be a plane sextic, and X the double cover of Open image in new window branched along Z. Then X has semilog 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 \(\vec t =(1,1)\) 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 \(\vec t\)(semi)stable if and only if the double cover X of Open image in new window branched at \(C+L_1+L_2\) has at worst simple singularities (respectively simple elliptic or cuspidal singularities).
Proof
The sextic Open image in new window (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 GITsemistable sextic curve if and only if X has slc singularities. The surface X is normal, as Z is reduced [6, Proposition 0.1.1]. In particular Open image in new window has hypersurface log canonical singularities away from the singular point Open image in new window , and by the classification of such singularities in [20, Table 1], they can only consist of either simple, simple elliptic or cuspidal singularities. If Z has only simple singularities then Z is GITstable by Lemma 2.12. Now suppose Z is GITstable and reduced. By [19, Theorem 1.3 and Remark 1.4] a GITsemistable 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.\(\square \)
Remark 2.14
Although, we will not discuss other polarizations. It is worth to notice that for \(\vec t=( \epsilon , \epsilon )\) 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.
Lemma 2.15
Proof
Let Open image in new window , parametrising tuples \((g,l_1)\) up to multiplication by scalar where g is a quintic homogeneous polynomial and \(l_1\) is a linear form. As we have seen in Proposition 2.8, we have a morphism Open image in new window defined by Open image in new window , and an orbit O of Open image in new window is closed if and only if the orbit \(\phi _2(O)\) of Open image in new window is closed.
Hence the points which compactify Open image in new window into Open image in new window corresponding to closed orbits of Open image in new window are mapped via \(\phi _2\) onto points in Open image in new window corresponding to closed orbits in Open image in new window . Hence we just need to identify closed orbits in Open image in new window . Our result is a straight forward identification of these orbits in the classification of Open image in new window in [17, Proposition 3.22] (Fig. 2).\(\square \)
Lemma 2.15, together with Proposition 2.8 gives us the following compactification which will be of interest for the next section:
Corollary 2.16

the points Open image in new window are the images of the points Open image in new window , and

the rational curves Open image in new window , Open image in new window are the images of the rational curves Open image in new window .
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 \(\mathbb {Z}\)module L together with a symmetric bilinear form \((,)\). The basic invariants of a lattice are its rank and signature. A lattice is even if \((x,x) \in 2\mathbb {Z}\) for every \(x \in L\). The direct sum Open image in new window of two lattices \(L_1\) and \(L_2\) is always assumed to be orthogonal, which will be denoted by Open image in new window . For a lattice \(M \subset L\), \(M_L^{\perp }\) denotes the orthogonal complement of M in L. Given two lattices L and \(L'\) and a lattice embedding \(L \hookrightarrow L'\), we call it a primitive embedding if Open image in new window is torsion free.
We shall use the following lattices: the (negative definite) root lattices \(A_n\) (\(n \geqslant 1\)), \(D_m\) (\(m \geqslant 4\)), \(E_r\) (\(r = 6,7,8\)) and the hyperbolic plane U. Given a lattice L, L(n) denotes the lattice with the same underlying \(\mathbb {Z}\)module as L but with the bilinear form multiplied by n.
Notation 3.1

Open image in new window , the dual lattice;

Open image in new window , the discriminant group endowed with the induced quadratic form Open image in new window ;

Open image in new window : the determinant of the Gram matrix (i.e. the intersection matrix) with respect to an arbitrary \(\mathbb {Z}\)basis of L;

O(L): the group of isometries of L;

\(O(q_L)\): the automorphisms of \(A_L\) that preserve the quadratic form \(q_L\);

\(O_{}(L)\): the group of isometries of L of spinor norm 1 (see [29, Section 3.6]);

\(\widetilde{O}(L)\): the group of isometries of L that induce the identity on \(A_L\);

\(O^*(L) = O_{}(L) \cap \widetilde{O}(L)\);

\(\Delta (L)\): the set of roots of L (\(\delta \in L\) is a root if Open image in new window );

