604 research outputs found
On varieties defined by large sets of quadrics and their application to error-correcting codes
Let be a -dimensional subspace of quadratic forms
defined on with the property that does not
contain any reducible quadratic form. Let be the points of
which are zeros of all quadratic forms in .
We will prove that if there is a group which fixes and no line of
and spans
then any hyperplane of is incident with at most
points of . If is a finite field then the linear code
generated by the matrix whose columns are the points of is a
-dimensional linear code of length and minimum distance at least
. A linear code with these parameters is an MDS code or an almost MDS
code. We will construct examples of such subspaces and groups , which
include the normal rational curve, the elliptic curve, Glynn's arc from
\cite{Glynn1986} and other examples found by computer search. We conjecture
that the projection of from any points is contained in the
intersection of two quadrics, the common zeros of two linearly independent
quadratic forms. This would be a strengthening of a classical theorem of Fano,
which itself is an extension of a theorem of Castelnuovo, for which we include
a proof using only linear algebra
Darboux cyclides and webs from circles
Motivated by potential applications in architecture, we study Darboux
cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin
cyclides and quadrics, and they carry up to six real families of circles.
Revisiting the classical approach to these surfaces based on the spherical
model of 3D Moebius geometry, we provide computational tools for the
identification of circle families on a given cyclide and for the direct design
of those. In particular, we show that certain triples of circle families may be
arranged as so-called hexagonal webs, and we provide a complete classification
of all possible hexagonal webs of circles on Darboux cyclides.Comment: 34 pages, 20 figure
Real k-flats tangent to quadrics in R^n
Let d_{k,n} and #_{k,n} denote the dimension and the degree of the
Grassmannian G_{k,n} of k-planes in projective n-space, respectively. For each
k between 1 and n-2 there are 2^{d_{k,n}} \cdot #_{k,n} (a priori complex)
k-planes in P^n tangent to d_{k,n} general quadratic hypersurfaces in P^n. We
show that this class of enumerative problem is fully real, i.e., for each k
between 1 and n-2 there exists a configuration of d_{k,n} real quadrics in
(affine) real space R^n so that all the mutually tangent k-flats are real.Comment: 10 pages, 3 figures. Minor revisions, to appear in Proc. AM
Varieties with too many rational points
We investigate Fano varieties defined over a number field that contain
subvarieties whose number of rational points of bounded height is comparable to
the total number on the variety.Comment: 23 page
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