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European Journal of Mathematics

, Volume 4, Issue 3, pp 1000–1034 | Cite as

Compactifications of the moduli space of plane quartics and two lines

  • Patricio Gallardo
  • Jesus Martinez-GarciaEmail author
  • Zheng Zhang
Open Access
Research Article

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 curves 

Mathematics Subject Classification

14L24 14J28 14J17 32G20 14Q10 

1 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 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 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 bi-double 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 (XH) 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 M-polarized 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 M-polarized 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 non-generic 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 quasi-projective 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 .

Fig. 1

Incidence relations among the boundary components of the compactification of Open image in new window in Open image in new window . We denote \(A \rightarrow B\) when the boundary component B is contained in the closure of the boundary component A

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 (XH) 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 n-dimensional hypersurfaces of degree d together with k (labeled) hyperplanes

Let Open image in new window be the parameter scheme of tuples \((F_d,l_1,\ldots ,l_k)\), where \(F_d\) is a polynomial of degree d and \(l_1,\ldots , l_k\) are linear forms in variables \((x_0,\ldots , x_{n+1})\), modulo scalar multiplication. We havewhere \(N= {\left( {\begin{array}{c}n+1+d\\ d\end{array}}\right) }-1\) and natural projections Open image in new window , Open image in new window for \(i=1,\ldots , k\). The natural action of Open image in new window in Open image in new window extends to each of the factors in Open image in new window and therefore to Open image in new window itself. The set of G-linearizable line bundles Open image in new window is isomorphic to \(\mathbb {Z}^{n+1}\). Then a line bundle Open image in new window is ample if and only if \(a>0\), \(b_i>0\) for \(i=1,\ldots , k\), whereThe latter is a trivial generalization of [13, Lemma 2.1]. Hence, for Open image in new window , the GIT quotient is defined aswhere \(t_i={b_i}/{a}\). Next, we explain why it is enough to consider the vector Open image in new window instead of \((a;b_1,\ldots , b_k)\). Let us introduce some notation.
Given a maximal torus Open image in new window , we can choose projective coordinates Open image in new window such that T is diagonal in G. Hence, any one-parameter subgroup Open image in new window is a diagonal matrix with diagonal entries \(s^{r_i}\) where \(r_i\in \mathbb Z\) for all i and Open image in new window . We say that \(\lambda \) is normalized if \(r_0\geqslant \cdots \geqslant r_{n+1}\) and \(\lambda \) is not trivial. Any homogeneous polynomial g of degree d can be written as Open image in new window , where Open image in new window , Open image in new window , Open image in new window and Open image in new window . The support of g is Open image in new window . We have a natural pairing Open image in new window , which we use to introduce the Hilbert–Mumford function for homogeneous polynomials:Definewhich is piecewise linear on \(\lambda \) for fixed \((f,l_1,\ldots ,l_k)\). Since the Hilbert–Mumford function is functorial [25, Definition 2.2, cf. p. 49], we can generalise [13, Lemma 2.2] to show that a tuple \((f,l_1,\ldots ,l_k)\) is (semi-)stable with respect to a polarisation Open image in new window if and only ifis negative (respectively, non-positive) for any normalized non-trivial one-parameter subgroup \(\lambda \) of any maximal torus T of G. Hence the stability of a tuple is independent of the scaling of Open image in new window and as such, we may define:

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 non-trivial normalized one-parameter 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 one-parameter subgroups—a finite set—and we showed that if \(k=1\) it was sufficient to consider the one-parameter 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

The fundamental set\(S_{n,d}\)of one-parameter subgroups\(\lambda \in T\) consists of all elements Open image in new window wheresatisfying the following:

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

A tuple \((X,H_1,\ldots ,H_k)\) given by equations \((f,l_1,\ldots ,l_k)\) is not \(\vec t\)-stable (respectively not \(\vec t\)-semistable) if and only if there is \(g \in G\) satisfyingMoreover Open image in new window .

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 one-parameter subgroups in the coordinate system induced by T are the intersection of \(\sum r_i=0\) and the convex hull of \(r_i-r_{i+1}\geqslant 0\), 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 one-parameter 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

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 Open image in new window . Since Open image in new window is a product of vector spaces (and hence a Mori dream space), Open image in new window is also a rational polyhedron. It is possible to precisely describe it and we will do this later for Open image in new window . Moreover, there is a finite number of non-isomorphic GIT compactifications Open image in new window as Open image in new window in varies. Therefore we have a natural division of Open image in new window 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 [9, Theorem 0.2.3]. The quotient Open image in new window is constant as \(\vec t\) moves in the interior of a face or chamber. It is possible to find these walls explicitly by means of Lemma 2.3 (see [13, Theorem 1.1]) for given (ndk), since all walls of dimension \(k-1\) should be a subset of the finite set of equationsAnother interesting feature is that the the \(\vec t\)-stability of tuples \((X,H_1,\ldots , H_k)\) is equivalent of the t-stability of reducible GIT hypersurfaces of higher degree. Indeed:

Lemma 2.5

Let Open image in new window where Open image in new window for all \(i=1,\ldots , k\). Let Open image in new window , Open image in new window such that Open image in new window and let Open image in new window . Let Open image in new window . A tuple \((X,H_1,\ldots ,H_k)\) is \(\vec t\)-(semi)stable if and only if the tupleis \(\vec {t'}\)-(semi)stable.