W(L): the Weyl group, i.e. the group of isometries generated by reflections \(s_{\delta }\) in root \(\delta \), where Open image in new window .
For a surface X, the intersection form gives a natural lattice structure on the torsionfree part of \(H^2(X, \mathbb {Z})\) and on the Néron–Severi group Open image in new window . For a K3 surface S, we have Open image in new window , and hence Open image in new window . Both \(H^2(S, \mathbb {Z})\) and Open image in new window are torsionfree and the natural map Open image in new window is a primitive embedding. Given any K3 surface S, \(H^2(S,\mathbb {Z})\) is isomorphic to Open image in new window , the unique even unimodular lattice of signature (3, 19). We shall use \(O(S),\Delta (S),W(S)\), etc. to denote the corresponding objects of the lattice Open image in new window . We also denote by \(\Delta ^+(S)\) and \(V^+(S)\) the set of effective \((2)\) divisor classes in Open image in new window and the Kähler cone of S respectively.
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 [17]) 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 [24, p. 40] or [17, Theorem 4.8, Proposition 4.9]).
Theorem 3.2
Let H be a nef and big divisor on a K3 surface S. The linear system H has base points if and only if there exists a divisor D such that Open image in new window and \(D^2=0\).
3.2 The K3 surfaces associated to a generic triple
We first consider the K3 surfaces arising as a double cover of Open image in new window 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 by \(\overline{S}_{(C,L_1,L_2)}\) the double cover of Open image in new window 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 nine singular points of \(\overline{S}_{(C,L_1,L_2)}\). Let Open image in new window be the natural morphism. Note that Open image in new window also factors as the composition of the blowup of Open image in new window at the singularities of \(C+L_1+L_2\) and the double cover of the blowup branched along the strict transforms of C, \(L_1\) and \(L_2\) (see [4, Section III.7]).
Notation 3.3
We set \(h = \gamma + \xi \). Observe that \(\jmath (h)\) is linearly equivalent to the pullback of a line in Open image in new window via \(\pi \) and therefore, it is a base point free polarization. In particular, we have \((h,h) = 2\), \((h,l_1') = (h, l_2') = 1\), and Open image in new window for Open image in new window . We also let Open image in new window and Open image in new window .
Let us compute the discriminant group \(A_M\) and the quadratic form Open image in new window .
Lemma 3.4
The discriminant group Open image in new window is isomorphic to \((\mathbb {Z}/2\mathbb {Z})^{\oplus 6}\).
Proof
Remark 3.5
Proposition 3.6
Let S be a K3 surface. If Open image in new window is a lattice embedding such that \(\jmath (h)\) is a base point free polarization and Open image in new window , and Open image in new window (\(1 \leqslant i,j \leqslant 3\)) all represent irreducible curves, then \(\jmath \) is a primitive embedding.
Proof
Assume that \(\jmath \) is not primitive. Then the embedding \(\jmath \) must factor through the saturation Open image in new window of M which is a nontrivial even overlattice of M: Open image in new window . By [26, Proposition 1.4.1], there is a bijection between even overlattices of M and isotropic subgroups of Open image in new window (which are generated by isotropic elements, i.e. \(v\in A_M\) such that \(q_M(v)=0\)). Using Lemma 3.4 and Remark 3.5, it is easy to classify the isotropic elements of \(A_M\). As Open image in new window and Open image in new window in \(A_M\), there are only three cases to consider. We drop the embedding \(\jmath \) in the rest of the proof.
Case 1. The isotropic element is \(\gamma ^*\). From the columns of \(G_M^{1}\) we see that \(\gamma ^* = \xi /2 \in A_M\). Hence, we have \(\xi = 2x\) for some Open image in new window . But then Open image in new window and Open image in new window which would imply that h is not base point free by Theorem 3.2.
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 Open image in new window be the lattice embedding given in Notation 3.3. Then \(\jmath \) 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 Open image in new window be a lattice embedding. If none of \(\jmath (\xi )\), Open image in new window or Open image in new window (\(1 \leqslant i, i'\!, j, j' \leqslant 3\)) is divisible by 2 in Open image in new window , then the embedding \(\jmath \) is primitive.
Proposition 3.9
