On the mixed-twist construction and monodromy of associated Picard-Fuchs systems

We begin by recalling some basic notions of elliptic fibrations and the associated Weierstrass models. Let $X$ and $S$ be normal complex algebraic varieties and $\pi:X\to S$ an elliptic fibration, that is, $\pi$ is proper surjective morphism with connected fibers such that the general fiber is a nonsingular elliptic curve. Moreover, we assume that $\pi$ is smooth over an open subset $S_{0}\subset S$, whose complement in $S$ is a divisor with at worst normal crossings. Thus, the local system $H^{i}_{0}:=R^{i}\pi_{*}\underline{\mathbb{Z}}_{X}{|}_{S_{0}}$ forms a variation of Hodge structure over $S_{0}$.

Elliptic fibrations possess the following canonical bundle formula: on $S$, the fundamental line bundle denoted $\mathcal{L}:=(R^{1}\pi_{*}\mathcal{O}_{X})^{-1}$ and the canonical bundles $\boldsymbol{\omega}_{X}:=\wedge^{\mathrm{top}}\,T^{*(1,0)}X$, $\boldsymbol{\omega}_{S}:=\wedge^{\mathrm{top}}\,T^{*(1,0)}S$ are related by

where $D$ is a certain effective divisor on $X$ depending only on divisors on $S$ over which $\pi$ has multiple fibers, and divisors on $X$ giving $(-1)$-curves of $\pi$. When $\pi:X\to S$ is a Jacobian elliptic fibration, that is, when there is a section $\sigma:S\to X$, the case of multiple fibers is prevented. We may avoid the presence of $(-1)$-curves in the following way: For $X$ an elliptic surface, we assume that the fibration is relatively minimal, meaning that there are no $(-1)$-curves in the fibers of $\pi$. When $X$ is an elliptic threefold, we additionally assume that no contraction of a surface is compatible with the fibration.

Assuming these minimality constraints, we have $D=0$, thus the canonical bundle formula (2.1) simplifies to $\boldsymbol{\omega}_{X}\cong\pi^{*}(\boldsymbol{\omega}_{S}\otimes\mathcal{L})$. In particular, for $\mathcal{L}\cong\boldsymbol{\omega}_{S}^{-1}$ we obtain $\boldsymbol{\omega}_{X}\cong\mathcal{O}_{X}$. Recall that $X$ is a Calabi-Yau manifold if $\boldsymbol{\omega}_{X}\cong\mathcal{O}_{X}$ and $h^{i}(X,\mathcal{O}_{X})=0$ for $0<i<n=\dim(X)$. In this article we will be concerned with Jacobian elliptic fibrations on Calabi-Yau manifolds. It is well known that for $X$ an elliptic Calabi-Yau threefold, the base surface can have at worst log-terminal orbifold singularities. We will take the base surface $S$ to be a Hirzebruch surface $\mathbb{F}_{k}$ (or its blowup).

It is well known that Jacobian elliptic fibrations admit Weierstrass models, i.e., given a Jacobian elliptic fibration $\pi:X\to S$ with section $\sigma:S\to X$, there is a complex algebraic variety $W$ together with a proper, flat, surjective morphism $\hat{\pi}:W\to S$ with canonical section $\hat{\sigma}:S\to W$ whose fibers are irreducible cubic plane curves, together with a birational map $X\dashrightarrow W$ compatible with the sections $\sigma$ and $\hat{\sigma}$; see [55]. The map from $X$ to $W$ blows down all components of the fibers that do not intersect the image $\sigma(S)$. If $\pi:X\to S$ is relatively minimal, the inverse map $W\dashrightarrow X$ is a resolution of the singularities of $W$.