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

Let \(\lambda \) be a normalized one-parameter subgroup, m be a positive integer and \(g=\sum g_Ix^I\) be a homogeneous polynomial. Let J be such thatThen, since \(\lambda \) is normalized, Open image in new window .
Let \((f,l_1,\ldots ,l_k)\) be the equations of \((X,H_1,\ldots , H_k)\) under some system of coordinates and let \(\lambda \) be a normalized one-parameter subgroup. Using the above observation, the lemma follows from:

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

(cf. [12, Corollary 1.2]) If the locus of stable points is not empty, and \(d \geqslant 3\), then

Proof

From [27, Theorem 2.1], any hypersurface Open image in new window where f is a homogeneous polynomial of degree \(d\geqslant 3\) has Open image in new window . Hence, for any tuple \(p=(X,H_1,\dots , H_K)\) such that X is smooth and \(X\cap H_i\) has simple normal crossings, its stabilizer Open image in new window satisfieswhere the last equality follows from [27, Theorem 2.1]. The result follows from [7, Corollary 6.2]:
Now let us consider the case of the symmetric polarization of Open image in new window . In order to do so, observe that the group \(S_k\) acts on Open image in new window by defining the action of \(h\in S_k\) as
$$\begin{aligned} h:(f,l_1,\ldots ,l_k)\mapsto (f,l_{h(1)},\ldots , l_{h(k)}). \end{aligned}$$
Define Open image in new window , which parametrizes classes of tuples \([(f,l_1,\ldots , l_k)]\) up to multiplication by a scalar and permutation of Open image in new window , i.e. Open image in new window for \(g\in S_k\) and Open image in new window . Hence, we parameterize the same elements as in Open image in new window but we forget the ordering of the linear forms. In particular \(\mathscr {R}'_{n,d,k}\) parametrizes pairs \([(X,H_1,\ldots , H_k)]\) formed by a hypersurface Open image in new window of degree d and k unordered hyperplanes. The quotient morphism Open image in new window is G-equivariant. Let Open image in new window such that \(({b_1}/{a},\ldots ,{b_k}/{a})=(1,\ldots ,1)\) (i.e. we are considering \(\vec t\)-stability with respect to Open image in new window ). If the conditionholds, then a tuple \((f,l_1,\ldots , l_k)\) is \(t_1\)-(semi)stable if and only if Open image in new window is stable with respect to Open image in new window by [25, Theorem 1.1 and p. 48]. Hence, it is natural to define the GIT quotientwhich is the GIT quotient of unordered tuples\((X,H_1,\cdots , H_k)\)with respect to the polarization Open image in new window . We have a commutative diagramWe want to determine all the orbits represented in Open image in new window from the orbits represented in Open image in new window via Open image in new window .
Choose \(1\leqslant j_1< \cdots < j_l\leqslant k\) and define the G-equivariant morphism Open image in new window given byBy Lemma 2.5, we have a commutative diagram

Proposition 2.8

Let \(1\leqslant j_1< \cdots <j_l\leqslant k\) and suppose that (2) holds. An unordered tuple \([(X,H_1,\dots ,H_k)]\)—where X is a hypersurface of degree d in Open image in new window and \(H_1,\ldots , H_k\) are k unordered hyperplanes—is (semi)stable with respect to Open image in new window if and only if—a pair represented by a tuple in Open image in new window —is \(\vec {t_*}\)-(semi)stable. Moreover an orbit Open image in new window is closed if and only if and only if Open image in new window is closed. In addition, an orbit Open image in new window is closed if and only if Open image in new window is closed.

Proof

Since (2) holds, all the spaces in the above diagram are non-empty. 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 \(k-l\) 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

The space of GIT stability conditions isIn particular, (2) holds.

Proof

Let Open image in new window be a vector and \((C,L_1,L_2)\) be a \(\vec t\)-semistable tuple. By choosing an appropriate change of coordinates, we may assumeLet Open image in new window . Then, as \(t_1\geqslant 0, t_2\geqslant 0\), we haveSimilarly, by taking a change of coordinates such that Open image in new window , Open image in new window , we may show that Open image in new window .

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 (dd). 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 t-stability 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 (dd)-semistable. Let Open image in new window . By Lemma 2.5, such a pair is t-semistable 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 (nd) 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 one-dimensional 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 t-semistable.