Assume that S is a K3 surface such that Open image in new window is isomorphic to the lattice M. Then S is the double cover of Open image in new window 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.
Proof
By assumption there exist Open image in new window satisfying the numerical conditions in Notation 3.3. Without loss of generality, we assume that h is nef (this can be achieved by acting by \(\pm W(S)\)). Then \(l_1'\) and \(l_2'\) are both effective (as Open image in new window , \((h,l_i')=1\)). We further assume that Open image in new window (\(1 \leqslant i,j \leqslant 4\)) and \(\gamma \) are effective (apply \(s_{\alpha _i}\) or Open image in new window or \(s_{\gamma }\) if necessary).
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 \(\gamma , l_1', \alpha _1, \ldots , \alpha _3, l_2', \beta _1, \ldots , \beta _3\) and \(\xi \), with coefficients \(c_1,\ldots , c_{10}\). 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 \(\gamma , l_1', \alpha _1, \ldots , \alpha _3, l_2', \beta _1, \ldots , \beta _3,\xi \) (as what we did at the beginning of this subsection) and consider their linear combination \(D'\) with coefficients \(c_1,\ldots , c_{10}\), 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 \(h= \gamma + \xi \). Again, by Theorem 3.2 the pullback h of Open image in new window in \(S_{(Q,L,L')}\) has base points, which gives a contradiction. So the linear system of h defines a degree two map Open image in new window . Since S is a K3 surface of degree 2, the branching locus must be a sextic curve B.
Because \((h'\!, l_1')=1\), the class \(l_1'\) is represented by an irreducible curve. Similarly, \(l_2',\alpha _i\) and Open image in new window (\(1 \leqslant i,j \leqslant 3\)) all correspond to irreducible curves. It follows that the irreducible rational curves \(\alpha _1, \ldots , \alpha _4, \beta _1, \ldots , \beta _4\) are contracted by \(\pi \) to ordinary double points of the sextic B. Let \(L_1'\) (respectively \(L_2'\)) be the unique irreducible curve in S corresponding to the class \(l_1'\) (respectively \(l_2'\)) and set \(L_1 = \pi (L_1')\) (respectively \(L_2 = \pi (L_2')\)). Since Open image in new window , the projection formula implies that \(L_1\) is a line. Moreover, the line \(L_1\) has to pass through four ordinary double points of the branched curve B since Open image in new window . Similarly, \(L_2\) is also a line passing through four different ordinary double points of B. (Note that both \(L_1\) and \(L_2\) pass through the singularity of B corresponding to \(\gamma \).) By Bezout’s theorem, the two lines \(L_1\) and \(L_2\) are both components of B (otherwise we have contradictions: Open image in new window and analogously for \(L_2\)).\(\square \)
Corollary 3.10
For a sufficiently general triple \((C, L_1, L_2)\) (i.e. outside the union of a countable number of proper subvarieties of the moduli space), the Picard lattice Open image in new window coincides with M via the embedding \(\jmath \).
Proof
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 Open image in new window 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 Open image in new window such that \(\jmath (h)\) is a base point free degree two polarization.
Proof
Thanks to the transversal intersection, we define the embedding \(\jmath \) as in the generic case. In particular, the morphism Open image in new window is defined by \(\jmath (h)\). The embedding \(\jmath \) is primitive by Proposition 3.6.\(\square \)
3.3 Mpolarized 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 Open image in new window and define a period map for generic triples \((C, L_1, L_2)\) via the periods of Mpolarized K3 surfaces \(S_{(C, L_1, L_2)}\).
Definition 3.12
Let M be the lattice defined in Notation 3.3. An MpolarizedK3 surface is a pair \((S, \jmath )\) such that Open image in new window is a primitive lattice embedding. The embedding \(\jmath \) is called the Mpolarization of S. We will simply say that S is an MpolarizedK3 surface when no confusion about \(\jmath \) is likely.
We now determine the lattice M and show that it admits a unique primitive embedding into the K3 lattice \(\Lambda _{K3}\).
Proposition 3.13
Let M be the lattice defined in Notation 3.3. Then M is isomorphic to the lattice \(U(2) \perp A_1^{2} \perp D_6\) and admits a unique primitive embedding (up to isometry) \(M \hookrightarrow \Lambda _{K3}\) into the K3 lattice \(\Lambda _{K3}\). The orthogonal complement Open image in new window with respect to the embedding is isometric to \(U \perp U(2) \perp A_1^{2} \perp D_6\).
Proof

