Volume 7, Issue 4

A long-standing conjecture is that any transitive finite projective plane is Desarguesian. We make a contribution towards a proof of this conjecture by showing that a group acting transitively on the points of a non-Desarguesian projective plane must not contain any components.

We prove that if G is a string C-group of rank 4 and G ≅ L 2 ( q ) with q a prime power, then q must be 11 or 19. The polytopes arising are Grünbaum's 11-cell of type {3, 5, 3} for L 2 (11) and Coxeter's 57-cell of type {5, 3, 5} for L 2 (19), each a locally projective regular 4-polytope.

Hilbert constructed an early example of a non-desarguesian plane by modifying the lines of the real plane inside an ellipse. We employ methods from elementary algebraic geometry to show that the group of automorphisms of this plane leaves the ellipse invariant (unless the ellipse is a circle). Using this result, we determine the group of automorphisms of Hilbert's plane and some generalizations.

Every nondesarguesian symplectic spread is also symplectic over its kernel. Any equivalence of nondesarguesian symplectic spreads preserves the resulting symplectic structures over the kernels.

The primitive curves are the multiple curves that can be locally embedded in smooth surfaces (we will always suppose that the associated reduced curves are smooth). These curves have been defined and studied by C. Bănică and O. Forster in 1984. In 1995, D. Bayer and D. Eisenbud gave a complete description of the double curves. We give here a parametrization of primitive curves of arbitrary multiplicity. Let Z n = spec(ℂ[ t ]/( t n )). The curves of multiplicity n are obtained by taking an open cover ( U i ) of a smooth curve C and by glueing schemes of type U i × Z n using automorphisms of U ij × Z n that leave U ij invariant. This leads to the study of the sheaf of nonabelian groups G n of automorphisms of C × Z n that leave the reduced curve invariant, and to the study of its first cohomology set. We prove also that in most cases it is the same to extend a primitive curve C n of multiplicity n to one of multiplicity n + 1, and to extend the quasi locally free sheaf D n of derivations of C n to a rank 2 vector bundle on C n .

We present polyhedral models for isoparametric families in the sphere with at most three principal curvatures. Each member of the family (including the analogues of the focal sets) is tight in the boundary complex of an ambient convex polytope. In particular, the tube around the real (or complex) Veronese surface is represented as a tight polyhedron in 5-space (or 8-space). The examples are based on a certain Bier sphere triangulation of S 4 or S 7 , respectively. In the 4-dimensional case there are simplicial branched coverings of these triangulations in the complex projective plane and in S 2 × S 2 which are branched precisely along the polyhedral analogues of the Veronese surface. Moreover, we introduce a notion of PL-tautness and discuss its relationship with tightness of polyhedra. In particular, each member of our polyhedral isoparametric family is PL-taut. For an extended abstract see [W. Kühnel, Discrete models of isoparametric hypersurfaces in spheres. Oberwolfach Reports 12 , 665–667 (2006).].