Lemma 2.11

Suppose that C is a plane curve of degree d whose only singular point \(p \in C\) is a linearly semi-quasihomogeneous singularity [17, Definition 2.21] with respect to the weights \(\vec w=(w_1,w_2)\), \(w_1 \geqslant w_2>0\). Suppose further that \(C+L_1+L_2\) have simple normal crossings in Open image in new window . Let f be the localization of the equation of f at p and \(\vec w(f)\) be its weighted degree with respect to \(\vec w\).
  1. (a)
    Suppose that \(p\not \in L_1\cup L_2\). Then
     
  2. (b)
    Suppose that \(p\not \in L_2\) and \(p\in L_1\cap C\). Then
     

Proof

We may choose a coordinate system such that Open image in new window is the singular point of C. We consider the one-parameter subgroup \(\lambda = (2w_1-w_2, 2w_2-w_1,-w_1-w_2)\) 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.

Let \(l_1(x_0,x_1)+x_2\) and \(l_2(x_0,x_1)+x_2\) be the equations of the lines \(L_1\) and \(L_2\), respectively, where \(l_1,l_2\) are linear forms. We haveTherefore, \(\mu _{t_1,t_2}(C,L_1, L_2, \lambda ) >0\) and the triple is destabilized by \(\lambda \).
(b) 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 Open image in new window , as \(w_1\geqslant w_2\), we have

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 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 \(\vec t =(1,1)\) 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 GIT-semistable 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 GIT-stable by Lemma 2.12. Now suppose Z is GIT-stable and reduced. By [19, 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.\(\square \)

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.

Let Open image in new window and Open image in new window be the set of \(\vec t\)-stable and \(\vec t\)-semistable tuples \((f,l_1,l_2)\), respectively. LetLet \(\vec t=(1,1)\). By Lemma 2.12, Open image in new window . Let Open image in new window , Open image in new window and recall that Open image in new window . Then Open image in new window . We are interested in describing the compactification of Open image in new window by Open image in new window . We use the notation in [19].

Lemma 2.15

The quotient Open image in new window is the compactification of Open image in new window 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 \(a \ne 0,1, \infty \).

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).\(\square \)

Fig. 2

Triples parametrized by Open image in new window . The dotted and dashed lines represent the lines \(L_1\) and \(L_2\), respectively (see Lemma 2.15)

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

Let Open image in new window be the GIT compactification of a quartic plane curve and two unlabeled lines and let Open image in new window be the open loci parametrizing triples \((C,L,L')\) such that \(C+L+L'\) is reduced and has at worst simple singularities. Then, Open image in new window is the union of three points, Open image in new window , Open image in new window , and five rational curves, Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , which are obtained as images—via the natural morphism Open image in new window —of points and rational curves described in Lemma 2.15, as follows:
Moreover, the boundary components Open image in new window and Open image in new window in Open image in new window are mapped onto the same boundary component Open image in new window in 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

Given any even lattice L, we define:
  • 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 torsion-free 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 torsion-free 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 blow-up of Open image in new window 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 [4, Section III.7]).

Let Open image in new window be the pullback of the class of a line in Open image in new window . The class h is a degree 2 polarization of \(S_{(C,L_1,L_2)}\). We assume that \(C \cap L_1 = \{p_1, p_2, p_3, p_4\}\), Open image in new window and Open image in new window . Denote the classes of the exceptional divisors corresponding to \(p_i\), \(q_i\), and r by \(\alpha _i\), \(\beta _i\) and \(\gamma \) respectively (\(1 \leqslant i \leqslant 4\)). Let us also denote by \(l_1'\) (respectively \(l_2'\)) the class of the strict transform of \(L_1\) (respectively \(L_2\)). Note that the morphism Open image in new window is given by the classIt is straightforward to check that Open image in new window , Open image in new window , Open image in new window for \(1 \leqslant i,j \leqslant 4\). Clearly, we have Open image in new window , Open image in new window , and Open image in new window for \(1 \leqslant i,j \leqslant 4\).
Consider the sublattice of the Picard lattice of \(S_{(C,L_1,L_2)}\) generated by the curve classes \(\gamma ,l_1',\alpha _1, \ldots , \alpha _4,l_2',\beta _1, \ldots , \beta _4\). LetIt follows from (5) that Open image in new window forms a \(\mathbb {Z}\)-basis of the sublattice. The Gram matrix with respect to this basis is computed as follows:
$$\begin{aligned} \left( {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} -2 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 2 \\ 1 &{} -2 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} -2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} -2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} -2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} -2 &{} 1 &{} 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -2 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} -2 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} -2 &{} 0 \\ 2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ \end{array} } \right) _{\textstyle .} \end{aligned}$$

Notation 3.3

Let M be the abstract lattice of rank 10 spanned by an ordered basiswith the intersection form given by the above Gram matrix, which we will call \(G_M\). Notice that M is an even lattice. If \(S_{(C,L_1,L_2)}\) is a K3 surface obtained as above from a smooth quartic C and two lines \(L_1,L_2\) such that \(C+L_1+L_2\) has simple normal crossings, then there is a natural lattice embedding Open image in new window as described before.

