1,704 research outputs found

    A geometrical construction for the polynomial invariants of some reflection groups

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    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

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    A Nikulin configuration is the data of 1616 disjoint smooth rational curves on a K3 surface. According to a well known result of Nikulin, if a K3 surface contains a Nikulin configuration C\mathcal{C}, then XX is a Kummer surface X=Km(B)X=Km(B) where BB is an Abelian surface determined by C\mathcal{C}. Let BB be a generic Abelian surface having a polarization MM with M2=k(k+1)M^{2}=k(k+1) (for k>0k>0 an integer) and let X=Km(B)X=Km(B) be the associated Kummer surface. To the natural Nikulin configuration C\mathcal{C} on X=Km(B)X=Km(B), we associate another Nikulin configuration C′\mathcal{C}'; we denote by B′B' the Abelian surface associated to C′\mathcal{C}', so that we have also X=Km(B′)X=Km(B'). For k≥2k\geq2 we prove that BB and B′B' are not isomorphic. We then construct an infinite order automorphism of the Kummer surface XX that occurs naturally from our situation. Associated to the two Nikulin configurations C,\mathcal{C}, C′\mathcal{C}', there exists a natural bi-double cover S→XS\to X, which is a surface of general type. We study this surface which is a Lagrangian surface in the sense of Bogomolov-Tschinkel, and for k=2k=2 is a Schoen surface.Comment: 22 pages, refereed versio

    Symmetries of order four on K3 surfaces

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    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
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