M has rank 10 and signature (1, 9).

The Gram matrix \(G_M\) has determinant \((64)\).

The discriminant group is Open image in new window with quadratic form Open image in new window , where \(u,w_{2,1}^{1}\) and \(w_{2,1}^{1}\) are the discriminant forms associated to \(U(2),E_7\) and \(A_1\) respectively (cf. [5, Section 1.5 and Appendix A] and references therein). Note that Open image in new window is isomorphic to the discriminant form of \(D_6\).
Remark 3.14
Note that both M and T are even indefinite 2elementary lattices (a lattice L is 2elementary if Open image in new window for some k). One could also invoke Nikulin’s classification [26, 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 Open image in new window .
Let L be an even lattice. Recall that any \(g \in O(L)\) naturally induces Open image in new window by \(g^*\varphi :v \mapsto \varphi (g^{1}v)\) (which further defines an automorphism of \(A_L\) preserving \(q_L\), therefore giving a natural homomorphism \(r_L:O(L)\rightarrow O(q_L)\)).
Lemma 3.15
The homomorphisms \(r_M:O(M) \rightarrow O(q_M)\) and \(r_T:O(T) \rightarrow O(q_T)\) are both surjective.
In particular, we have \(O(M) \twoheadrightarrow O(q_M) \cong O(q_T) \twoheadleftarrow O(T)\). By [26, Theorem 1.6.1, Corollary 1.5.2], an automorphism \(g_M \in O(M)\) can be extended to an automorphism of \(\Lambda _{K3}\) if and only if Open image in new window . In our case, any automorphism \(g_M \in O(M)\) can be extended to an element in \(O(\Lambda _{K3})\).
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 \(g \in O(\Lambda _{K3})\) with \(g_T = g_T\). The same statement holds for \(g_T\).
Proof
The proof is similar to that for [15, Proposition 14.2.6]. Take any \(x = x_M + x_T \in \Lambda _{K3}\) with \(x_M \in M^*\) and Open image in new window . View \(\Lambda _{K3}\) as an overlattice of Open image in new window . The corresponding isotropic subgroup (cf. [26, Section 1.4]) of Open image in new window is Open image in new window . Since \(x \in \Lambda _{K3}\), \(\overline{x}_M + \overline{x}_T\) is contained in Open image in new window (where \(\overline{x}_M\) denotes the corresponding element of \(x_M\) in \(A_M\) and similarly for \(\overline{x}_T\)). Consider Open image in new window . Note that the image of \(g_M(x_M) + g_T(x_T)\) under the map Open image in new window is \(r_M(g_M)(\overline{x}_M) + r_T(g_T)(\overline{x}_T)\). Recall that \(A_M\) and \(A_T\) are identified via the natural projections Open image in new window . Because \(r_M(g_M) = r_T(g_T)\), \(r_M(g_M)(\overline{x}_M) + r_T(g_T)(\overline{x}_T)\) is contained in Open image in new window . In other words, we have \(g_M(x_M) + g_T(x_T) \in \Lambda _{K3}\).\(\square \)
Let \(\Sigma _{\alpha } \subset O(M)\) (respectively Open image in new window ) be the subgroup which permutes Open image in new window (respectively the subgroup which permutes \(\{\beta _1, \ldots , \beta _4\}\)). We seek automorphisms of T which can be extended to automorphisms of \(\Lambda _{K3}\) whose restrictions to M belong to \(\Sigma _\alpha \) or \(\Sigma _\beta \). We observe that there is a natural inclusion Open image in new window .
Lemma 3.17
The composition Open image in new window is injective.
Proof
First let us describe the automorphisms of \(A_M\) induced by the transpositions in \(\Sigma _\alpha \) and \(\Sigma _{\beta }\). We consider \(\Sigma _\alpha \) and the case of \(\Sigma _\beta \) is analogous. The image of the transposition \((\alpha _i\alpha _i')\) (with \(1 \leqslant i \ne i' \leqslant 3\)) defines the element \(r_M((\alpha _i\alpha _i'))\) in \(O(q_M)\) given by \(\alpha _i^* \mapsto \alpha _{i'}^*\), \(\alpha _{i'}^* \mapsto \alpha _i^*\), leaving \(\gamma ^*\!,\alpha _{i''}^*\) ( Open image in new window ), Open image in new window (\(1 \leqslant j \leqslant 3\)) and \(\xi ^*\) invariant.

