Parallelisms of Quadrics

1. Parallelisms of Quadrics Bill Cherowitzo University of Colorado Denver Norm Johnson University of Iowa 4th Pythagorean Conference, Corfu Greece 31 May 2010

2. 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.

3. 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.

4. 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.

5. 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).

6. 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.

7. 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).

8. 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.

9. 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.

10. 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

11. The Spread Connection The “conical” translation plane π is mapped to an isomorphic “conical” translation plane by any of the elements of the group:

12. 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.

13. 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.

14. 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.

15. 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.

16. An unashamed plug! Coming on 17 June 2010 CRC Press