1,704 research outputs found
A geometrical construction for the polynomial invariants of some reflection groups
In these notes we investigate the rings of real polynomials in four
variables, which are invariant under the action of the reflectiongroups [3,4,3]
and [3,3,5]. It is well known that they are rationally generated in degree
2,6,8,12 and 2,12,20,30. We give a different proof of this fact by giving
explicit equations for the generating polynomials.Comment: 10 page
Construction of Nikulin configurations on some Kummer surfaces and applications
A Nikulin configuration is the data of disjoint smooth rational curves
on a K3 surface. According to a well known result of Nikulin, if a K3 surface
contains a Nikulin configuration , then is a Kummer surface
where is an Abelian surface determined by . Let
be a generic Abelian surface having a polarization with (for
an integer) and let be the associated Kummer surface. To the
natural Nikulin configuration on , we associate another
Nikulin configuration ; we denote by the Abelian surface
associated to , so that we have also . For we
prove that and are not isomorphic. We then construct an infinite order
automorphism of the Kummer surface that occurs naturally from our
situation. Associated to the two Nikulin configurations
, there exists a natural bi-double cover , which is a
surface of general type. We study this surface which is a Lagrangian surface in
the sense of Bogomolov-Tschinkel, and for is a Schoen surface.Comment: 22 pages, refereed versio
Symmetries of order four on K3 surfaces
We study automorphisms of order four on K3 surfaces. The symplectic ones have
been first studied by Nikulin, they are known to fix six points and their
action on the K3 lattice is unique. In this paper we give a classification of
the purely non-symplectic automorphisms by relating the structure of their
fixed locus to their action on cohomology, in the following cases: the fixed
locus contains a curve of genus g>0; the fixed locus contains at least a curve
and all the curves fixed by the square of the automorphism are rational. We
give partial results in the other cases. Finally, we classify non-symplectic
automorphisms of order four with symplectic square.Comment: Final version, to appear in J. Math. Soc. Japa
- …