If \(\overline{g}(\xi ^*) = \xi ^*\), then \(g_\alpha \) (respectively \(g_\beta \)) fixes \(\alpha _4\) (respectively \(\beta _4\)) by the description of the permutations above and \(g_\alpha \) (respectively \(g_\beta \)) can be recovered from the action of \(\overline{g}\) on the set Open image in new window (respectively \(\{\beta _1^*, \beta _2^*, \beta _3^*\}\)).

If \(\overline{g}(\xi ^*) = \xi ^* + \alpha _i^*\) (\(1 \leqslant i \leqslant 3\)), then \(g_\alpha \) maps \(\alpha _4\) to \(\alpha _i\) and \(g_\beta \) fixes \(\beta _4\). Then \(g_\alpha \) (respectively \(g_\beta \)) is determined by the action of \(\overline{g}\) on the set Open image in new window (respectively \(\{\beta _1^*, \beta _2^*, \beta _3^*\}\)).

If Open image in new window (\(1 \leqslant j \leqslant 3\)), then \(g_\beta \) maps \(\beta _4\) to Open image in new window and \(g_\alpha \) fixes \(\alpha _4\). Then \(g_\alpha \) (respectively \(g_\beta \)) is determined by the action of \(\overline{g}\) on the set Open image in new window (respectively \(\{\xi ^*+ \beta _1^*, \xi ^*+\beta _2^*, \xi ^*+\beta _3^*, \xi ^*\}\)).

