4,536 research outputs found
Abelian Projective Planes of Square Order
We prove that, under certain conditions, multipliers of an abelian projective plane of square order have odd order modulo v*, where v* is the exponent of the underlying Singer group. As a consequence, we are able to establish the non-existence of an infinite number of abelian projective planes of square order
Nilpotent Singer groups
Let be a nilpotent group normal in a group . Suppose that acts transitively upon the points of a finite non-Desarguesian projective plane . We prove that, if has square order, then must act semi-regularly on .
In addition we prove that if a finite non-Desarguesian projective plane admits more than one nilpotent group which is regular on the points of then has non-square order and the automorphism group of has odd order
Central aspects of skew translation quadrangles, I
Except for the Hermitian buildings , up to a combination
of duality, translation duality or Payne integration, every known finite
building of type satisfies a set of general synthetic
properties, usually put together in the term "skew translation generalized
quadrangle" (STGQ). In this series of papers, we classify finite skew
translation generalized quadrangles. In the first installment of the series, as
corollaries of the machinery we develop in the present paper, (a) we obtain the
surprising result that any skew translation quadrangle of odd order is
a symplectic quadrangle; (b) we determine all skew translation quadrangles with
distinct elation groups (a problem posed by Payne in a less general setting);
(c) we develop a structure theory for root-elations of skew translation
quadrangles which will also be used in further parts, and which essentially
tells us that a very general class of skew translation quadrangles admits the
theoretical maximal number of root-elations for each member, and hence all
members are "central" (the main property needed to control STGQs, as which will
be shown throughout); (d) we solve the Main Parameter Conjecture for a class of
STGQs containing the class of the previous item, and which conjecturally
coincides with the class of all STGQs.Comment: 66 pages; submitted (December 2013
Heisenberg-invariant Kummer surfaces
We study the geometry of Nieto's quintic threefold (Barth & Nieto, J. Alg.
Geom. 3, 1994) and the Kummer and abelian surfaces that correspond to special
loci.Comment: Plain TeX, 17 pages. Final version, with minor corrections, to appear
in Proc. Edinburgh Math. So
Curves on Heisenberg invariant quartic surfaces in projective 3-space
This paper is about the family of smooth quartic surfaces that are invariant under the Heisenberg group . For a
very general such surface , we show that the Picard number of is 16 and
determine its Picard group. It turns out that the general Heisenberg invariant
quartic contains 320 smooth conics and that in the very general case, this
collection of conics generates the Picard group.Comment: Updated references, corrected typo
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