Algebraic K3 surfaces with finite automorphism groups

Abstract

The purpose of this paper is to give a proof to the result announced in [3]. Let X be an algebraic surface defined over C. X is called a K3 surface if its canonical line bundle Kx is trivial and dim H1(X, ϕX) = 0. It is known that the automorphism group Aut (X) of X is isomorphic, up to a finite group, to the factor group O(Sx)/Wx, where O(Sx) is the automorphism group of the Picard lattice of X (i.e. Sx is the Picard group of X together with the intersection form) and Wx is its subgroup generated by all reflections associated with elements with square (–2) of Sx ([11]). Recently Nikulin [8], [10] has completely classified the Picard lattices of algebraic K3 surfaces with finite automorphism groups.

Stable pair compactification of moduli of K3 surfaces of degree 2

“…R with the half spaces v • E ≥ 0 for the irreducible curves E with E 2 < 0. By [Kon89] the 24 curves E i are the only negative curves on S. We thus get the same inequalities that define a fundamental chamber K of F cox . The pullback of a negative curve is a sum of negative curves.…”

The moduli of curves of genus six and K3 surfaces

“…The diagram in our case is given in Figure 1 (see also Figure 5, [Ko1]). Note that the symmetry group of the diagram is Z 2 × S 3 and it can be easily seen that all symmetries can be realized by isometries in Γ.…”

The Kodaira dimension of some moduli spaces of elliptic K3 surfaces

“…Let X be a K3 surface with NS(X) = U ⊕ E 8 (−1). Then X has finite automorphism group and a finite number of irreducible (−2)-curves (see, for example, [14,19]). More precisely, if |E| denotes the unique elliptic fibration on X, then the irreducible (−2)-curves on X are the 9 curves C 2 , .…”