If Open image in new window (\(1 \leqslant i,j \leqslant 3\)), then \(g_\alpha \) maps \(\alpha _4\) to \(\alpha _i\) and \(g_\beta \) maps \(\beta _4\) to Open image in new window . Then \(g_\alpha \) (respectively \(g_\beta \)) can be recovered by the action of \(\overline{g}\) on the set Open image in new window (respectively Open image in new window ). \(\square \)
Recall that Open image in new window 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 Open image in new window 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 Open image in new window .
Proof
3.4 Mpolarization for nongeneric intersections
We will show in this section that the birational map Open image in new window in Proposition 3.18 extends to a birational morphism Open image in new window . To do this, we need to extend the construction of Mpolarization Open image in new window to the nongeneric triples \((C, L_1, L_2)\) and show that the construction fits in families. The idea is to use the normalized lattice polarization (cf. [17, Definition 4.24]) for degree 5 pairs constructed in [17, Section 4.2.3]. A degreedpair (D, L) consists of a degree d plane curve D and a line Open image in new window (see [17, 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 \(\gamma , l_1', \alpha _1, \ldots , \alpha _4\) (respectively \(\gamma , l_2', \beta _1, \ldots , \beta _4\)) using the degree 5 pair \((C+L_2, L_1)\) (respectively \((C+L_1, L_2)\)).
The procedure described above produces five divisors: \(D_0, \ldots , D_4\). 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 \(L'\!, D_0, \ldots , D_4\) 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 [17, p. 2141].
Lemma 3.19
 (a)
the polarization class of \(S_{(D,L)}\) is Open image in new window , and
 (b)
their intersections are Open image in new window , Open image in new window , \((L'\!, D_k) = 1\) for \(0 \leqslant k,k' \leqslant 4\).
Proof
See the proof of [17, Proposition 4.25].\(\square \)
Let us consider 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 worst simple singularities. Let \(\overline{S}_{(C,L_1,L_2)}\) be the double plane branched along \(C+L_1+L_2\) and \(S_{(C,L_1,L_2)}\) be the K3 surface obtained by taking the minimal resolution of \(\overline{S}_{(C,L_1,L_2)}\). Let Open image in new window be the natural morphism. To define a lattice embedding Open image in new window , one needs to specify the images of \(\gamma , l_1', \alpha _1, \ldots , \alpha _4, l_2', \beta _1, \ldots , \beta _4\) so that the intersection form is preserved. There is a compatibility condition induced by \(2l_1' + \alpha _1 + \ldots + \alpha _4 = 2l_2' + \beta _1 + \cdots + \beta _4\). Recall that \(h=\xi + \gamma = 2l_1' + \alpha _1 + \ldots + \alpha _4 +\gamma = 2l_2' + \beta _1 + \cdots + \beta _4 + \gamma \). We also require that \(\jmath (h)\) is the class of the base point free polarization Open image in new window .

\(\jmath (\gamma ) = R_0 = T_0\) (by our choice of the labelings, both \(R_0\) and \(T_0\) are the fundamental cycle associated to the singularity of \(C+L_1+L_2\) at the point \(L_1 \cap L_2\));

\(\jmath (\alpha _i) = R_i\) and Open image in new window for \(1 \leqslant i,j \leqslant 4\).

\(\jmath (h)\) is the class of the base point free polarization Open image in new window .

Open image in new window and Open image in new window are the classes of irreducible rational curves.

