Frank Sottile  (University of Toronto and MSRI)

           "Enumerative geometry for real varieties"

     Algebraic Geometry Seminar    Thursday, 26 September 1996.


Of the geometric figures in a given family satisfying real conditions,
some figures are real, while the rest occur in complex conjugate
pairs, and the distribution of the two types depends subtly upon the
configuration of the conditions.  In this talk, we describe an
approach to the question of when such an enumerative problem may have
all of its solutions be real.  Our particular focus is how one may use
the knowledge that one problem can have all of its solutions be real
to deduce that other, related problems may, as well.

The primary technique is to study deformations which transform
intersections of subvarieties into simpler cycles.  We also discuss
how these techniques may lead to algorithms to find explicit solutions
to such enumerative problems, such as the five lines meeting 6 given
planes in P^4.