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

Let us denote by Open image in new window (respectively Open image in new window ) the dual element of \(\alpha _i\in M\) (respectively Open image in new window , for \(1 \leqslant i,j \leqslant 3\)). Recall that \(\alpha _i^*\) is defined to be the unique element of \(M^*\) such that \((\alpha _i^*, \alpha _i) = 1\) and the pairing of \(\alpha _i^*\) with any other element of the basis Open image in new window is 0. We define Open image in new window and \(\xi ^*\) in a similar way. The coefficients of the dual elements \(\gamma ^*\!, l_1'^*, \alpha _1^*, \alpha _2^*, \alpha _3^*, l_2'^*, \beta _1^*, \beta _2^*, \beta _3^*, \xi ^*\) (with respect to the basis Open image in new window ) can be read from the rows or columns of the inverse matrix \(G_M^{-1}\) of the Gram matrix \(G_M\) of M:
$$\begin{aligned} G_M^{-1}= \left( {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{2} \\ 0 &{} -2 &{} -1 &{} -1 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} -1 &{} -1 &{} -\frac{1}{2} &{} -\frac{1}{2} &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{2} \\ 0 &{} -1 &{} -\frac{1}{2} &{} -1 &{} -\frac{1}{2} &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{2} \\ 0 &{} -1 &{} -\frac{1}{2} &{} -\frac{1}{2} &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{2} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -2 &{} -1 &{} -1 &{} -1 &{} 1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} -1 &{} -\frac{1}{2} &{} -\frac{1}{2} &{} \frac{1}{2} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} -\frac{1}{2} &{} -1 &{} -\frac{1}{2} &{} \frac{1}{2} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} -\frac{1}{2} &{} -\frac{1}{2} &{} -1 &{} \frac{1}{2} \\ \frac{1}{2} &{} 1 &{} \frac{1}{2} &{} \frac{1}{2} &{} \frac{1}{2} &{} 1 &{} \frac{1}{2} &{} \frac{1}{2} &{} \frac{1}{2} &{} -\frac{1}{2} \end{array} } \right) _{\textstyle .} \end{aligned}$$
For instance, Open image in new window , where we identify each element of M with its image in \(M^*\). By abuse of notation, we also use Open image in new window to denote the corresponding elements in Open image in new window . Observe that \(l_1'^* = l_2'^* \equiv 0 \in A_M\). It is straightforward to verify that \(A_M\) can be generated by Open image in new window and hence \(A_M\) is isomorphic to \((\mathbb {Z}/2\mathbb {Z})^{ \oplus 6}\). Indeed, this follows from observing from the columns of \(G_M^{-1}\) that \(\alpha ^*_3=\gamma ^* + \alpha _1^*+\alpha _2^*\in A_M\) and \(\beta ^*_3=\gamma ^* + \beta _1^*+\beta _2^*\in A_M\).\(\square \)

Remark 3.5

We derive a formula for the quadratic form \(q_M:A_M \rightarrow \mathbb {Q}/2\mathbb {Z}\):
$$\begin{aligned}&q_M\bigl (a\gamma ^* + b\alpha _1^* + c\alpha _2^* + d\beta _1^* + e\beta _2^* + f\xi ^*\bigr )\\ {}&\;\;\equiv b^2 + c^2 +bc + d^2 + e^2 + de + (a+b+c+d+e)f - \frac{1}{2}\, f^2 \in \mathbb {Q}/2\mathbb {Z}. \end{aligned}$$

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 non-trivial 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.

Case 2. The isotropic element is Open image in new window where \(1 \leqslant i,j \leqslant 3\). Let us take \(\alpha _1^* + \beta _1^*\) for example. The other cases are similar. Note that Open image in new window in \(A_M\). We have \(\alpha _2 + \alpha _3 + \beta _2 + \beta _3 = 2y\) for some Open image in new window . Because S is a K3 surface and Open image in new window , either y or \(-y\) is effective. Note that \(l_1'\), \(l_2'\), \(\alpha _i\) and Open image in new window (\(1 \leqslant i,j \leqslant 3\)) are irreducible curves (by the assumption), h is nef and Open image in new window . It follows that Open image in new window is nef. Since Open image in new window , y is effective. Because \((y,\alpha _2) = (y, \alpha _3) = (y, \beta _2) = (y, \beta _3) = -1\), we know \(\alpha _2,\alpha _3,\beta _2\) and \(\beta _3\) are in the support of y. Write \(y = m\alpha _2 + n\alpha _3 + k \beta _2 + l \beta _3 + D = (\alpha _2 + \alpha _3 + \beta _2 + \beta _3)/2\) where D is an effective divisor, Open image in new window and \(m,n,k,l \geqslant 1\). But then we have a contradictionCase 3. The isotropic element is Open image in new window where \(1 \leqslant i,j \leqslant 3\). Take \(\alpha _1^* + \beta _1^* + \gamma ^*\) for example. Since \(\alpha _1^* + \beta _1^* + \gamma ^* = \alpha _2 /2+ \alpha _3/2 + \beta _2/2 + \beta _3/2 + \xi /2\) in \(A_M\), there exists an element z of Open image in new window such that Open image in new window . Because S is a K3 surface, Open image in new window and \((z,h) = 1\), the class z represents an effective divisor. By the assumption \(l_1',l_2',\alpha _i\) and Open image in new window (\(1 \leqslant i,j \leqslant 3\)) represent irreducible curves. Note that Open image in new window . Let us write \(z = m\alpha _2 + n\alpha _3 + k\beta _2 + l\beta _3 + D\), where D is effective, Open image in new window and \(m,n,k,l > 0\). Then we havewhich implies that \((D, l_1') < 0\) and \((D, l_2') < 0\). Now we write
$$\begin{aligned} z = m\alpha _2 + n\alpha _3 + k\beta _2 + l\beta _3 + sl_1' + tl_2'+D' \end{aligned}$$
where \(D'\) is effective, Open image in new window and \(m,n,k,l,s,t \geqslant 1\). But this is impossible: \(2\leqslant s+t \leqslant (z, h) = 1\). \(\square \)