\(\jmath (\gamma ), \jmath (\alpha _1), \ldots , \jmath (\alpha _4)\) (respectively \(\jmath (\gamma ), \jmath (\beta _1), \ldots , \jmath (\beta _4)\)) are classes of effective divisors which are contracted by \(\pi \) to the points of the intersection \(C \cap L_1\) (respectively \(C \cap L_2\)). In particular, \(\jmath (\gamma )\) is contracted to the point \(L_1 \cap L_2\).
Lemma 3.20
The lattice embedding Open image in new window is primitive.
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.\(\square \)
Proposition 3.21
The birational map in Proposition 3.18 extends to a morphism Open image in new window . Moreover, the map Open image in new window 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 Mpolarized K3 surface \((S_{(C, L_1, L_2)}, \jmath )\) 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 \(\jmath \) is the lattice polarization constructed above. By [8], the Mpolarized K3 surface \((S_{(C, L_1, L_2)}, \jmath )\) corresponds to a point in Open image in new window . The polarization \(\jmath \) depends only on \((C, L_1, L_2)\), the ordering of \(C \cap L_1\) and the ordering of \(C \cap L_2\), and it is compatible with the action of Open image in new window . Consequently, we can associate to every triple \((C, L_1, L_2)\) a point in Open image in new window . In other words, we have a welldefined morphism Open image in new window extending the birational map in Proposition 3.18.
Choose a point Open image in new window (more precisely, \(\omega \) is a \(\Gamma \)orbit) which corresponds to an Mpolarization K3 surface \(S_{(C, L_1, L_2)}\). Lemma 3.16 allows us to extend an element of \(\Gamma \) to an isometry of the K3 lattice \(\Lambda _{K3}\). The global Torelli theorem for K3 surfaces implies that the period \(\omega \) 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 \(\Gamma \). It follows that the period point \(\omega \) uniquely expresses the K3 surface as a double cover of Open image in new window and determines two line components of the branched locus. Now we conclude that \(\omega \) determines uniquely the triple \((C, L_1, L_2)\). \(\square \)
3.5 Surjectivity of the period map
We will show in this section that the period map Open image in new window 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 Mpolarization 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 Open image in new window . Then there exists a plane quartic curve C and two different lines Open image in new window such that \(S \cong S_{(C, L_1, L_2)}\) and \(C+L_1+L_2\) has at worst simple singularities.
Proof
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 Open image in new window .
Proof
It suffices to prove that Open image in new window is surjective. Let Open image in new window be a period point. By the surjectivity of the period map of latticepolarized K3 surfaces (see [8, Theorem 3.1]) there exists an Mpolarized K3 surface Open image in new window corresponding to \(\omega \). By Proposition 3.22, the K3 surface S is the double cover of Open image in new window branched at a plane quartic curve C and two different lines \(L_1,L_2\). Moreover, let \(M_1\) and \(M_2\) be the primitive sublattices of M defined in the proof of Proposition 3.22. After choosing the Kähler cone \(V^+(S)\) and the set of effective \((2)\) curves \(\Delta ^+(S)\) as in the proof of [17, Theorem 4.1], we may assume that the restrictions \(\jmath _{M_1}\) and \(\jmath _{M_2}\) of the polarization \(\jmath \) to the sublattices \(M_1\) and \(M_2\) are both normalized embeddings, as defined in [17, Definition 4.24]. The embeddings \(\jmath _{M_1}\) and \(\jmath _{M_2}\) are unique up to permutation of the classes \(\alpha _1,\ldots , \alpha _4\) (\(\beta _1,\ldots , \beta _4\), respectively) thanks to [17, Lemma 4.29]. It follows that the polarization \(\jmath \) is unique up to action of Open image in new window and coincides with our construction in Sect. 3.4. By Propositions 3.18 and 3.21, the period map Open image in new window is a bijective birational morphism between normal varieties. As a result, Open image in new window is an isomorphism, by Zariski’s Main Theorem.\(\square \)
3.6 The period map for unlabeled triples
3.7 Comparison of the GIT and the Baily–Borel compactifications
Consider the moduli space Open image in new window 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 Open image in new window in Sect. 3.6 and have shown that it is an isomorphism. There are two natural ways to compactify Open image in new window as the GIT quotient Open image in new window defined in (3) and described in Corollary 2.16, or as the Baily–Borel compactification [3]. We compare these two compactifications by applying some general results of Looijenga [21]. See also [17, Theorem 4.2].
Theorem 3.24
The period map Open image in new window extends to an isomorphism of projective varieties Open image in new window where Open image in new window denotes the Baily–Borel compactification of Open image in new window .
Proof
Question 3.25
Does the period map for labeled triples Open image in new window (cf. Theorem 3.23) preserve the natural polarizations?
A positive answer to this question would imply that the period map Open image in new window can be extended to an isomorphism Open image in new window . 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 Open image in new window and Open image in new window ).
3.8 The Baily–Borel compactification
The locally symmetric space Open image in new window admits a canonical minimal compactification, the Baily–Borel compactification Open image in new window (cf. [3]). The boundary components of Open image in new window are either 0dimensional (Type III components) or 1dimensional (Type II components), and they correspond to the primitive rank 1, respectively, rank 2 isotropic sublattices of T up to \(\Gamma '\)equivalence. Following the approach of [17, 29, 31], we determine the number of the Type III boundary components of Open image in new window and compute certain invariants for the Type II boundary components. Notice that by 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 Open image in new window described in Corollary 2.16.
We determine the 0dimensional components of Open image in new window using [29, Proposition 4.1.3]. The 0dimensional boundary components are in onetoone correspondence with the \(\Gamma '\)orbits of primitive isotropic rank 1 sublattices of T. By Proposition 3.13 and [26, Theorem 3.6.2] we have Open image in new window . (In particular, T contains two hyperbolic planes.) Write Open image in new window (see Sect. 3.6). Note that for \(v \in T\) one can associate a vector Open image in new window defined by Open image in new window \(\mathrm{mod}\,T\) (where Open image in new window is the divisor of v which is a positive integer such that Open image in new window ). If v is a primitive isotropic vector then \(\overline{v}\) is an isotropic element in \(A_T\). By [29, Proposition 4.1.3] the map \(\mathbb {Z}v \mapsto \overline{v}\) induces a bijection between the equivalence classes of primitive isotropic rank 1 sublattices of T and \(\Gamma ^*\)orbits of isotropic elements of \(A_T\). Because T is the orthogonal complement of M in \(\Lambda _{K3}\), one has Open image in new window . We have computed the discriminant quadratic form \(q_M\) in Lemma 3.4. In particular, there are 20 isotropic elements in \(A_M \cong A_T\): \(0, \gamma ^*\), Open image in new window and Open image in new window (\(1 \leqslant i,j \leqslant 3\)). The action of \(\Gamma ^*\) has been described in the proof of Lemma 3.17 and Sect. 3.6. It is easy to see that Open image in new window and Open image in new window (\(1 \leqslant i,j \leqslant 3\)) form one \(\Gamma ^*\)orbit. As a result, the Baily–Borel compactification Open image in new window consists of three 0dimensional boundary components.
Remark 3.26
Similarly, one can show that Open image in new window has three 0dimensional boundary components (compare Lemma 2.15).
Remark 3.27

