Moduli of K3 surfaces of degree 2 with four rational double points of type $D_{4}$

It is a classical fact that the maximum number of nodes of a quartic surface in $\mathbb{P}^{3}$ is sixteen. Indeed, if $Y$ is a quartic surface with a node $p$, then the projection from $p$ gives a double cover of $\mathbb{P}^{2}$. The branch locus is a sextic in $\mathbb{P}^{2}$, and the set of nodes of $Y$ other than $p$ is bijective with the set of nodes of the sextic. The maximum number of nodes of a sextic is fifteen, which is attained if the sextic is the union of six lines no three of which intersect at one point. The double cover of $\mathbb{P}^{2}$ branched along six lines comes from a quartic surface with sixteen nodes if and only if no three lines intersect at one point and all lines are tangent to a smooth conic.

A quartic surface with sixteen nodes is called a Kummer quartic surface. The moduli space of Kummer quartic surfaces, equipped with some additional combinatorial data (which is equivalent to a level 2 structure on the corresponding principally polarized Abelian surface, which in turn is equivalent to the choice of a total order on the set of six Weierstrass points of the corresponding genus 2 curve), is an open subset of the Igusa quartic. The Igusa quartic, also known as the Castelnuovo–Richmond quartic (see [Dol12]), is the Satake–Baily–Borel compactification of the Siegel modular variety $\mathcal{A}_{2}(2)$ [Igu64]. The Igusa quartic contains fifteen singular lines, and for a smooth point on the Igusa quartic, the corresponding Kummer quartic surface is the intersection of the Igusa quartic and its tangent space at that point, which has one node at that point and fifteen other nodes at the intersection points with singular lines.

In this paper, we classify Gorenstein K3 surfaces of degree 2 with the maximum number of rational double points of type $D_{4}$. Since the signature of the K3 lattice $\boldsymbol{\Lambda}_{\mathrm{K3}}$ is $(3,19)$, this number cannot be more than four, and it is easy to find a Gorenstein K3 surface of degree 2 with four rational double points of type $D_{4}$. Let $P$ be the Picard lattice of the minimal resolution of a very general Gorenstein K3 surface of degree 2 with four rational double points of type $D_{4}$, and $Q$ be the orthogonal complement of $P$ in the K3 lattice $\boldsymbol{\Lambda}_{\mathrm{K3}}$. Instead of the subspace of the moduli space of K3 surfaces of degree 2, we consider the moduli space of K3 surfaces polarized by the lattice $P$ in the sense of Nikulin (see Definition 5.1), which is an orthogonal modular variety for the lattice $Q$. The main result is Theorem 5.2, which describes this modular variety and the corresponding graded ring of automorphic forms. The proof of Theorem 5.2 occupies Sections 6, 7, 8, and 9. In Section 10, we classify K3 surfaces of degree 2 with two rational double points of type $E_{8}$. This gives a proof of a theorem of Igusa [Igu64] describing the graded ring of Siegel modular forms of genus 2 in terms of generators and relations. Note that the relation between lattice polarized K3 surfaces and Siegel modular forms is also discussed in [CD12] using the same lattice and a different model (i.e., a family of quartic surfaces which is related to the family appearing in this paper by a birational map).

A Gorenstein K3 surface of degree 2 is a pair $(Y,\mathcal{L})$ consisting of a Gorenstein projective surface $Y$ and an ample line bundle $\mathcal{L}$ of degree 2 on $Y$ such that the minimal model $\mu\colon\widetilde{Y}\to Y$ is a K3 surface. Set $\widetilde{\mathcal{L}}\coloneqq\mu^{*}\mathcal{L}$. If $(Y,\mathcal{L})$ is a Gorenstein K3 surface of degree 2, then it follows from [May72] (see also [Sha80]) that either

• •

  $|\mathcal{L}|$ is base point free and $\varphi_{L}\colon Y\to|\mathcal{L}|$ is a double cover of $\mathbb{P}^{2}$, or

• •

  the fixed component $s$ of $|\widetilde{\mathcal{L}}|$ is a $(-2)$-curve, and the free part is $2f$ where $f$ is a curve of self-intersection number $0$.