A Weierstrass model is constructed as follows: given a line bundle $\mathcal{L}\to S$, and sections $g_{2}$, $g_{3}$ of $\mathcal{L}^{4}$, $\mathcal{L}^{6}$ such that the discriminant $\Delta=g_{2}^{3}-27g_{3}^{2}$ as a section of $\mathcal{L}^{12}$ does not vanish, define a $\mathbb{P}^{2}$-bundle $p:\mathbf{P}\to S$ as $\mathbf{P}:=\mathbb{P}\left(\mathcal{O}_{S}\oplus\mathcal{L}^{2}\oplus\mathcal{L}^{3}\right)$ with $p$ the natural projection. Moreover, let $\mathcal{O}_{\mathbf{P}}(1)$ be the tautological line bundle. Denoting $x,y$ and $z$ as the sections of $\mathcal{O}_{\mathbf{P}}(1)\otimes\mathcal{L}^{2}$, $\mathcal{O}_{P}(1)\otimes\mathcal{L}^{3}$ and $\mathcal{O}_{\mathbf{P}}(1)$ that correspond to the natural injections of $\mathcal{L}^{2}$, $\mathcal{L}^{3}$ and $\mathcal{O}_{S}$ into $\pi_{*}\mathcal{O}_{\mathbf{P}}(1)=\mathcal{O}_{S}\oplus\mathcal{L}^{2}\oplus\mathcal{L}^{3}$, the Weierstrass model $W$ from above is given by the the sub-variety of $\mathbf{P}$ defined by the equation

The canonical section $\sigma:S\to W$ is given by the point $[x:y:z]=[0:1:0]$ in each fiber, such that $\Sigma:=\sigma(S)\subset W$ is a Cartier divisor whose normal bundle is isomorphic to the fundamental line bundle $\mathcal{L}$ via $p_{*}\mathcal{O}_{\mathbf{P}}(-\Sigma)\cong\mathcal{L}$. It follows that $W$ inherits the properties of normality and Gorenstein if $S$ possesses these. Thus, the canonical bundle formula  (2.1) reduces to

The Jacobian elliptic fibration $p:W\to S$ then has a Calabi-Yau total space if $\mathcal{L}\cong\boldsymbol{\omega}_{S}^{-1}=\mathcal{O}_{S}(-K_{S})$ (misusing notation slightly to denote the projection map $p$ the as the projection from the ambient $\mathbb{P}^{2}$-bundle).

For a Jacobian elliptic fibration $X$ the canonical bundle $\boldsymbol{\omega}_{X}$ is determined by the discriminant $\Delta=g_{2}^{3}-27g_{3}^{2}$. For example, if $\pi:X\to S$ is a Jacobian elliptic fibration for a smooth algebraic surface $X$ and $S=\mathbb{P}^{1}$ with homogeneous coordinates $[t:s]$, then $X$ is a rational elliptic surface if the $\Delta$ is a homogeneous polynomial of degree 12 (meaning that $\mathcal{L}=\mathcal{O}(1)$), and $X$ is a K3 surface when $\Delta$ is a homogeneous polynomial of degree 24 (meaning that $\mathcal{L}=\mathcal{O}(2)$); these results follow readily from adjunction and Noether’s formula. The nature of the singular fibers and their effect on the canonical bundle was established by the seminal work of Kodaira  [42, 43, 41].

Of particular interest in this article are multi-parameter families of elliptic Calabi-Yau $n$-folds over a base $B$, a quasi-projective variety of dimension $r$, denoted by $\pi:~{}X\to B$. Hence, each $X_{p}=\pi^{-1}(p)$ is a compact, complex $n$-fold with trivial canonical bundle. Moreover, each $X_{p}$ is elliptically fibered with section over a fixed normal variety $S$. This means that we have a multi-parameter family of minimal Weierstrass models $p_{b}:W_{b}\to S$ representing a family of Jacobian elliptic fibrations $\pi_{b}:X_{b}\to S$. We denote the collective family of Weierstrass models as $p:W\to B$.

Working within affine coordinates for $B$ and $S$ we set $u=(u_{1},\dots,u_{n-1})\in\mathbb{C}^{n-1}\subset S$ and $b=(b_{1},\dots,b_{r})\in\mathbb{C}^{r}\subset B$. We then may write the Weierstrass model $W_{b}$ in the form

where for each fiber we have chosen the affine chart of $W_{b}$ given by $z=1$ in Equation (2.2).