If \(\overline{v}=0\), then Open image in new window .

If \(\overline{v}=\gamma ^*\), then Open image in new window .

If Open image in new window or Open image in new window (\(1 \leqslant i,j \leqslant 3\)), then Open image in new window .
To determine the 1dimensional components of Open image in new window , 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 [33]. Specifically, for each of the equivalence classes of primitive isotropic rank 1 sublattices \(\mathbb {Z}v\) of T we apply Vinberg’s algorithm to the hyperbolic lattice Open image in new window (with respect to the action by the stabilizer \(\Gamma '_v\) of v).
Now we briefly recall Vinberg’s algorithm [33] (see also [31, Section 4.3]). Let N be a hyperbolic lattice of signature (1, n). (In our case we take Open image in new window .) The algorithm starts by fixing an element \(h \in N\) of positive square. Then one needs to inductively choose roots \(\delta _1, \delta _2, \ldots \) such that the distance function \({(h,\delta )^2}/{(\delta ,\delta )}\) is minimized. The algorithm stops with the choice of \(\delta _N\) if every connected parabolic subdiagram (i.e. the extended Dynkin diagram of a root system) of the Dynkin diagram \(\Sigma \) associated to the roots \(\delta _1, \delta _2, \ldots , \delta _N\) is a connected component of some parabolic subdiagram of rank \(n1\). If the algorithm stops then the W(N)orbits of the isotropic lines in N correspond to the parabolic subdiagrams of rank \(n1\) of \(\Sigma \) (N.B. the isomorphism classes of Open image in new window , 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 \(\Sigma \) into consideration.

If \(\overline{v}=0\), then Open image in new window has at least three equivalence classes of isotropic vectors which correspond to Open image in new window , Open image in new window and Open image in new window respectively.

If \(\overline{v}=\gamma ^*\), then Open image in new window has at least two equivalence classes of isotropic vectors which correspond to Open image in new window and \(D_8\) respectively.

If Open image in new window or Open image in new window (\(1 \leqslant i,j \leqslant 3\)), then Open image in new window has at least three equivalence classes of isotropic vectors which correspond to Open image in new window , Open image in new window and \(D_8\) respectively.
Remark 3.28
GIT boundary  Baily–Borel boundary 

\(D_8\)  
Notes
Acknowledgements
We thank Radu Laza and Gregory Pearlstein for useful discussions.
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