1,950 research outputs found
k-nets embedded in a projective plane over a field
We investigate k-nets with k ≥ 4 embedded in the projective plane P G(2, K) defined over a field K; they are line configurations in P G(2, K) consisting of k pairwise disjoint line-sets, called compo- nents, such that any two lines from distinct families are concurrent with exactly one line from each component. The size of each com- ponent of a k-net is the same, the order of the k-net. If K has zero characteristic, no embedded k-net for k ≥ 5 exists; see [10, 13]. Here we prove that this holds true in positive characteristic p as long as p is sufficiently large compared with the order of the k-net. Our ap- proach, different from that used in [10, 13], also provides a new proof in characteristic zero
3-nets realizing a diassociative loop in a projective plane
A \textit{-net} of order is a finite incidence structure consisting of
points and three pairwise disjoint classes of lines, each of size , such
that every point incident with two lines from distinct classes is incident with
exactly one line from each of the three classes. The current interest around
-nets (embedded) in a projective plane , defined over a field
of characteristic , arose from algebraic geometry. It is not difficult to
find -nets in as far as . However, only a few infinite
families of -nets in are known to exist whenever , or .
Under this condition, the known families are characterized as the only -nets
in which can be coordinatized by a group. In this paper we deal with
-nets in which can be coordinatized by a diassociative loop
but not by a group. We prove two structural theorems on . As a corollary, if
is commutative then every non-trivial element of has the same order,
and has exponent or . We also discuss the existence problem for such
-nets
Group-labeled light dual multinets in the projective plane (with Appendix)
In this paper we investigate light dual multinets labeled by a finite group
in the projective plane defined over a field .
We present two classes of new examples. Moreover, under some conditions on the
characteristic , we classify group-labeled light dual multinets
with lines of length least
Light dual multinets of order six in the projective plane
The aim of this paper is twofold: First we classify all abstract light dual
multinets of order which have a unique line of length at least two. Then we
classify the weak projective embeddings of these objects in projective planes
over fields of characteristic zero. For the latter we present a computational
algebraic method for the study of weak projective embeddings of finite
point-line incidence structures
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