Part of the utility of a Weierstrass model is the explicit construction of the holomorphic $n$-form on each $X_{b}$, up to fiberwise scale, allowing for the detailed study of the Picard-Fuchs operators underlying a variation of Hodge structure. In fact, consider the holomorphic sub-system $H\to B$ of the local system $V=R^{n}\pi_{*}\underline{\mathbb{C}}_{X}\to B$, whose fibers are given as the line $H^{0}(\omega_{X_{b}})\subset H^{n}(X_{b},\mathbb{C})$. Here, $\underline{\mathbb{C}}\to X$ is the constant sheaf whose stalks are $\mathbb{C}$. Griffiths showed [28, 27, 29, 30] that $\mathcal{V}=V\otimes_{\mathbb{C}}\mathcal{O}_{B}$ is a vector bundle carrying a canonical flat connection $\nabla$, the Gauss-Manin connection. A meromorphic section of $\mathcal{H}=H\otimes_{\mathbb{C}}\mathcal{O}_{B}\subset\mathcal{V}$ is given fiberwise by the holomorphic $n$-form $\eta_{b}\in H^{0}(\boldsymbol{\omega}_{X_{b}})\subset H^{n}(X_{b},\mathbb{C})$

where we denote the collective section as $\eta\in\Gamma(\mathcal{V},B)$. It is natural to consider local parallel sections of the dual bundle $\mathcal{H}^{*}=H^{*}\otimes_{\mathbb{C}}\mathcal{O}_{B}$, where $H^{*}$ is the local system dual to $H$; these are generated by transcendental cycles $\Sigma_{b}\in H_{n}(X_{b},\mathbb{R})$ that vary continuously with $b\in B$, writing the collective section as $\Sigma\in\Gamma(\mathcal{V}^{*},B)$. The sections are covariantly constant since the local system $V=R^{n}\pi_{*}\underline{\mathbb{C}}_{X}$ is locally topologically trivial, and thus local sections of the dual $V^{*}$ are as well. Utilizing the natural fiberwise de Rham pairing

we obtain the period sheaf $\Pi\to B$, whose stalks are given by the local analytic function $b\mapsto\omega(b)=\langle\Sigma_{b},\eta_{b}\rangle$. The function $\omega(b)$ is called a period integral (over $\Sigma_{b}$) and satisfies a system of coupled linear PDEs in the variables $b_{1},\dots,b_{r}$ – the so called Picard-Fuchs system - whose rank is that of the period sheaf $\Pi\to B$, or the number of linearly independent period integrals of the family.

Given the affine local coordinates $(b_{1},\dots,b_{r})\in\mathbb{C}^{r}\subset B$, fix the meromorphic vector fields $\partial_{j}=\partial/\partial b_{j}$ for $j=1,\dots,r$. Then each $\partial_{j}$ induces a covariant derivative operator $\nabla_{\partial_{j}}$ on $\mathcal{V}$. Since $\nabla$ is flat, the curvature tensor $\Omega=\Omega_{\nabla}$ vanishes, and hence, for all meromorphic vector fields $U,V$ on $B$ we have

Substituting in the commuting coordinate vector fields $\partial_{i},\partial_{j}$, we conclude

This integrability condition is crucial in obtaining a system of PDEs from the Gauss-Manin connection. Since $\mathcal{V}$ has rank $m=\dim H^{n}(X_{b},\mathbb{C})$, each sequence of parallel sections $\nabla^{i_{1}}_{\partial_{k_{1}}}\cdots\nabla^{i_{\hat{m}}}_{\partial_{k_{r}}}\eta$, for $i_{1}+\cdots+i_{\hat{m}}=0,1,\dots,\hat{m}$ and $1\leq k_{1},\dots,k_{r}\leq r$ form the linear dependence relations

for some integer $0<\hat{m}\leq m$, where $a^{k_{1}\cdots k_{r}}_{i_{1}\cdots i_{\hat{m}}}(b)$ are meromorphic. Here, it is understood that $\nabla^{0}=\mathrm{id}$. As $\nabla$ annihilates the transcendental cycle $\Sigma$ and is compatible with the pairing $\langle\Sigma,\eta\rangle$, we may “differentiate under the integral sign” to obtain

It follows that the period integral $\omega(b)$ satisfies the system of linear PDEs of rank $\mathsf{r}\geq 1$, given by

Equation (2.6) is the Picard-Fuchs system of the multi-parameter family $\pi:X\to B$ of Calabi-Yau $n$-folds. The resulting system is then known to be a linear Fuchsian system, i.e., the system with at worst regular singularities. This is due to analytical results of Griffiths [29] and Deligne [17] who utilized Hironaka’s resolution of singularities [32, 33] to estimate the growth of solutions of the system.

