On the Automorphisms of Some K3 Surface Double Cover of the Plane Federica Galluzzi and Giuseppe Lombardo

In this paper we study the automorphisms group of some K3 surfaces which are double covers of the projective plane ramified over a smooth sextic plane curve. More precisely, we study the case of a K3 surface of Picard rank two such that there is a rational curve of degree d which is tangent to the sextic in d points. Introduction K3 surfaces which are double covers of the plane ramified over a plane sextic are classical objects. In this paper we determine the automorphisms group of some of these surfaces. More precisely, we restrict to the case of Picard rank two. Moreover we suppose that there is a rational curve of degree d which is tangent to the sextic in d points. The result is that K3 surfaces of this type with d odd have automorphisms group isomorphic to Z2. We first analize the case d = 3 in full details to better explain the results and the methodology we have used. Then we study the case with d odd for which the situation is not substantially different from the previous one. The automorphisms of a K3 surface are given by the Hodge isometries that preserve the Kähler cone (see [2], VIII.11). Thus, the strategy to study the automorphism of a K3 surface X is to determine its Kähler cone and the Hodge isometries of H(X,Z) which preserve it. To do this, we first determine the isometries of the Néron-Severi lattice NS(X) which preserve the Kähler cone and then we study the gluing conditions on the transcendental lattice T (X), since H(X,Z) = NS(X)⊕ T (X). The first author is supported by Progetto di Ricerca Nazionale COFIN 2004 ”Geometria sulle Varietà Algebriche”. 1 2 FEDERICA GALLUZZI AND GIUSEPPE LOMBARDO In the preliminary Section 1 we introduce some basic material on lattices and K3 surfaces. In section 2 we give explicitly the Néron-Severi lattice of the K3 we are interested in. In section 3 we start our work on the case d = 3 following the strategy explained before. We determine the Kähler cone in Prop.3.1 and the automorhisms group of the NéronSeveri lattice in Prop.3.2 using some basic facts on the generalized Pell equation a − 13b = −4. Finally, we derive in Theorem3.3 that the automorhisms group of the surface is isomorphic to Z2 analyzing the gluing condition on the transcendental lattice. Using similar methods we obtain the same result for d odd in Theorem4.1. 1. Preliminaries 1.1. Lattices. A lattice is a free Z-module L of finite rank with a Zvalued symmetric bilinear form <,> . A lattice is called even if the quadratic form associated to the bilinear form has only even values, odd otherwise. The discriminant d(L) is the determinant of the matrix of the bilinear form. A lattice is called non-degenerate if the discriminant is non-zero and unimodular if the discriminant is ±1. If the lattice L is non-degenerate, the pair (s+, s−), where s± denotes the multiplicity of the eigenvalue ±1 for the quadratic form associated to L⊗R, is called signature of L. Finally, we call s+ + s− the rank of L. Given a lattice (L,<,>) we can construct the lattice (L(m) <,>m), that is the Z-module L with form < x, y >m= m < x, y > . An isometry of lattices is an isomorphism preserving the bilinear form. Given a sublattice L →֒ L′, the embedding is primitive if L ′ L is free. Two even lattices S, T are orthogonal if there exist an even unimodular lattice L and a primitive embedding S →֒ L for which (S)L ∼= T. 1.2. K3 surfaces. A K3 surface is a compact Kähler surface with trivial canonical bundle and such that its first Betti number is equal to zero. Let U be the lattice of rank two with quadratic form given by the matrix (