2,664 research outputs found
Cayley-Bacharach Formulas
The Cayley-Bacharach Theorem states that all cubic curves through eight given
points in the plane also pass through a unique ninth point. We write that point
as an explicit rational function in the other eight.Comment: 13 pages, 4 figure
Incidence structures from the blown-up plane and LDPC codes
In this article, new regular incidence structures are presented. They arise
from sets of conics in the affine plane blown-up at its rational points. The
LDPC codes given by these incidence matrices are studied. These sparse
incidence matrices turn out to be redundant, which means that their number of
rows exceeds their rank. Such a feature is absent from random LDPC codes and is
in general interesting for the efficiency of iterative decoding. The
performance of some codes under iterative decoding is tested. Some of them turn
out to perform better than regular Gallager codes having similar rate and row
weight.Comment: 31 pages, 10 figure
On the arithmetic of a family of degree-two K3 surfaces
Let denote the weighted projective space with weights
over the rationals, with coordinates and ; let
be the generic element of the family of surfaces in
given by \begin{equation*}
X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*} The surface
is a K3 surface over the function field . In this paper, we
explicitly compute the geometric Picard lattice of , together with
its Galois module structure, as well as derive more results on the arithmetic
of and other elements of the family .Comment: 20 pages; v2 with some all additions and clarifications suggested by
the refere
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