625 research outputs found
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
The classification of flag-transitive Steiner 3-designs
We solve the long-standing open problem of classifying all 3-(v,k,1) designs
with a flag-transitive group of automorphisms (cf. A. Delandtsheer, Geom.
Dedicata 41 (1992), p. 147; and in: "Handbook of Incidence Geometry", ed. by F.
Buekenhout, Elsevier Science, Amsterdam, 1995, p. 273; but presumably dating
back to 1965). Our result relies on the classification of the finite
2-transitive permutation groups.Comment: 27 pages; to appear in the journal "Advances in Geometry
Almost simple groups with socle acting on Steiner quadruple systems
Let , {}, a prime power, be a projective linear
simple group. We classify all Steiner quadruple systems admitting a group
with N \leq G \leq \Aut(N). In particular, we show that cannot act as a
group of automorphisms on any Steiner quadruple system for .Comment: 5 pages; to appear in: "Journal of Combinatorial Theory, Series A
On surfaces of general type with
The moduli space of surfaces of general type with (where is the genus of the Albanese fibration) was constructed by
Catanese and Ciliberto in \cite{CaCi93}. In this paper we characterize the
subvariety corresponding to surfaces
containing a genus 2 pencil, and moreover we show that there exists a
non-empty, dense subset which parametrizes
isomorphism classes of surfaces with birational bicanonical map.Comment: 35 pages. To appear in Collectanea Mathematic
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