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 pull-back 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.

Consider Open image in new window . Note that \((h'\!, h')>0\) and \((h'\!, h) > 0\). We can write any effective divisor aswhere \(a_i, b_i, c\in \mathbb Z\), where \(0\leqslant i\leqslant 4\). Let \(k_i= a_0/2-a_i\), \(l_i= b_0/2-b_i\) where \(1\leqslant i\leqslant 4\). It follows thatLet \(D\in \Delta (S)\) as in (6). Then Open image in new window implies thatFirst note that when \(a_0+b_0=(D,h)=0\), then (8) gives that either Open image in new window and Open image in new window or \(c=0\), and all coefficients in Open image in new window but one equal 0 and Open image in new window . In particular, Open image in new window . If \(D\in \Delta ^+(S)\cap \langle h \rangle ^{\perp }_M\), then \((D,h)=a_0+b_0=0\) which in turn implies that Open image in new window .
Now suppose that \(D\in \Delta ^+(S)\) and \((h,D)>0\). Then (7) implies that \(a_0+b_0>0\) and (8) gives \(c\geqslant 0\). Then, by the arithmetic-geometric mean inequality and (8), we getwhere the latter inequality follows from observing that the first summand is positive andHence \((h'\!,D)>0\) for all \(D\in \Delta ^+(S)\). Moreover, if \(D \subset S\) is rational and \( D \not \in \Delta ^+(S)\), then \(\pi _*(D)\ne 0\) and Open image in new window . Hence, by [15, Corollary 8.1.7], \(h'\) is ample.

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

The argument in [16, Corollary 6.19] works here. Alternatively, let \(L_1\) and \(L_2\) be given by linear forms \(l_1\) and \(l_2\), respectively, and consider the elliptic fibration Open image in new window defined by the function Open image in new window . If \((C, L_1, L_2)\) is sufficiently general, then the pencil of lines generated by \(L_1\) and \(L_2\) only consists of lines intersecting C normally or lines tangent to C at a point. As a result, the elliptic fibration contains two reducible singular fibers of type \(I_0^*\) (i.e. with 5 components) and 12 singular fibers of type \(I_1\) (i.e. with one nodal component), where we follow Kodaira’s notation as in [4, Section V.7], [15, Section 11.1.3]. Note that the fibration admits a 2-section \(\gamma \). Consider the associated Jacobian fibration Open image in new window (see for example [15, Section 11.4]). By the Shioda–Tate formula [15, Corollaries 11.3.4 and 11.4.7], the K3 surface \(S_{(C,L_1,L_2)}\) has Picard number 10 which equals the rank of M. Moreover, we have [15, Section 11, (4.5)]:It is easy to compute that the Gram matrix \(G_M\) has determinant \((-64)\). The proposition then follows from the following standard fact on lattices (which implies Open image in new window ):As Open image in new window and they have the same rank and discriminant, then Open image in new window .\(\square \)

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 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 Open image in new window 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-polarizedK3 surface is a pair \((S, \jmath )\) such that Open image in new window is a primitive lattice embedding. The embedding \(\jmath \) is called the M-polarization of S. We will simply say that S is an M-polarizedK3 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

By [26, 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).
  • 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 \(D_6\).

By [26, Theorem 1.14.4] the lattice M admits a unique primitive embedding into \(\Lambda _{K3}\). The claim on the orthogonal complement T follows from [26, Proposition 1.6.1].\(\square \)

Remark 3.14

Note that both M and T are even indefinite 2-elementary lattices (a lattice L is 2-elementary 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 .

The moduli space of M-polarized K3 surfaces is a quotient Open image in new window for a certain Hermitian symmetric domain Open image in new window of type IV and some arithmetic group \(\Gamma \) (see [8]). Fix the (unique) embedding \(M \hookrightarrow \Lambda _{K3}\) and defineto be one of the two connected components. Note that Open image in new window can also be identified withTo specify the moduli of M-polarized K3 surfaces, one also needs to determine the arithmetic group \(\Gamma \). In the standard situation considered in [8] it is required that the M-polarization is pointwise fixed by the arithmetic group and one takes \(\Gamma \) to be \(O^*(T)\). In our geometric context the choice is different. Specifically, the permutations among \(\alpha _1, \ldots , \alpha _4\) and among \(\beta _1, \ldots , \beta _4\) 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 \(g \in O(L)\) 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.

