Correspondence among families of Gorenstein K3 surfaces in certain Q-Fano 3-folds

A Gorenstein K3 surface is a complete complex surface with a trivial canonical bundle and irregularity zero and having at worst ADE singularities, which is birational to a K3 surface. A 3-dimensional normal algebraic variety X is called a Q-Fano 3-fold if X is Q-Gorenstein, that is, there exists an integer r > 0, such that rK_X is a Cartier divisor and X has an ample anticanonical divisor. Three-dimensional weighted projective spaces, which are Q-Fano 3-folds with abelian quotient singularities, whose general anticanonical members are K3 surfaces are classified into 95 by Yonemura [40] and Iano-Fletcher [20]. Smooth Fano 3-folds with the second Betti number ≥ 2 are classified into 88 classes by Mori-Mukai [26][27], and smooth toric Fano 3-folds are classified into 18 classes by Batyrev [2] and Watanabe-Watanabe [39] independently. For a K3 surface S, we denote by Pic(S) the Picard group equipped with a cup product inherited from the second cohomology group with integer coefficient H^2(S;Z), which we call the Picard lattice of S. Denote ρ(S) the Picard number of S, that is, the rank of the Picard lattice. For a family F of K3 surfaces, the Picard lattice Pic(F) of F is defined to be the Picard lattice of a generic member in F. Remark 1 The Picard lattice of arbitrary family of K3 surfaces may not make sense. However, in our case (see below for setting), the notion is welldefined. The Picard lattices of 95 families of weighted K3 hypersurfaces are computed by Belcastro [10], according to which, there are several families whose Picard lattices are isometric, even if the weights themselves are distinct. So does arise a natural question whether or not such families are isomorphic. Generally, we consider a following problem for families F, G of K3 surfaces: if the Picard lattices Pic(F) and Pic(G) of the families F, G are isometric, then, does there exist a birational correspondence between generic members in F and G? If the Picard numbers are large enough, the statement of the problem may be proved by Nikulin’s lattice theory and Torelli-type theorem for K3 surfaces. However, if the Picard numbers are small, for example, for families of K3 surfaces in smooth Fano 3-folds, whose Picard numbers are almost ≤ 5 it may not be true. Family of Calabi-Yau manifolds has become one of the central targets of the study in mirror symmetry theory. For example, Picard-Fuchs differential equations for certain families of weighted K3 hypersurfaces with Shioda-Inose structure have been recently studied. However, such studies are not much done for families of non-toric K3 hypersurfaces. We only consider the problem for families of weighted K3 hypersurfaces and families of K3 surfaces in smooth Fano 3-folds. In particular, we treat the following special families F_1 and F_2, whose Picard lattices are isometric: let l and C be a line and an irreducible plane cubic in P^3, and X' and X be the blow-up of P^3 along the line l, and C, and let F_1, F_2 be the families of K3 surfaces in X', X, respectively. X' is a smooth toric Fano 3-fold and X is a smooth non-toric Fano 3-fold. Let ­i be the moduli space of ample M_i-polarised K3 surfaces that are the minimal models of Gorenstein K3 surfaces in F_i; i = 1, 2. We state the main results (Theorems 1,2 and Propositions 1,2). Theorem 1 Let a and b be weights such that the families of K3 surfaces in P(a) and P(b) have the isometric Picard lattices. Then there exist subspace D_a (resp. D_b) of the complete anticanonical linear system of P(a) (resp. P(b)) and an isomorphism ϕ: D_a→D_b with the following properties. (1) If S ∈ D_a is Gorenstein, then S and ϕ(S) ∈ D_b are birational. (2) The Picard lattices of the families Da and Db are isometric to the Picard lattices of families of K3 surfaces in P(a) and P(b). Proposition 1 Under the assumption of the theorem, there exists a group isomorphism M(a) <～_-> M(b), and a common reflexive subpolytope Δ of polytopes Δ(a) and Δ(b), with the following properties. (1) The associated birational maps ϕ_a : P(Δ)-- → P(a) and ϕ_b : P(Δ)--→ P(b) send general anticanonical members of P(Δ) to those of P(a) and P(b) (2) The Picard lattices of families of K3 surfaces in P(a) and P(b), and in P(Δ) are isometric. Remark 2 By Theorem 1, the families of weighted K3 hypersurfaces are essentially reduced to be 75 families. We next consider families of K3 surfaces in smooth Fano 3-folds. Proposition 2 The Picard lattices of families of K3 surfaces in smooth toric Fano 3-folds are mutually distinct. Remark 3 By Proposition 2, families of K3 surfaces in smooth toric Fano 3-folds are mutually distinct in the sense that there is no birational correspondence between generic members in any these families. Theorem 2 The moduli space Ω_1 and Ω_2 are isomorphic. Remark 4 Theorem 1 is not due to Torelli-type theorem but by reminding the fact that those K3 surfaces are hypersurfaces in toric varieties, we construct explicit monomial transformations of Laurent polynomials. Since X ...