Parallelisms of
Quadrics
Bill Cherowitzo
University of Colorado Denver
Norm Johnson
University of Iowa
4th Pythagorean Conference, Corfu Greece
31 May 2010
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General Parallelisms

Given a finite set X of size n and
the set F, of all the subsets of X of
size t ( t ≤ n), a parallelism is a
partition of F into subsets, each of
which is a partition of X. The
divisibility condition, t|n, is a
necessary and sufficient condition
for the existence of a parallelism.

When the set F is restricted in any
way, the existence guarantee is
lost.
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General Parallelisms
While modeled by the parallel line
structure of an affine plane, the
general form of a parallelism with
restrictions on the set F has been
useful in many areas of
combinatorics … graph theory
(factorizations), design theory
(resolutions), and other geometric
settings where F does not
necessarily consist of lines.
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Parallelisms of Quadrics



X will consist of the points (or
almost all of the points) of a nondegenerate quadric Q in PG(3,K).
F shall consist of the planes which
intersect Q in conics (or these
conics of intersection themselves).
A partition of Q (or almost all of the
points of Q) by elements of F is
called a flock of Q.
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Hyperbolic Quadrics


In PG(3,q), all flocks of hyperbolic
quadrics are known (Thas, BaderLunardon).
Every finite flock lies in a transitive
parallelism (Bonisoli).
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Elliptic Quadrics



In PG(3,q), there are q3 + q nontangent (secant) planes.
A flock of an elliptic quadric
requires q-1 of these planes.
There are no finite parallelisms of
elliptic quadrics.
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The Infinite Cases

Let K be a field of characteristic ≠2
admitting a quadratic extension.
Parallelisms of elliptic quadrics Q
exist in PG(3,K) arising from linespread parallelisms coming from a
generalized line star of Q (BettenRiesinger).
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The Infinite Cases

For K an arbitrary field, let F be any
flock of a hyperbolic quadric H in
PG(3,K). Then F is contained in a
transitive parallelism of H.
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Quadratic Cones

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Flocks of quadratic cones have not
been classified.
A flock of a quadratic cone is a
partition of all the points except the
vertex into conics.
Equivalently, a flock consists of the
planes determined by these conics.
Normally, it doesn’t matter, but for
our result we require the plane
interpretation.
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The Spread Connection


The planes of a flock can be represented
in the form
x0t – x1f(t) + x2g(t) + x3 = 0, t  K.
There is an associated translation plane
π with spread set
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The Spread Connection
The “conical” translation plane π is
mapped to an isomorphic “conical”
translation plane by any of the elements
of the group:
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The Spread Connection

The new “conical” translation planes have
spread sets of the form:
 The planes of the corresponding flock are
disjoint from those of the original.
 The set of all images of π under G give
rise to a parallelism of the quadratic cone.
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Transitivity of Parallelism


The group G used to produce the
parallelism does not preserve the
original cone.
The set of all planes of PG(3,K),
not through the vertex of a given
cone C with flock F is partitioned
into flocks (including F) of cones
which are isomorphic to C.
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Can we do with less?


Over some infinite fields maximal
partial flocks of quadratic cones
exist.
We can use them in a transitive
parallelism of the quadratic cone.
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More General Cones


The argument that provided the
parallelism did not depend on the
nature of the cone, only the
connection with spreads.
Flocks of certain non-quadratic
cones, called flokki, also give rise
to spreads (Kantor-Penttila), so we
may use the same technique to
produce parallelisms of flokki.
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An unashamed plug!
Coming on 17 June 2010
CRC Press
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