Proof

The lemma follows from Lemma 3.4 and [26, Theorem 1.14.2].\(\square \)

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.

The automorphism of \(A_M\) induced by the transposition \((\alpha _1\alpha _4)\) between \(\alpha _1\) and \(\alpha _4\) is given by
$$\begin{aligned}&\alpha _2^* \mapsto \alpha _3^* + \gamma ^* \equiv \alpha _1^* + \alpha _2^*,&\alpha _3^* \mapsto \alpha _2^* + \gamma ^* \equiv \alpha _1^* + \alpha _3^*,&\xi ^* \mapsto \xi ^* + \alpha _1^*, \end{aligned}$$
and Open image in new window (\(1 \leqslant j \leqslant 3\)) are invariant by this action. The case of transpositions \((\alpha _2\alpha _4)\) and \((\alpha _3\alpha _4)\) is analogous. As it is well-known, the transpositions generate \(\Sigma _\alpha \) and \(\Sigma _\beta \). It is easy to compute the image of Open image in new window in \(O(q_M)\) using the previous descriptions.
Let \(g_\alpha \in \Sigma _\alpha \) and \(g_\beta \in \Sigma _\beta \). Now we describe how to univocally recover Open image in new window from the induced action \(\overline{g}\) on \(A_M\). In particular, this will show that the composed map Open image in new window is injective. Consider \(\overline{g}(\xi ^*)\). Because Open image in new window , the induced action \(\overline{g}\) sends \(\xi ^*\) to an element \(v^*\) satisfying Open image in new window . By Remark 3.5 such elements are \(\xi ^*\), \(\xi ^* + \alpha _i^*\), Open image in new window and Open image in new window (\(1 \leqslant i,j \leqslant 3\)).
  • 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 ). \(\square \)

By abuse of notation, we also use Open image in new window to denote its image in \(O(q_T) \cong O(q_M)\). There exists a natural exact sequence \(1 \rightarrow \widetilde{O}(T) \rightarrow O(T) \xrightarrow {r_T} O(q_T) \rightarrow 1\) which also induces \(1 \rightarrow O^*(T) \rightarrow O_-(T) \rightarrow O(q_T) \rightarrow 1\). We define \(\Gamma \subset O_{-} (T)\subset O(T)\) as the following extension:By Lemmas 3.16 and 3.17, the group \(\Gamma \) fixes M but may permute the elements \(\alpha _1, \ldots , \alpha _4\) (respectively \(\beta _1, \ldots , \beta _4\)). Note that \(O^*(T)\) is a normal subgroup of \(\Gamma \) and Open image in new window . Also, \(\Gamma \) and O(T) are commensurable and hence \(\Gamma \) is an arithmetic group. There is a natural action of \(\Gamma \) on Open image in new window (see the description of Open image in new window in (10)).

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

Let U be an open subset of Open image in new window 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 \(\Sigma _4\) to be the permutation group of four elements and note that \(\Sigma _\alpha \cong \Sigma _\beta \cong \Sigma _4\). Let \(\widetilde{U}\) be the Open image in new window -cover of U that parametrizes quintuples \((C, L_1, L_2, \sigma _1,\sigma _2)\) where Open image in new window , \(k=1,2\), is a labeling of the intersection points of \(C \cap L_k\). Note that the monodromy group acts as the permutation group \(\Sigma _4\) on the four points of intersection \(C \cap L_k\). By Corollary 3.7, \(\sigma _1\) and \(\sigma _2\) determine an M-polarization \(\jmath \) of the K3 surface \(S_{(C, L_1, L_2)}\). Therefore there is a well-defined mapBy the global Torelli theorem and the surjectivity of the period map for K3 surfaces (see also Proposition 3.9), the map Open image in new window is a birational morphism. The group Open image in new window acts naturally on both \(\widetilde{U}\) and Open image in new window as \(\Gamma \) is an extension of Open image in new window and \(O^*(T)\). Essentially, the actions are induced by the permutation of the labeling of the intersection points \(C \cap L_k\) (\(k=0,1\)). It follows that Open image in new window is Open image in new window -equivariant and descends to the birational map Open image in new window (see also Lemmas 3.16 and 3.17).\(\square \)

3.4 M-polarization for non-generic 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 M-polarization Open image in new window 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. [17, Definition 4.24]) for degree 5 pairs constructed in [17, Section 4.2.3]. A degreedpair (DL) 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 \((C+L_1, L_2)\)).

