J. A. Thas (Ghent University)
Eggs with applications to generalized quadrangles, flocks, planes, ovoids and hyperovals

In PG(2n+m-1,q) consider a set O(n,m,q) of q^m+1 (n-1)-dimensional subspaces PG⁽⁰⁾(n-1,q),PG⁽¹⁾(n-1,q),..., PG^(qm)(n-1,q), every three of which generate a PG(3n-1,q) and such that each element PG⁽ⁱ⁾(n-1,q) of O(n,m,q) is contained in a PG⁽ⁱ⁾(n+m-1,q) having no point in common with any PG^(j)(n-1,q) for j not equal to i. It is easy to check that PG⁽ⁱ⁾(n+m-1,q) is uniquely determined, i=0,1,..., q^m. The space PG⁽ⁱ⁾(n+m-1,q) is called the tangent space of O(n,m,q) at PG⁽ⁱ⁾(n-1,q). For n=m such a set O(n,n,q) is called a pseudo-oval or a generalized oval or a [n-1]-oval of PG(3n-1,q); a generalized oval of PG(2,q) is just an oval of PG(2,q). For n not equal to m such a set O(n,m,q) is called a pseudo-ovoid. or a generalized ovoid or a [n-1]-ovoid or an egg of PG(2n+m-1,q); a [0]-ovoid of PG(3,q) is just an ovoid of PG(3,q).

The following hold for any O(n,m,q) : (i) either n=m or n(a+1)=ma with a odd, and (ii) if q is even, then either n=m or m=2n.

Eggs have applications to generalized quadrangles, flocks of quadratic cones, projective planes, ovoids of Q(4,q) and hyperovals.

A survey of important old and new results will be given.