576 research outputs found
Lines in higgledy-piggledy position
We examine sets of lines in PG(d,F) meeting each hyperplane in a generator
set of points. We prove that such a set has to contain at least 1.5d lines if
the field F has more than 1.5d elements, and at least 2d-1 lines if the field F
is algebraically closed. We show that suitable 2d-1 lines constitute such a set
(if |F| > or = 2d-1), proving that the lower bound is tight over algebraically
closed fields. At last, we will see that the strong (s,A) subspace designs
constructed by Guruswami and Kopparty have better (smaller) parameter A than
one would think at first sight.Comment: 17 page
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)
Embeddings of Line-grassmannians of Polar Spaces in Grassmann Varieties
An embedding of a point-line geometry \Gamma is usually defined as an
injective mapping \epsilon from the point-set of \Gamma to the set of points of
a projective space such that \epsilon(l) is a projective line for every line l
of \Gamma, but different situations have lately been considered in the
literature, where \epsilon(l) is allowed to be a subline of a projective line
or a curve. In this paper we propose a more general definition of embedding
which includes all the above situations and we focus on a class of embeddings,
which we call Grassmman embeddings, where the points of \Gamma are firstly
associated to lines of a projective geometry PG(V), next they are mapped onto
points of PG(V\wedge V) via the usual projective embedding of the
line-grassmannian of PG(V) in PG(V\wedge V). In the central part of our paper
we study sets of points of PG(V\wedge V) corresponding to lines of PG(V)
totally singular for a given pseudoquadratic form of V. Finally, we apply the
results obtained in that part to the investigation of Grassmann embeddings of
several generalized quadrangles
The geometry of the moduli space of odd spin curves
We describe the birational geometry of the moduli space S_g^{-} of odd spin
curves (theta-characteristics) for all genera g. The odd spin moduli space is a
uniruled variety for g<12, and of general type for g at least 12. Furthermore,
for g<9 we use the existence of Mukai models of the moduli space of curves, to
prove that S_g^{-} is unirational. Our results show that in genus 8, the odd
spin moduli space in unirational, whereas its even counterpart is of Calabi-Yau
type.Comment: 34 pages. Final version, to appear in the Annals of Mathematic
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