The rank $\mathsf{r}$ and order $\hat{m}$ of the system depends on the parameter space $B$ and algebro-geometric data of the generic fiber $X_{b}$. For example, let $\pi:X\to B$ be a family of Jacobian elliptic K3 surfaces which is polarized by a lattice¹¹1For the definition of lattice polarized K3 surface, see §3.2. $L$ of rank $\rho\leq 18$ such that $B$ defines an $n=20-\rho$ dimensional family of L-polarized K3 surfaces. By results due to Dolgachev [19], there is a coarse moduli space $\mathcal{M}_{L}$ of all lattice polarized K3 surfaces of dimension $n$; in this case, we are requiring that $B$ be a top dimensional family of $L$-polarized K3 surfaces. It then follows from the general program of Sasaki and Yoshida [65] on orbifold uniformizing differential equations that the Picard-Fuchs system (2.6) is a linear system of order $\hat{m}=2$ and rank $\mathsf{r}=n+2$ in $n$ variables, the latter coming from the local coordinates in $B$. Naturally, there are sub-loci of such parameter spaces $B$ where the lattice polarization extends to higher Picard rank and the rank of the Picard-Fuchs system drops accordingly. This behavior was studied, for example, by Doran et al. in [21], and coined the differential rank-jump property therein. In the sequel, we will analyze it by studying corresponding Weierstrass model $p:W\to B$. Moreover, we will see that the Picard-Fuchs system can be explicitly computed from the geometry of the elliptic fibrations and the presentation of the associated period integrals as generalized Euler integrals using GKZ systems  [24].

It is commonplace in the literature to study the Picard-Fuchs equations of one parameter families of Calabi-Yau $n$-folds; in this case, the base $B$ is a punctured complex plane with local affine coordinate $t\in\mathbb{C}\subset B$, and an analogous construction leads to a regular Fuchsian ODE of order $\leq m$ with $m=\dim H^{n}(X_{t},\mathbb{C})$ for the general fiber $X_{t}$. In the construction of Doran and Malmendier [22], this is the central focus, with $B=\mathbb{P}^{1}-\{0,1,\infty\}$ and $B=\mathbb{P}^{1}-\{0,1,p,\infty\}$. We will show that the restriction of the multi-parameter Picard-Fuchs system (2.6) above leads to the Picard-Fuchs ODE operators and families of lattice polarized K3 surfaces of Picard rank $\rho=19$, for example the mirror partners of the classic deformed Fermat quartic K3.

We first recall the generalized functional invariant of the mixed-twist construction studied by Doran and Malmendier [22], first introduced by Doran [20]. A generalized functional invariant is a triple $(i,j,\alpha)$ with $i,j\in\mathbb{N}$ and $\alpha\in\left\{\frac{1}{2},1\right\}$ such that $1\leq i,j\leq 6$. To this end, the generalized functional invariant encodes a 1-parameter family of degree $i+j$ covering maps $\mathbb{P}^{1}\to\mathbb{P}^{1}$, which is totally ramified over $0$, ramified to degrees $i$ and $j$ over $\infty$, and simply ramified over another point $\tilde{t}$. For homogeneous coordinates $[v_{0}:v_{1}]\in\mathbb{P}^{1}$, this family of maps (parameterized by $\tilde{t}\in\mathbb{P}^{1}-\{0,1,\infty\}$) is given by

for some constant $c_{ij}\in\mathbb{C}^{\times}$. For a family $\pi:X\to B$ with Weierstrass models given by Equation (2.4) with complex $n$-dimensional fibers and a generalized functional invariant $(i,j,\alpha)$ such that

Doran and Malmendier showed that a new family $\tilde{\pi}:\tilde{X}\to B$ can be constructed such that the general fiber $\tilde{X}_{\tilde{t}}=\tilde{\pi}^{-1}(\tilde{t})$ is a compact, complex $(n+1)$-manifold equipped with a Jacobian elliptic fibration over $\mathbb{P}^{1}\times S$. In the coordinate chart $\{[v_{0}:v_{1}],(u_{1},\dots,u_{n-1})\}\in\mathbb{P}^{1}\times S$ the family of Weierstrass models $W_{\tilde{t}}$ is given by

with $c_{ij}=(-1)^{i}i^{i}j^{j}/(i+j)^{i+j}$. The new family is called the twisted family with generalized functional invariant $(i,j,\alpha)$ of $\pi:X\to B$. It follows that conditions (2.8) guarantee that the twisted family is minimal and normal if the original family is. Moreover, they showed that if the Calabi-Yau condition is satisfied for the fibers of the twisted family if it is satisfied for the fibers of the original.

