Lattices and \(K3\) surfaces of degree 6
type: Item
instance of: scholarly article
title: Lattices and \(K3\) surfaces of degree 6
zbMATH Open document ID: 0874.14026
DOI: 10.1016/0024-3795(95)00150-P
author: Hans Sterk
published in: Linear Algebra and its Applications
publication date: 1997-11-10T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 06C10
Mathematics Subject Classification ID: 14D05
zbMATH DE Number: 803745
zbMATH Keywords: lattice degeneration
zbMATH Keywords: Del Pezzo surface
zbMATH Keywords: 3-dimensional complex manifold
zbMATH Keywords: type II degeneration of K3 surfaces
zbMATH Keywords: unimodular lattice
reviewed by: Kenji Ueno
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 803745