Let us briefly review the construction of normalized lattice polarization for degree 5 pairs. See [17, Section 4.2.3] for more details. Let (DL) be a degree 5 pair such that Open image in new window has at worst simple singularities. Let \(\overline{S}_{(D,L)}\) be the normal surface obtained as the double cover of Open image in new window branched along B. Let \(S_{(D,L)}\) be the minimal resolution of \(\overline{S}_{(D,L)}\), called the K3 surface associated to (DL). The surface \(S_{(D,L)}\) can also be obtained as the canonical resolution of \(\overline{S}_{(D,L)}\), see [4, Section III.7]. Namely, there exists a commutative diagram:where \(S'\) is obtained by an inductive process. Start with Open image in new window and \(B_{-1} = B = D+ L\). Simultaneously blow up all the singular points of B. Let Open image in new window be the resulting surface and set \(B_0\) to be the strict transform of B together with the exceptional divisors of \(\epsilon _0\) reduced mod 2. Repeat the process for \(S_0\) and \(B_0\) until the resulting divisor \(B_N\) is smooth. Let \(S' = S_N\), \(B' = B_N\) and Open image in new window . Now take the double cover \(\pi ':S_{(D, L)} \rightarrow S'\) branched along the smooth locus \(B'\).
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 Open image in new window satisfying Open image in new window for every \(p \in D \cap L\). As argued in [17, 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: Open image in new window , and the labeling determines the order of these first five blow-ups. Let \(\{p_k\}_{k=0}^4\) be the centers of these blow-ups and \(E_k\) be the exceptional divisors. Note that \(p_k \in S_{k-1}\) (the image of \(p_k\) under the contraction to Open image in new window is a point of intersection \(D \cap L\)) and \(E_k\) is a divisor on \(S_k\) for \(k = 0, \ldots , 4\). We define the following divisors:for \(0 \leqslant k \leqslant 4\). The divisor \(D_k\) on \(S_{(D,L)}\) is Artin’s fundamental cycle (see for example [4, p. 76]) associated to the simple singularity of the curve \(B_{k-1}\) at the point \(p_k\).

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 (CL), 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

For a pair (DL) and the surface \(S_{(D,L)}\) as described above the following statements hold:
  1. (a)

    the polarization class of \(S_{(D,L)}\) is Open image in new window , and

     
  2. (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 .

Given a triple \((C, L_1, L_2)\), one has two associated degree 5 pairs: Open image in new window and Open image in new window , which induce the same K3 surface \(S_{(D,L)}\), constructed as above. Let us fix two labelings Open image in new window and Open image in new window such that Open image in new window . Every degree 5 pair produces six divisors as described above. For the pair \((C+L_2, L_1)\) (respectively \((C+L_1, L_2)\)), we denote the 6 divisors by \(L_1', R_0, \ldots , R_4\) (respectively \(L_2', T_0, \ldots , T_4\)). Note that Open image in new window . We define Open image in new window as follows:
After the first blow-up \(\epsilon _0\), the strict transforms of the two lines \(L_1\) and \(L_2\) are disjoint and hence, we have Open image in new window for \(1 \leqslant i,j \leqslant 4\). Using Lemma 3.19, it is straightforward to verify that \(\jmath \) is a well-defined lattice embedding. The embedding \(\jmath \) also satisfies the following geometric properties and fits well in families (cf. [17, Section 4.2.3], especially the last paragraph on p. 2141).
  • \(\jmath (h)\) 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 \(L_1 \cap L_2\).

To conclude that \(\jmath \) is an M-polarization we also need the following lemma.

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 M-polarized 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 M-polarized 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 well-defined 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 M-polarization 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 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 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

We apply [17, Proposition 4.31] (see also [17, Lemmas 4.27, 4.28, 4.30]). The idea is to consider the primitive sublattices \(M_1\) and \(M_2\) of M generated by \(l_1', \gamma , \alpha _1, \ldots , \alpha _4\) and \(l_2', \gamma , \beta _1, \ldots , \beta _4\) respectively. Both of the sublattices have the following Gram matrix
$$\begin{aligned} \left( {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} -2 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} -2 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} -2 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} -2 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} -2 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} -2 \\ \end{array} } \right) _. \end{aligned}$$
Hence they are isomorphic to the lattice considered in [17, Notation 4.11] for degree 5 pairs. In particular, S is both \(M_1\)-polarized and \(M_2\)-polarized. Indeed, recall thatwhich coincides with Laza’s lattice (for both \(M_1\) and \(M_2\)). Using [17, Proposition 4.31], we find two degree 5 pairs \((D_1, L_1)\) and \((D_2, L_2)\) (where, a priori, \(D_1\) and \(D_2\) may be irreducible) such that \(D_1 + L_1 = D_2 + L_2\) has at worst simple singularities. The two morphisms Open image in new window associated to each degree 5 pair are both defined by \(\jmath (h)\), and hence they are the same. Because \(L_1\) and \(L_2\) are both contained in the branch locus of the map Open image in new window , \(D_1=L_2+C\) and \(D_2=L_1+C\), where C is a quartic plane curve such that \(C+L_1+L_2\) has at worst simple singularities. \(\square \)