The twisting associated with the generalized functional invariant above is referred to as the pure twist construction; we may extend this notion to that of a mixed twist construction. This means that one combines a pure twist from above with a rational map $B\to B$, thus allowing one to change locations of the singular fibers and ramification data. This was studied in [22, Sec. 8] for linear and quadratic base changes. We may also perform a multi-parameter version of the mixed twist construction for a generalized functional invariant $(i,j,\alpha)=(1,1,1)$. For us, it will be enough to consider the two-parameter family of ramified covering maps given by

such that for $a,b\in\mathbb{P}^{1}-\{0,1,\infty\}$ with $a\not=b$ the map in Equation (2.10) is totally ramified over $a$ and $b$. We will apply the mixed twist construction to certain (families of) rational elliptic surfaces $X\to\mathbb{P}^{1}$. In [22, Sec. 5.5] the authors showed that the twisted family with generalized functional invariant $(1,1,1)$ in this case is birational to a quadratic twist family of $X\to\mathbb{P}^{1}$. We will explain the relationship in more detail and utilize it in the construction of the associated Picard-Fuchs operators in the next section.

A two-parameter family of rational elliptic surfaces $S_{c,d}\to\mathbb{P}^{1}$ is given by the affine Weierstrass model

where $g_{2}(t)$ and $g_{3}(t)$ are the following polynomials of degree four and six, respectively,

where $t$ is the affine coordinate on the base curve. Assuming general parameters $c,d$, Equation (3.1) defines a rational elliptic surface with 6 singular fibers of Kodaira type $I_{2}$ over $t=0,1,\infty,c,c+d,$ and $c/(d-1)$. We have the following:

A quadratic twist applied to a rational elliptic surface can be identified with Doran and Malmendier’s mixed-twist construction with generalized functional invariant $(i,j,\alpha)=(1,1,1)$. The two-parameter family of ramified covering maps in Equation (2.10) is totally ramified over $a,b\in\mathbb{P}^{1}-\{0,1,\infty\}$ with $a\not=b$. We apply the mixed-twist construction to the rational elliptic surface $S_{c,d}$:

A direct computation for the Weierstrass model yields the following:

Equation (3.4) provides a model for the K3 surfaces $\mathbf{X}$ as double covers of the projective plane branched on the union of six lines. In general, we call a K3 surface $\mathcal{X}$ a double-sextic surface if it is the minimal resolution of a double cover of the projective plane $\mathbb{P}^{2}$ branched along the union of six lines, which we denote by $\boldsymbol{\ell}=\{\ell_{1},\dots,\ell_{6}\}$. In weighted homogeneous coordinates $[t_{1}:t_{2}:t_{3}:z]\in\mathbb{P}(1,1,1,3)$ such a double-sextic is given by the equation

where the lines $\ell_{i}=\left\{[t_{1}:t_{2}:t_{3}]\;|\;a_{i1}t_{1}+a_{i2}t_{2}+a_{i3}t_{3}=0\right\}\subset\mathbb{P}^{2}$ for parameters $a_{ij}\in\mathbb{C}$, $i=1,\dots,6$, $j=1,2,3$ are assumed to be general. Let $A=(a_{ij})\in\mathrm{Mat}(3,6;\mathbb{C})$ be the matrix whose entries are the coefficients encoding the six-line configuration $\boldsymbol{\ell}$. Let $M$ be the configuration space of six lines $\boldsymbol{\ell}$ whose minimal resolution is a K3 surface. Then isomorphic K3 surfaces are obtained if we act on elements $A\in M$ by matrices induced from automorphisms of $\mathbb{P}^{2}$ on the left and overall scale changes of each line $\ell_{i}\in\boldsymbol{\ell}$ on the right. Thus, we are led to consider the four-dimensional quotient space

and $\mathcal{M}$ in Equation (3.5) can be identified with the open subspace of $\mathcal{M}_{6}$, given by elements $[A]\in\mathcal{M}_{6}$ of the form

with $(a,b,c,d)\in\mathcal{M}$ and $t_{1}=x,t_{2}=-t,t_{3}=-1$.