In the former case, the branch locus must be a sextic. In the latter case, the morphism $\varphi_{f}\colon\widetilde{Y}\to|f|\cong\mathbb{P}^{1}$ is an elliptic fibration with a section $s$. In both cases, the surface can be modeled by a complete intersection of bidegree $(2,6)$ in $\mathbb{P}(1,1,1,2,3)$ of the form

where $a_{0}$, $g_{2}$, and $f_{6}$ are homogeneous polynomials of degrees 0, 2, and 6 (the linear term in $x_{5}$ is eliminated by a coordinate transformation). If $a_{0}\neq 0$, then $Y$ is the double cover of $\mathbb{P}^{2}_{[x_{1}:x_{2}:x_{3}]}$ branched along the sextic curve defined by $f_{6}(x_{1},x_{2},x_{3},g_{2}(x_{1},x_{2},x_{3})/a_{0})$. If $a_{0}=0$, then the projection to $\mathbb{P}^{2}_{[x_{1}:x_{2}:x_{3}]}$ gives a rational map to the conic defined by $g_{2}=0$, which becomes regular by blowing up the point $\left[0:0:0:1:\sqrt{f_{6}(0,0,0,1)}\right]$. The exceptional divisor of the blow-up is the section $s$.

Let

be a set of four point on $\mathbb{P}^{2}$ in general position, which is unique up to the action of $\operatorname{Aut}\mathbb{P}^{2}\cong\operatorname{PGL}_{3}(\mathbb{C})$.

Let

be the space of homogeneous polynomials of degree 2 vanishing at $B$. The projective line $\mathbb{P}(Q_{B})$ can be identified with the pencil $|2H-B|$ of conics passing through $B$. The pencil contains three singular members

each of which is the union of two lines.

We use the following notations in this paper:

• •

  For a pair $(m,n)$ of natural numbers, the lattice $\mathbf{I}_{m,n}$ is the free abelian group of rank $m+n$ equipped with the inner product

  (4.1)

  $\displaystyle((a_{i})_{i},(b_{i})_{i})=\sum_{i=1}^{m}a_{i}b_{i}-\sum_{i=m+1}^{n+m}a_{i}b_{i}.$

• •

  For a pair $(m,n)$ of positive integers satisfying $m\equiv n\mod 8$, the lattice $\mathbf{I}\!\mathbf{I}_{m,n}$ is the unique even unimodular indefinite lattice of signature $(m,n)$.

• •

  The root lattice of type $X_{k}$ is denoted by $\mathbf{X}_{k}$.

• •

  For an integer $k$, the lattice $\left\langle k\right\rangle$ is generated by a single element with self-intersection $k$.

• •

  The lattice $\mathbf{U}$ is the even unimodular hyperbolic plane $\mathbf{I}\!\mathbf{I}_{1,1}$.

• •

  Given a lattice $L$, the lattice $L(k)$ has the same underlying free abelian group as $L$ and the inner product which is $k$ times that of $L$.

• •

  The K3 lattice $\boldsymbol{\Lambda}_{\mathrm{K3}}\cong\mathbf{I}\!\mathbf{I}_{3,19}\cong\mathbf{E}_{8}^{\bot 2}\bot\mathbf{U}^{\bot 3}$ is the second integral cohomology group of a K3 surface equipped with the intersection form.

• •

  Given a lattice $L$, the set of $(-2)$-elements is denoted by $\Delta(L)\coloneqq\left\{\delta\in L\mathrel{}\middle|\mathrel{}(\delta,\delta)=-2\right\}.$

The discriminant group of a non-degenerate lattice $L$ is the quotient $D_{L}$ of the dual free abelian group $L^{\vee}\coloneqq\operatorname{Hom}(L,\mathbb{Z})$ by the image of the natural map $L\hookrightarrow L^{\vee},$ $l\mapsto(l,-).$ The $\ell$-invariant $\ell_{L}$ is defined as the minimal number of generators of the finite abelian group $D_{L}$. A non-degenerate lattice $L$ is said to be 2-elementary if $D_{L}\cong(\mathbb{Z}/2\mathbb{Z})^{\ell_{L}}$. If the lattice $L$ is even, then the discriminant form $q_{L}$ is defined as the quadratic form on the discriminant group sending $x\in D_{L}$ to $[(x,x)]\in\mathbb{Q}/2\mathbb{Z}.$ The parity invariant of $L$ is defined by

By Nikulin [Nik79b] (cf. also [Dol83, Theorem 1.5.2]), the isomorphism class of an even indefinite 2-elementary lattice is determined uniquely by the parity invariant, the $\ell$-invariant, and the signature.

