31,640 research outputs found
Abstract configurations in algebraic geometry
An abstract -configuration is a pair of finite sets of
cardinalities and with a relation on the product of the sets such that
each element of the first set is related to the same number of elements
from the second set and each element of the second set is related to the same
number of elements in the first set. An example of an abstract
configuration is a finite geometry. In this paper we discuss some examples of
abstract configurations and, in particular finite geometries, which one
encounters in algebraic geometry.Comment: 39 pages, 12 figure
On the order three Brauer classes for cubic surfaces
We describe a method to compute the Brauer-Manin obstruction for smooth cubic
surfaces over \bbQ such that \Br(S)/\Br(\bbQ) is a 3-group. Our approach is
to associate a Brauer class with every ordered triplet of Galois invariant
pairs of Steiner trihedra. We show that all order three Brauer classes may be
obtained in this way. To show the effect of the obstruction, we give explicit
examples.Comment: Extended introduction, discussion of the example of Cassels and Gu
Luigi Cremona and cubic surfaces
We discuss Luigi Cremona's contribution to the early development of the
theory of cubic surfaces.Comment: 12 page
Configurations of lines and models of Lie algebras
The automorphism groups of the 27 lines on the smooth cubic surface or the 28
bitangents to the general quartic plane curve are well-known to be closely
related to the Weyl groups of and . We show how classical
subconfigurations of lines, such as double-sixes, triple systems or Steiner
sets, are easily constructed from certain models of the exceptional Lie
algebras. For and we are lead to
beautiful models graded over the octonions, which display these algebras as
plane projective geometries of subalgebras. We also interpret the group of the
bitangents as a group of transformations of the triangles in the Fano plane,
and show how this allows to realize the isomorphism in terms of harmonic cubes.Comment: 31 page
Cubic surfaces with a Galois invariant pair of Steiner trihedra
We present a method to construct non-singular cubic surfaces over \bbQ with
a Galois invariant pair of Steiner trihedra. We start with cubic surfaces in a
form generalizing that of A. Cayley and G. Salmon. For these, we develop an
explicit version of Galois descent
A linear system on Naruki's moduli space of marked cubic surfaces
Allcock and Freitag recently showed that the moduli space of marked cubic
surfaces is a subvariety of a nine dimensional projective space which is
defined by cubic equations. They used the theory of automorphic forms on ball
quotients to obtain these results. Here we describe the same embedding using
Naruki's toric model of the moduli space. We also give an explicit
parametrization of the tritangent divisors, we discuss another way to find
equations for the image and we show that the moduli space maps, with degree at
least ten, onto the unique quintic hypersurface in a five dimensional
projective space which is invariant under the action of the Weyl group of the
root system E_6.Comment: 23 page
Polynomial cubic differentials and convex polygons in the projective plane
We construct and study a natural homeomorphism between the moduli space of
polynomial cubic differentials of degree d on the complex plane and the space
of projective equivalence classes of oriented convex polygons with d+3
vertices. This map arises from the construction of a complete hyperbolic affine
sphere with prescribed Pick differential, and can be seen as an analogue of the
Labourie-Loftin parameterization of convex RP^2 structures on a compact surface
by the bundle of holomorphic cubic differentials over Teichmuller space.Comment: 64 pages, 5 figures. v3: Minor revisions according to referee report.
v2: Corrections in section 5 and related new material in appendix
Structure of Cubic Lehman Matrices
A pair of square -matrices is called a \emph{Lehman pair} if
for some integer . In this case and
are called \emph{Lehman matrices}. This terminology arises because Lehman
showed that the rows with the fewest ones in any non-degenerate minimally
nonideal (mni) matrix form a square Lehman submatrix of . Lehman
matrices with are essentially equivalent to \emph{partitionable graphs}
(also known as -graphs), so have been heavily studied as part
of attempts to directly classify minimal imperfect graphs. In this paper, we
view a Lehman matrix as the bipartite adjacency matrix of a regular bipartite
graph, focusing in particular on the case where the graph is cubic. From this
perspective, we identify two constructions that generate cubic Lehman graphs
from smaller Lehman graphs. The most prolific of these constructions involves
repeatedly replacing suitable pairs of edges with a particular -vertex
subgraph that we call a -rung ladder segment. Two decades ago, L\"{u}tolf \&
Margot initiated a computational study of mni matrices and constructed a
catalogue containing (among other things) a listing of all cubic Lehman
matrices with of order up to . We verify their catalogue
(which has just one omission), and extend the computational results to matrices. Of the cubic Lehman matrices (with ) of order
up to , only two do not arise from our -rung ladder
construction. However these exceptions can be derived from our second
construction, and so our two constructions cover all known cubic Lehman
matrices with
On quartics with three-divisible sets of cusps
We study the geometry and codes of quartic surfaces with many cusps. We apply
Gr\"obner bases to find examples of various configurations of cusps on
quartics.Comment: 15 page
The exterior splash in PG(6,q): Transversals
Let be an order--subplane of that is exterior to
. Then the exterior splash of is the set of points
on that lie on an extended line of . Exterior splashes are
projectively equivalent to scattered linear sets of rank 3, covers of the
circle geometry , and hyper-reguli in . In this article we
use the Bruck-Bose representation in to investigate the structure of
, and the interaction between and its exterior splash. In ,
an exterior splash has two sets of cover planes (which are
hyper-reguli) and we show that each set has three unique transversals lines in
the cubic extension . These transversal lines are used to
characterise the carriers of , and to characterise the sublines of
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