187 research outputs found
On Invariant Notions of Segre Varieties in Binary Projective Spaces
Invariant notions of a class of Segre varieties \Segrem(2) of PG(2^m - 1,
2) that are direct products of copies of PG(1, 2), being any positive
integer, are established and studied. We first demonstrate that there exists a
hyperbolic quadric that contains \Segrem(2) and is invariant under its
projective stabiliser group \Stab{m}{2}. By embedding PG(2^m - 1, 2) into
\PG(2^m - 1, 4), a basis of the latter space is constructed that is invariant
under \Stab{m}{2} as well. Such a basis can be split into two subsets whose
spans are either real or complex-conjugate subspaces according as is even
or odd. In the latter case, these spans can, in addition, be viewed as
indicator sets of a \Stab{m}{2}-invariant geometric spread of lines of PG(2^m
- 1, 2). This spread is also related with a \Stab{m}{2}-invariant
non-singular Hermitian variety. The case is examined in detail to
illustrate the theory. Here, the lines of the invariant spread are found to
fall into four distinct orbits under \Stab{3}{2}, while the points of PG(7,
2) form five orbits.Comment: 18 pages, 1 figure; v2 - version accepted in Designs, Codes and
Cryptograph
Veldkamp-Space Aspects of a Sequence of Nested Binary Segre Varieties
Let be a Segre variety that is -fold direct product of projective
lines of size three. Given two geometric hyperplanes and of
, let us call the triple the
Veldkamp line of . We shall demonstrate, for the sequence , that the properties of geometric hyperplanes of are fully
encoded in the properties of Veldkamp {\it lines} of . Using this
property, a complete classification of all types of geometric hyperplanes of
is provided. Employing the fact that, for , the
(ordinary part of) Veldkamp space of is , we shall
further describe which types of geometric hyperplanes of lie on a
certain hyperbolic quadric that
contains the and is invariant under its stabilizer group; in the
case we shall also single out those of them that correspond, via the
Lagrangian Grassmannian of type , to the set of 2295 maximal subspaces
of the symplectic polar space .Comment: 16 pages, 8 figures and 7 table
Aspects of the Segre variety S_{1,1,1}(2)
We consider various aspects of the Segre variety S := S_{1,1,1}(2) in
PG(7,2), whose stabilizer group G_S < GL(8, 2) has the structure N {\rtimes}
Sym(3), where N := GL(2,2)\times GL(2,2)\times GL(2,2). In particular we prove
that S determines a distinguished Z_3-subgroup Z < GL(8, 2) such that AZA^{-1}
= Z, for all A in G_S, and in consequence S determines a G_S-invariant spread
of 85 lines in PG(7,2). Furthermore we see that Segre varieties S_{1,1,1}(2) in
PG(7,2) come along in triplets {S,S',S"} which share the same distinguished
Z_3-subgroup Z < GL(8,2). We conclude by determining all fifteen G_S-invariant
polynomial functions on PG(7,2) which have degree < 8, and their relation to
the five G_S-orbits of points in PG(7,2)
On the dimension of contact loci and the identifiability of tensors
Let be an integral and non-degenerate variety. Set
. We prove that if the -secant variety of has (the
expected) dimension and is not uniruled by lines, then
is not -weakly defective and hence the -secant variety satisfies
identifiability, i.e. a general element of it is in the linear span of a unique
with . We apply this result to many Segre-Veronese
varieties and to the identifiability of Gaussian mixtures . If is
the Segre embedding of a multiprojective space we prove identifiability for the
-secant variety (assuming that the -secant variety has dimension
, this is a known result in many cases), beating several
bounds on the identifiability of tensors.Comment: 12 page
Four lectures on secant varieties
This paper is based on the first author's lectures at the 2012 University of
Regina Workshop "Connections Between Algebra and Geometry". Its aim is to
provide an introduction to the theory of higher secant varieties and their
applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in
Mathematics & Statistics), Springer/Birkhause
Algebraic varieties representing group-based Markov processes on trees
In this paper we complete the results of Sullivant and Sturmfels proving that
many of the algebraic group-based models for Markov processes on trees are
pseudo-toric. We also show in which cases these varieties are normal. This is
done by the generalization of the discrete Fourier transform approach. In the
next step, following Sullivant and Sturmfels, we describe a fast algorithm
finding a polytope associated to these algebraic models. However in our case we
apply the notions of sockets and networks extending the work of Buczynska and
Wisniewski who introduced it for the binary case
Tetrads of lines spanning PG(7,2)
Our starting point is a very simple one, namely that of a set L_4 of four
mutually skew lines in PG(7,2): Under the natural action of the stabilizer
group G(L_4) < GL(8,2) the 255 points of PG(7,2) fall into four orbits omega_1,
omega_2, omega_3 omega_4; of respective lengths 12, 54, 108, 81: We show that
the 135 points in omega_2 \cup omega_4 are the internal points of a hyperbolic
quadric H_7 determined by L_4; and that the 81-set omega_4 (which is shown to
have a sextic equation) is an orbit of a normal subgroup G_81 isomorphic to
(Z_3)^4 of G(L_4): There are 40 subgroups (isomorphic to (Z_3)^3) of G_81; and
each such subgroup H < G_81 gives rise to a decomposition of omega_4 into a
triplet of 27-sets. We show in particular that the constituents of precisely 8
of these 40 triplets are Segre varieties S_3(2) in PG(7,2): This ties in with
the recent finding that each Segre S = S_3(2) in PG(7,2) determines a
distinguished Z_3 subgroup of GL(8,2) which generates two sibling copies S'; S"
of S.Comment: Some typos correcte
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