The stable orthogonal group $\widetilde{\operatorname{O}}(L)$ is defined as the kernel of the natural projection from $\operatorname{O}(L)$ to the finite orthogonal group $\operatorname{O}(D_{L},q_{L}).$

Let $P$ be the Picard lattice of the minimal resolution of a very general Gorenstein K3 surface of degree 2 with four rational double points of type $D_{4}$, and $Q$ be the orthogonal complement of $P$ in the K3 lattice $\boldsymbol{\Lambda}_{\mathrm{K3}}$.

Let $\widetilde{\mathcal{D}}$ be a connected component of $\left\{\Omega\in Q\otimes\mathbb{C}\mathrel{}\middle|\mathrel{}(\Omega,\Omega)=0,\ (\Omega,{\overline{\Omega}})>0\right\}$ and $\mathcal{D}$ be its image in $\mathbb{P}(Q\otimes\mathbb{C})$, which is a symmetric domain of type IV. The subgroup of the orthogonal group $\operatorname{O}(Q)$ of index two preserving the connected component $\widetilde{\mathcal{D}}$ will be denoted by $\operatorname{O}^{+}(Q)$. Given a subgroup $\Gamma$ of $\operatorname{O}^{+}(Q)$, an automorphic form of weight $k\in\mathbb{Z}$ is a holomorphic function $f\colon\widetilde{\mathcal{D}}\to\mathbb{C}$ satisfying

1. (i)

   $f(\alpha z)=\alpha^{-k}f(z)$ for any $\alpha\in\mathbb{C}^{\times}$, and

2. (ii)

   $f(\gamma z)=f(z)$ for any $\gamma\in\Gamma$.

The space $A_{k}(\Gamma)$ of automorphic forms of weight $k$ constitute the graded ring $A(\Gamma)\coloneqq\bigoplus_{k=0}^{\infty}A_{k}(\Gamma),$ whose projective spectrum ${\overline{\mathcal{M}}}(\Gamma)\coloneqq\operatorname{Proj}A(\Gamma)$ is the Satake–Baily–Borel compactification of the modular variety $\mathcal{M}(\Gamma)\coloneqq\mathcal{D}/\Gamma.$

We can choose a subset $\Delta(P)^{+}$ of $\Delta(P)$ satisfying

1. (1)

   $\Delta(P)=\Delta(P)^{+}\coprod(-\Delta(P)^{+})$ and

2. (2)

   if $\delta_{1},\delta_{2}\in\Delta(P)^{+}$ and $\delta_{1}+\delta_{2}\in\Delta(P)$, then $\delta_{1}+\delta_{2}\in\Delta(P)^{+}$.

The choice of $\Delta(P)^{+}$ is unique up to the action of $O(P)$. Define

For a projective variety $Y$, we set

where $C(Y)^{\circ}\subset H^{1,1}(Y)\cap H^{2}(Y;\mathbb{R})$ is the Kähler cone of $Y$ and $C(Y)$ is its closure.