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 lattice-polarized K3 surfaces (see [8, Theorem 3.1]) there exists an M-polarized 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

Consider the compact space Open image in new window and Open image in new window , 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 Open image in new window —for an appropriately chosen arithmetic group \(\Gamma '\)—and show that Open image in new window is an isomorphism. We use the same approach taken to define Open image in new window (Propositions 3.18 and 3.21) and to prove Theorem 3.23. The modification one needs to do is to choose a different arithmetic group \(\Gamma '\). We follow the same notation as in the previous subsections, especially regarding the description of the subgroup Open image in new window in Lemma 3.17. Consider the subgroup Open image in new window where the factor \(\mathbb {Z}/2\mathbb {Z}\) corresponds to the swap of \(\alpha _i\)’s and \(\beta _i\)’s for \(0 \leqslant i \leqslant 4\) (the induced action on \(A_M\) exchanges \(\alpha _i^*\) with \(\beta _i^*\) for \(1 \leqslant i \leqslant 3\) and fixes \(\gamma ^*\) and \(\xi ^*\)). As in the proof of Lemma 3.17, we can verify that the composition Open image in new window is injective. Now we define \(\Gamma '\) to be the following extension:For Open image in new window (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 Open image in new window branched along \(C+L+L'\). Because \(S_{(C,L,L')}\) is polarized by the lattice M, the period corresponds to a point in Open image in new window . The lattice polarization depends on the labeling of L and \(L'\), the ordering of \(C \cap L\) and the ordering of \(C \cap L'\), and thus is compatible with the action of Open image in new window . Therefore we have a well-defined period map Open image in new window (see also Propositions 3.18 and 3.21). Moreover, the same argument in Proposition 3.21 and Theorem 3.23 allows us to prove that the period map Open image in new window is an isomorphism.

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

We apply a general framework of comparing GIT compactifications to certain compactifications of the period domain developed by Looijenga. Specifically, by [21, Theorem 7.6] an isomorphism Open image in new window (typically coming from a period map) between a geometric quotient Open image in new window and a complement of an arithmetic hyperplane arrangement Open image in new window in a type IV domain \(\Omega \) extends to an isomorphism Open image in new window between the GIT compactification Open image in new window and the Looijenga compactification \(\widetilde{\Omega /\Lambda }\) associated to Open image in new window if their polarizations agree and Open image in new window . We have Open image in new window . The hyperplane arrangement is empty and the associated Looijenga compactification is the Baily–Borel compactification Open image in new window . Moreover, by Corollary 2.16 and Lemma 2.7, we haveand 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 [21, Section 8]). Hence, by [21, Theorem 7.6] Open image in new window is an isomorphism for polarized varieties. \(\square \)

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 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 \(\Gamma '\)-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 0-dimensional components of Open image in new window using [29, Proposition 4.1.3]. The 0-dimensional boundary components are in one-to-one 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 0-dimensional boundary components.

Remark 3.26

Similarly, one can show that Open image in new window has three 0-dimensional boundary components (compare Lemma 2.15).

Remark 3.27

As discussed in [17, Section 4.4.1], one important invariant for the \(O_-(T)\)-equivalence class of isotropic sublattices E of T is the isomorphism classes of Open image in new window (and we shall use it to label E). Let us compute the isomorphism classes of Open image in new window (where \(\mathbb {Z}v\) is a primitive isotropic rank 1 sublattice of T). Observe that Open image in new window . One could compute the Gram matrix of Open image in new window explicitly. Alternatively, we consider Open image in new window (cf. [17, Section 4.4.1]) which is an isotropic subgroup of \(A_T \cong (\mathbb {Z}/2\mathbb {Z})^6\) and the discriminant group \(A_{v^{\perp }/\mathbb {Z}v} \cong H_v^{\perp }/H_v\). In our case, \(H_v\) equals either 0 or \(\mathbb {Z}/2\mathbb {Z}\). The lattice Open image in new window is an even hyperbolic (N.B. the signature is (1, 9)) 2-elementary lattice. By a direct computation we get the following (see also [26, Theorem 3.6.2]).

To determine the 1-dimensional 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 \(n-1\). 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 \(\Sigma \) (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.

In our case, a straightforward application of Vinberg’s algorithm allows us to compute the isomorphism classes of Open image in new window .
By Theorem 3.24 and Corollary 2.16 we conclude that the Baily–Borel compactification Open image in new window consists of five 1-dimensional boundary components labeled by Open image in new window , Open image in new window , Open image in new window , Open image in new window and \(D_8\) respectively.

Remark 3.28

Notes

Acknowledgements

We thank Radu Laza and Gregory Pearlstein for useful discussions.

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Authors and Affiliations

  • Patricio Gallardo
    • 1
  • Jesus Martinez-Garcia
    • 2
    Email author
  • Zheng Zhang
    • 3
  1. 1.Department of MathematicsWashington University in St. LouisSt. LouisUSA
  2. 2.Department of Mathematical SciencesUniversity of BathBathUK
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA

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