510 research outputs found
On Frobenius incidence varieties of linear subspaces over finite fields
We define Frobenius incidence varieties by means of the incidence relation of
Frobenius images of linear subspaces in a fixed vector space over a finite
field, and investigate their properties such as supersingularity, Betti numbers
and unirationality. These varieties are variants of the Deligne-Lusztig
varieties. We then study the lattices associated with algebraic cycles on them.
We obtain a positive-definite lattice of rank 84 that yields a dense sphere
packing from a 4-dimensional Frobenius incidence variety in characteristic 2.Comment: 24 pages, no figures; Introduction is changed. New references are
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Transcendental lattices and supersingular reduction lattices of a singular surface
A (smooth) K3 surface X defined over a field k of characteristic 0 is called
singular if the N\'eron-Severi lattice NS (X) of X over the algebraic closure
of k is of rank 20. Let X be a singular K3 surface defined over a number field
F. For each embedding \sigma of F into the complex number field, we denote by
T(X^\sigma) the transcendental lattice of the complex K3 surface X^\sigma
obtained from X by \sigma. For each prime ideal P of F at which X has a
supersingular reduction X_P, we define L(X, P) to be the orthogonal complement
of NS(X) in NS(X_P). We investigate the relation between these lattices
T(X^\sigma) and L(X, P). As an application, we give a lower bound of the degree
of a number field over which a singular K3 surface with a given transcendental
lattice can be defined.Comment: 40 pages, revised version, to appear in Transactions of the American
Mathematical Societ
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