As explained in [Dol96, Remark 3.4], the coarse moduli space $\mathcal{M}$ of pseudo-ample $P$-polarized K3 surfaces is isomorphic to $\mathcal{M}\left(\widetilde{\operatorname{O}}^{+}(Q)\right)$ where $\widetilde{\operatorname{O}}^{+}(Q)\coloneqq\operatorname{O}^{+}(Q)\cap\widetilde{\operatorname{O}}(Q)$. The coarse moduli space of ample $P$-polarized K3 surfaces is the complement $\mathcal{M}\setminus\mathcal{H}$ of the union $\mathcal{H}=\bigcup_{\delta\in\Delta^{+}(Q)}\mathcal{H}_{\delta}$ of the divisors $\mathcal{H}_{\delta}\coloneqq\left.\widetilde{\operatorname{O}}^{+}(Q)\,{\delta}^{\bot}\middle/\,\widetilde{\operatorname{O}}^{+}(Q)\right.$ where ${\delta}^{\bot}\coloneqq\{[\Omega]\in\mathcal{D}\mid(\Omega,\delta)=0\}$ is the reflection hyperplane.

The main result of this paper is the following:

Let $\mathcal{H}_{10}$ be the subvariety of the moduli space $\mathcal{M}$ of pseudo-ample $P$-polarized K3 surfaces $(Y,j)$ parametrizing those containing a $(-2)$-curve $s$ and an elliptic curve $f$ such that $\mathcal{L}=\mathcal{O}_{Y}(s+2f)$. The notation $\mathcal{H}_{10}$ is in anticipation of the divisors $\mathcal{H}_{i}$ for $i=1,\ldots,9$ appearing in Section 7 below. For such $(Y,j)$, the sublattice $j\left(\left(\mathbf{D}_{4}\right)^{\bot 4}\right)$ of $\operatorname{Pic}Y$ must be generated by irreducible components of fibers, so that $Y$ must have at least four singular fibers of Kodaira type $I_{0}^{*}$. This implies that $Y$ is a Kummer surface of product type, i.e., the minimal model of the quotient $(E\times E^{\prime})/(-1)\times(-1)$ of the product of a pair of elliptic curves, and the elliptic fibration comes from the projection to $E/(-1)\cong\mathbb{P}^{1}$. Such a surface is modeled in (2.1) by $a_{0}=0$, $g_{2}\in|2H-B|\setminus\{D_{i}\}_{i=1}^{3}$, and

where $(\lambda_{1},\lambda_{2},\lambda_{3})$ is a configuration of three ordered points on $\mathbb{A}^{1}$ (the order determines and is determined by the $P$-polarization, and one can set $(\lambda_{1},\lambda_{2},\lambda_{3})=(0,1,\lambda)$ by translation and rescaling of $x_{4}$), and $g_{2}^{\prime}$ is any element of $Q_{B}$ linearly independent from $g_{2}$. The choice of $g_{2}^{\prime}$ is irrelevant since $Q_{B}/\mathbb{C}\cdot g_{2}$ is one-dimensional, and one can choose the defining equation $d_{1}$ of $D_{1}$ as $g_{2}^{\prime}$. The transcendental lattice $T_{Y}$ of $Y$ is isometric to $T_{E\times E^{\prime}}(2)$, which is contained in $\mathbf{U}(2)^{\bot 2}$. The class of the $(-2)$-curve $s$ in $Q\subset H^{2}(Y;\mathbb{Z})$ is a $(-2)$-element, which we write as $v_{10}$. The orthogonal lattice of $v_{10}$ in $Q$ is $\mathbf{U}(2)^{\bot 2}$, and $\mathcal{H}_{10}$ is the divisor associated with $v_{10}$. The divisor $\mathcal{H}_{10}$ is an orthogonal modular variety for the lattice $\mathbf{U}(2)^{\bot 2}$, which is the product of two copies of the modular curve $X(2)\cong\mathbb{P}^{1}\setminus\{0,1,\infty\}$ of level 2.

