Title: Common transversals and tangents in P^3

CRANTS Seminar,
SUNY Albany
20 March 2002.

Frank Sottile

University of Massachusetts, Amherst

Abstract:
  The following geometric problem has its origins in
  computational vision: Determine the (degenerate)
  configurations of two lines l_1 and l_2 and two spheres
  in R^3 for which there are infinitely m_any lines
  simultaneously transversal to l_1 and l2 and tangent
  to both spheres. We generalize this, replacing the
  spheres by quadric surfaces in P^3. In this setting,
  the question has an amazing answer. Fixing the two
  lines to be skew and one quadric, the set of degenerate
  second quadrics is a curve of degree 24 in the P^9 of
  quadrics which is in fact the union of 12 plane conics!
  Moreover, there are examples where all 12 degenerate
  families are real. We describe the symbolic computation
  behind this result and give some vivid pictures.

  This is joint work with Gabor Megyesi and Thorsten Theobald