791 research outputs found
On hyperovals of polar spaces
We derive lower and upper bounds for the size of a hyperoval of a finite polar space of rank 3. We give a computer-free proof for the uniqueness, up to isomorphism, of the hyperoval of size 126 of H(5, 4) and prove that the near hexagon E-3 has up to isomorphism a unique full embedding into the dual polar space DH(5, 4)
Splines in geometry and topology
This survey paper describes the role of splines in geometry and topology,
emphasizing both similarities and differences from the classical treatment of
splines. The exposition is non-technical and contains many examples, with
references to more thorough treatments of the subject.Comment: 18 page
Equivariant -theory of GKM bundles
Given a fiber bundle of GKM spaces, , we analyze the
structure of the equivariant -ring of as a module over the equivariant
-ring of by translating the fiber bundle, , into a fiber bundle of
GKM graphs and constructing, by combinatorial techniques, a basis of this
module consisting of -classes which are invariant under the natural holonomy
action on the -ring of of the fundamental group of the GKM graph of .
We also discuss the implications of this result for fiber bundles where and are generalized partial flag varieties and show how
our GKM description of the equivariant -ring of a homogeneous GKM space is
related to the Kostant-Kumar description of this ring.Comment: 15 page
Torus graphs and simplicial posets
For several important classes of manifolds acted on by the torus, the
information about the action can be encoded combinatorially by a regular
n-valent graph with vector labels on its edges, which we refer to as the torus
graph. By analogy with the GKM-graphs, we introduce the notion of equivariant
cohomology of a torus graph, and show that it is isomorphic to the face ring of
the associated simplicial poset. This extends a series of previous results on
the equivariant cohomology of torus manifolds. As a primary combinatorial
application, we show that a simplicial poset is Cohen-Macaulay if its face ring
is Cohen-Macaulay. This completes the algebraic characterisation of
Cohen-Macaulay posets initiated by Stanley. We also study blow-ups of torus
graphs and manifolds from both the algebraic and the topological points of
view.Comment: 26 pages, LaTeX2e; examples added, some proofs expande
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