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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 non-degenerate 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, Bader-Lunardon).

Every finite flock lies in a transitive parallelism (Bonisoli).

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Elliptic Quadrics

In PG(3,q), there are q3 + q non-tangent (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 line-spread parallelisms coming from a generalized line star of Q (Betten-Riesinger).

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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

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