Maximally Inflected (Real) Rational Curves

Frank Sottile
University of Massachusetts-Amherst

  A map from P^1 to P^n of degree d necessarily has (n+1)(d-n+1) 
points where it is ramified, counted with multiplicities.   We 
call these inflection points, as in the image curve, a point of 
simple ramification is an inflection point.   A maximally inflected
curve is a real curve, all of whose inflection points are also real.
The existence of such curves is guaranteed by a result in the real
Schubert calculus, and these curves are closely related to an 
important conjecture in that field.   

   Maximally inflected plane curves satisfy some topological restrictions
given by the Klein and Pluecker formulas.  They also satisfy some more 
subtle restrictions whose existence was discovered experimentally.

   This talk will introduce these objects, then discuss symbolic and numerical
techniques to generate examples of maximally inflected curves, and lastly
describe the known topological restrictions, including those whose existence
is only suspected via experiments.