Construction and Applications of Spreads: An Introduction

Hans Havlicek (Institute of Geometry, Vienna University of Technology)

A spread S in three-dimensional (real projective) space is a set of line such that each point is on exactly one line of S. The classical example (dating back to the 19th century) is the elliptic linear congruence of lines. It can be described in various ways, both algebraic and geometric, and we will present some of them. Also, we report on the widespread literature concerning the projection of the space to a plane by means of such a congruence of lines.

The interest in arbitrary spreads arose in the fifties and sixties of the 20th century in connection with the problem to construct certain geometric structures, namely so-called translation planes. In this context the elliptic congruence of lines is also called a regular spread.

The essential problem is to find spreads that are non-regular. It is very easy to find such spreads by modifying a regular spread. However, in general simple constructions yield non-continuous spreads. It is still not too difficult to find non-regular continuous spreads, but it is highly non-trivial to find smooth spreads other than the regular ones. In particular, just recently the first examples of non-regular algebraic spreads have been discovered. In this lecture we will illustrate some of these results by giving explicit examples.