739 research outputs found
Transitive and Co-Transitive Caps
A cap in PG(r,q) is a set of points, no three of which are collinear. A cap
is said to be transitive if its automorphism group in PGammaL(r+1,q) acts
transtively on the cap, and co-transitive if the automorphism group acts
transtively on the cap's complement in PG(r,q). Transitive, co-transitive caps
are characterized as being one of: an elliptic quadric in PG(3,q); a
Suzuki-Tits ovoid in PG(3,q); a hyperoval in PG(2,4); a cap of size 11 in
PG(4,3); the complement of a hyperplane in PG(r,2); or a union of Singer orbits
in PG(r,q) whose automorphism group comes from a subgroup of GammaL(1,q^{r+1}).Comment: To appear in The Bulletin of the Belgian Mathematical Society - Simon
Stevi
Subgroups and Orbits by Companion Matrix in Three Dimensional Projective Space
الهدف من هذا البحث هو انشاء زمر جزئية دورية من الزمرة الخطية العامة الإسقاطية على الحقل من المصفوفة المصاحبة ، ثم تكوين أغطية بدرجات مختلفة في . تم إعطاء الخصائص الهندسية لهذه الأغطية كتوزيعات القطع و توزيعات الدليل ، وتحديد فيما إذا كانت كاملة. كذلك, تجزئة الى خطوط غير متقاطعة تم دراسته.The aim of this paper is to construct cyclic subgroups of the projective general linear group over from the companion matrix, and then form caps of various degrees in . Geometric properties of these caps as secant distributions and index distributions are given and determined if they are complete. Also, partitioned of into disjoint lines is discussed
On rational homology disk smoothings of valency 4 surface singularities
Thanks to the recent work of Bhupal, Stipsicz, Szabo, and the author, one has
a complete list of resolution graphs of weighted homogeneous complex surface
singularities admitting a rational homology disk ("QHD") smoothing, i.e., one
with Milnor number 0. They fall into several classes, the most interesting of
which are the three classes whose resolution dual graph has central vertex with
valency 4. We give a uniform "quotient construction" of the QHD smoothings for
these classes; it is an explicit Q-Gorenstein smoothing, yielding a precise
description of the Milnor fibre and its non-abelian fundamental group. This had
already been done for two of these classes in a previous paper; what is new
here is the construction of the third class, which is far more difficult. In
addition, we explain the existence of two different QHD smoothings for the
first class.
We also prove a general formula for the dimension of a QHD smoothing
component for a rational surface singularity. A corollary is that for the
valency 4 cases, such a component has dimension 1 and is smooth. Another
corollary is that "most" H-shaped resolution graphs cannot be the graph of a
singularity with a QHD smoothing. This result, plus recent work of
Bhupal-Stipsicz, is evidence for a general
Conjecture: The only complex surface singularities with a QHD smoothing are
the (known) weighted homogeneous examples.Comment: 28 pages: title changed, typos fixed, references and small
clarifications adde
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