Families of \(K3\) surfaces and curves of \((2,3)\)-torus type
Curves of \((2,3)\)-torus type are plane sextic curves defined by a polynomial of the form \(F=F_2^3+F3^2\) with \(F_i\) homogeneous of degree \(i\), and have at most simple singularities. This note studies families of \(K3\) surfaces that are double covers of the projective plane curves of \((2,3)\)-torus type, in particular, their Picard lattices. Let \(X\to\mathbb{P}^2\) be a non-Galois triple cover branched along a curve of \((2,3)\)-torus type \(B\), and let \(\hat{X}\) be the Galois closure. If \(Z\) is a minimal model of the double cover \(D(X/\mathbb{P}^2)\) of \(\mathbb{P}^2\) branching along \(B\), then \(\hat{X}\) is obtained as the cyclic triple cover of \(Z\). In this case, a cyclic triple cover \(\hat{X}\to Z\) is obtained where \(Z\) is a \(K3\) surface, and \(\hat{X}\) is an abelian surface (with singularities), or a Gorenstein \(K3\) surface (a \(K3\) surface with at most simple singularities). A Gorenstein \(K3\) surface \(D(X/\mathbb{P}^2)\) has the defining equation \(W^2-F(X,Y,Z)=0\) in the weighted projective space \(\mathbb{WP}(1,1,1,3)\). These two cases are distinguished by the invariant \(\delta\) which takes values \(9\) in the first case, and \(6\) in the latter. Indeed, \(\delta=9\) if and only if \(\text{Sing}(B)=9A_2\), and \(\delta=6\) if \(\text{Sing}(B)\) is one of the 19 cases. This paper is focused in the latter case, namely \(\delta=6\). A Gorenstein \(K3\) surface \(D(X/\mathbb{P}^2)\) has the defining equation \(W^2-F(X,Y,Z)=0\) in the weighted projective space \(\mathbb{WP}(1,1,1,3)\), where \(F\) is the equation for the plance curve \(B\). This gives rise to families of Gorenstein \(K3\) surfaces. In particular, the main result is concerned with the structure of the Picard lattices of such families. Three families \(\mathcal{F}_i,\,i=1,2,3\) of \(K3\) surfaces are constructed by toric method corresponding to reflexive polytopes \(\Delta_i,\,i=1,2,3\), respectively, which are parametrized by the complete anticannonical linear systems of toric Fano threefolds. Theorem: Let \(\mathcal{F}_i,\,i=1,2,3\) be families of \(K3\) surfaces. (1) The Picard lattice of \(\mathcal{F}_1\) is isomorphic to \(U \oplus <-2>\oplus <-4>\). (2) The Picard lattice of \(\mathcal{F}_2\) is isomorphic to \(U\oplus A_5\). (3) The Picard lattice of \(\mathcal{F}_3\) is isomorphic to \(<-2>\oplus <2>\). Furthermore, \(\mbox{Pic}(\Delta_3^*)\oplus U=\mbox{Pic}(\Delta_3)\), that is, the Picard lattice of the family \(\mathcal{F}_3\) satisfies the mirror duality. Here \(\Delta_3^*\) is the dual polytope of \(\Delta_3\). The proof is computational with intersection matrices, and explicit descrition of defining equations with prescribed type of singularities.
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