Lattices and \(K3\) surfaces of degree 6
Let \(X\) be a 3-dimensional complex manifold and \(\pi:X\to \Delta=\{t\in\mathbb{C}\mid|t|<\varepsilon\}\) be a surjective proper holomorphic mapping with connected fibres. Assume that for each \(t\in \Delta^*=\Delta\setminus \{0\}\), the fibre \(X_t= \pi^{- 1}(t)\) is an algebraic K3 surface of degree 6, that is the K3 surface \(X_t\) carries a nef divisor \(D\) with self-intersection number \(D^2=6\). Such a divisor defines a polarization of degree 6. The fibre \(X_0= \pi^{-1}(0)\) is called a degeneration of K3 surfaces. In the present paper the author considers the case in which \(X_0\) consists of two smooth rational surfaces \(V_1, V_2\) glued along an elliptic curve \(E\), and \(X_0\) carries a Cartier divisor \(D_0\) with \(D_0^2=6\). Such a degeneration is called a type II degeneration of K3 surfaces of degree 6. By the intersection pairing, for \(t\in \Delta^*\) the second cohomology group \(H^2(X_t,\mathbb{Z})\) is a unimodular lattice with signature \((3,19)\). Since the K3 surface \(X_t\) carries a nef divisor \(D\) with \(D^2=6\), \(D\) defines an element \(h\in H^2(X_t,\mathbb{Z})\). The orthogonal complement of \(h\) is denoted by \(L_6\), which is a lattice of signature \((2,19)\). To a type II degeneration of K3 surfaces of degree 6 we can associate a rank 2 totally isotropic sublattice of \(L_6\). In the present paper the author shows that there are 10 types of primitive rank 2 totally isotropic sublattices of \(L_6\) up to \(O(L_6)\) equivalence, and he also constructs an example of type II degeneration to each sublattice.
scientific article; zbMATH DE number 803745
scientific article