Recall from the proof of Lemma 4.1 that $\ell$ is the generator of $\left\langle 2\right\rangle$ in $(\mathbf{D}_{4})^{\bot 4}\bot\left\langle 2\right\rangle\subset P$. Let $(\widetilde{Y},j)$ be a $P$-polarized K3 surface such that $\widetilde{Y}\to|j(\ell)|\cong\mathbb{P}^{2}$ is a morphism of degree two. Then $\widetilde{Y}$ has four $D_{4}$-configurations of $(-2)$-curves which goes to $D_{4}$-singularities of the branch curve in $\mathbb{P}^{2}$. The four singular points must be in general position by Lemma 3.2, so that they can be set to $B$ defined in (3.1) by the action of $\operatorname{Aut}\mathbb{P}^{2}$. Note also that the order of the four singular points is determined by the $P$-polarization. The branch locus must be the union of three distinct conics $Q_{1},Q_{2},Q_{3}\in|2H-B|\cong\mathbb{P}^{1}$, and the order of these three conics is also determined by the $P$-polarization. Conversely, any ordered triple $(Q_{1},Q_{2},Q_{3})\in|2H-B|^{3}\setminus\Delta_{\mathrm{big}}$ determines a $P$-polarized K3 surface, where $\Delta_{\mathrm{big}}\subset|2H-B|^{3}$ is the big diagonal where at least two of the three conics are equal, so that

Define divisors $\mathcal{H}_{i}$ of $\mathcal{M}$ for $i=1,\ldots,9$ as the closures in $\mathcal{M}$ of

for $j,k\in\{1,2,3\}$, where $D_{k}$ is the union of two lines defined in (3.6). The Picard lattice of a very general K3 surface on the divisor $\mathcal{H}_{i}$ is an extension of $P\bot\mathbf{A}_{1}$ by $\mathbb{Z}/2\mathbb{Z}$, where $\mathbf{A}_{1}$ comes from the resolution of the $A_{1}$-singularity above the node of $D_{k}$, and the extension comes from half the strict transform of an irreducible component of $D_{k}$ just as in Lemma 4.1. The class of the exceptional divisor for the resolution of the $A_{1}$-singularity gives a $(-2)$-element $v_{i}$ in $Q\subset H^{2}(Y,\mathbb{Z})$, and $\mathcal{H}_{i}$ is the divisor of $\mathcal{M}$ associated with $v_{i}$. The transcendental lattice $v_{i}^{\bot}\subset Q$ is the even 2-primary lattice $\mathbf{I}_{2,2}(2)$ characterized by $\delta=1$, $\ell=4$, and the signature (2,2).

Models of $P$-polarized K3 surfaces in a neighborhood of $\mathcal{H}_{10}$ can be parametrized by $(a_{0},g_{2},\lambda)\in\mathbb{C}\times\left(|2H-B|\setminus\{D_{i}\}_{i=1}^{3}\right)\times\left(\mathbb{P}^{1}\setminus\{0,1,\infty\}\right)$ as

When $a_{0}\neq 0$, one can eliminate $x_{4}$ so that the corresponding $P$-polarized K3 surface corresponds to the point

which goes to the small diagonal

as $a_{0}$ goes to 0. It follows that the moduli space $\mathcal{M}$ can be embedded in the blow-up $(|2H-B|^{3})^{\sim}$ of $|2H-B|^{3}$ along $\Delta_{\mathrm{small}}$. The complement $(|2H-B|^{3})^{\sim}\setminus\mathcal{M}\cong(\Delta_{\mathrm{big}})^{\sim}\setminus(\Delta_{\mathrm{small}})^{\sim},$ where $\Delta_{\mathrm{big}}^{\sim}$ is the pull-back of the big diagonal and $\Delta_{\mathrm{small}}^{\sim}\cong\mathcal{H}_{10}$ is the exceptional divisor, consists of three irreducible components. By contracting each of these three components to $\mathbb{P}^{1}$, one obtains $\mathbb{P}^{3}$. Hence $\mathcal{M}$ is an open subvariety of $\mathbb{P}^{3}$, and the complement $\mathbb{P}^{3}\setminus\mathcal{M}$ is the union of three lines. Since the complement is of codimension 2, $\mathbb{P}^{3}$ must be isomorphic to the Satake–Baily–Borel compactification ${\overline{\mathcal{M}}}$.

To be explicit, choose a homogeneous coordinate $[s:t]$ of $|2H-B|$ in such a way that $d_{1}$, $d_{2}$, and $d_{3}$ corresponds to $[1:0]$, $[1:1]$, and $[0:1]$. Then one can choose a homogeneous coordinate $[u_{1}:u_{2}:u_{3}:u_{4}]$ of ${\overline{\mathcal{M}}}\cong\mathbb{P}^{3}$ in such a way that the birational map $|2H-B|^{3}\dashrightarrow{\overline{\mathcal{M}}}$ sends $([s_{1}:t_{1}],[s_{2}:t_{2}],[s_{3}:t_{3}])$ to

The big diagonal is contracted as

and the small diagonal is blown up to

Under the embedding $\mathcal{M}\subset\mathbb{P}^{3}$, the divisor $\mathcal{H}_{i}$ is a hyperplane if $i\in\{1,\ldots,9\}$, and the quadric (8.8) if $i=10$.

Let $\mathbb{M}\coloneqq\left[\mathcal{D}\middle/\widetilde{\operatorname{SO}}^{+}(Q)\right]\cong\left[\widetilde{\mathcal{D}}\middle/\widetilde{\operatorname{SO}}^{+}(Q)\times\mathbb{C}^{\times}\right]$ be the orbifold quotient, where we use $\widetilde{\operatorname{SO}}^{+}(Q)\coloneqq\widetilde{\operatorname{O}}^{+}(Q)\cap\operatorname{SO}(Q)$ instead of $\widetilde{\operatorname{O}}^{+}(Q)$, since the subgroup $\{\pm\operatorname{id}_{Q}\}\subset\widetilde{\operatorname{O}}^{+}(Q)$ acts trivially on $\mathcal{D}$, so that one has $A_{k}\left(\widetilde{\operatorname{O}}^{+}(Q)\right)=A_{k}\left(\widetilde{\operatorname{SO}}^{+}(Q)\right)$ while $\left[\mathcal{D}\middle/\widetilde{\operatorname{O}}^{+}(Q)\right]$ has $\{\pm\operatorname{id}_{Q}\}$ as the generic stabilizer. For any $k\in\mathbb{Z}$, the line bundle $\mathcal{O}_{\mathcal{D}}(k)\coloneqq\mathcal{O}_{\mathbb{P}(Q)}(k)|_{\mathcal{D}}$ is invariant under the action of $\widetilde{\operatorname{SO}}^{+}(Q)$, and hence descends to a line bundle on $\mathbb{M}$, which we write as $\mathcal{O}_{\mathbb{M}}(k)$. Then the line bundle $\mathcal{O}_{\mathbb{M}}(k)$ is associated with the trivial line bundle on $\widetilde{\mathcal{D}}$ equipped with the character $\widetilde{\operatorname{SO}}^{+}(Q)\times\mathbb{C}^{\times}\ni(g,\alpha)\mapsto\alpha^{k},$ so that one has

Recall that the canonical bundle of a hypersurface $X$ of degree $d$ in $\mathbb{P}^{n}$ is given by $\omega_{X}\cong\mathcal{O}_{X}(d-n-1)$. Since $\mathcal{D}$ is an open subvariety defined by $(\Omega,{\overline{\Omega}})>0$ of the quadric hypersurface defined by $(\Omega,\Omega)=0$ in the 4-dimensional projective space $\mathbb{P}(Q\otimes\mathbb{C})$, one has

Since the isomorphism (9.2) is equivariant with respect to the action of $\widetilde{\operatorname{SO}}(Q)$, one has

The structure morphism $\phi\colon\mathbb{M}\to\mathcal{M}$ to the coarse moduli space is an isomorphism in codimension one outside the image $\mathbb{H}\coloneqq\bigcup_{\delta\in\Delta(Q)}\mathbb{H}_{\delta}$ of the reflection hyperplanes in $\mathcal{D}$. Locally near a smooth point of $\mathbb{H}$, the orbifold $\mathbb{M}$ is the quotient of the double cover of $\mathcal{M}$ branched along $\mathcal{H}$ by the group of deck transformations (known as the root construction [Cad07, AGV08] of $\mathcal{M}$ along $\mathcal{H}$), so that

where

and $\mathcal{O}_{\mathbb{M}}(\mathbb{H})$ is the ‘square root’ of $\mathcal{O}_{\mathcal{M}}(\mathcal{H})$ (which is the raison d’etre of the root construction);