The family of double-sextics in Equation (3.6) has been studied in the literature, for example by Matsumoto [51], and Matsumoto et al. [52, 53, 54]. One takeaway from their work is that the family of double sextic K3 surfaces is, in many ways, analogous to the Legendre pencil of elliptic curves which is realized as double covers of $\mathbb{P}^{1}$ branching over four points. More recently, the double-sextic family $\mathcal{X}$ and closely related K3 surfaces have been studied in the context of string dualities [47, 49, 13, 46, 10]. In Clingher et al. [13], the authors showed that four different elliptic fibrations on $\mathcal{X}$ have interpretations in F-theory/heterotic string duality. Similar constructions are relevant to anomaly cancellations [48], studied by the authors of the present article. In [10], the authors classified all Jacobian elliptic fibrations on the Shioda-Inose surface associated with $\mathcal{X}$. Finally, Hosono et al.  in [35, 35] constructed compactifications of $\mathcal{M}_{6}$ from GKZ data and toric geometry, suitable for the study of the Type IIA/Type IIB string duality.

In the following we will use the following standard notations for lattices: $L_{1}\oplus L_{2}$ is orthogonal sum of the two lattices $L_{1}$ and $L_{2}$, $L(\lambda)$ is obtained from the lattice $L$ by multiplication of its form by $\lambda\in\mathbb{Z}$, $\langle R\rangle$ is a lattice with the matrix $R$ in some basis; $A_{n}$, $D_{m}$, and $E_{k}$ are the positive definite root lattices for the corresponding root systems, $H$ is the unique even unimodular hyperbolic rank-two lattice. A lattice $L$ is two-elementary if its discriminant group $A_{L}$ is a two-elementary abelian group, namely $A_{L}\cong(\mathbb{Z}/2\mathbb{Z})^{\ell}$ with $\ell$ being the minimal number of generators of the discriminant group $A_{L}$, also called the length of the lattice $L$. Even, indefinite, two-elementary lattices $L$ are uniquely determined by the rank $\rho$, the length $\ell$, and the parity $\delta$ – which equals $1$ unless the discriminant form $q_{L}(x)$ takes values in $\mathbb{Z}/2\mathbb{Z}\subset\mathbb{Q}/2\mathbb{Z}$ for all $x\in A_{L}$ in which case it is $0$; this is a result by Nikulin [63, Thm. 4.3.2].

Let $\mathbf{X}$ be a smooth algebraic K3 surface over the field of complex numbers. Denote by $\operatorname{NS}(\mathbf{X})$ the Néron-Severi lattice of $\mathbf{X}$. This is known to be an even lattice of signature $(1,\rho_{\mathbf{X}}-1)$, where $p_{\mathbf{X}}$ denotes the Picard number of $\mathbf{X}$, with $1\leq\rho_{\mathbf{X}}\leq 20$. In this context, a lattice polarization [59, 60, 61, 62, 18] on $\mathbf{X}$ is, by definition, a primitive lattice embedding $i\colon L\hookrightarrow\operatorname{NS}(\mathbf{X})$, with $i(L)$ containing a pseudo-ample class, i.e., a numerically effective class of positive self-intersection in the Néron-Severi lattice $\mathrm{NS}(\mathbf{X})$. Here, $L$ is a choice of even lattice of signature $(1,\rho)$, with $1\leq\rho\leq 20$ that admits a primitive embeddings into the K3 lattice $\Lambda_{K3}\cong H^{\oplus 3}\oplus E_{8}(-1)^{\oplus 2}$. Two $L$-polarized K3 surfaces $(\mathbf{X},i)$ and $(\mathbf{X}^{\prime},i^{\prime})$ are said to be isomorphic²²2Our definition of isomorphic lattice polarizations coincides with the one used by Vinberg [72, 73, 74]. It is slightly more general than the one used in [19, Sec. 1]., if there exists an analytic isomorphism $\alpha\colon\mathbf{X}\rightarrow\mathbf{X}^{\prime}$ and a lattice isometry $\beta\in O(L)$, such that $\alpha^{*}\circ i^{\prime}=i\circ\beta$, where $\alpha^{*}$ is the appropriate morphism at cohomology level. In general, $L$-polarized K3 surfaces are classified, up to isomorphism, by a coarse moduli space $\mathcal{M}_{L}$, which is known [19] to be a quasi-projective variety of dimension $20-\rho$. A general $L$-polarized K3 surface $(\mathbf{X},i)$ satisfies $i(L)=\operatorname{NS}(\mathbf{X})$.

We have the following result:

The Picard-Fuchs system for the family can also be determined: