Math Seminars.
Gary Kennedy (U of M visitor),
Contact formulas for rational plane curves.
Two algebraic plane curves are said to have a contact of
order n at a point if both curves are smooth there and if
the intersection number is n. If X is a family of such
curves depending on s parameters, then one expects to find
finitely many members of the family satisfying s specified
conditions, where a "condition" means either passing through
a specified point or having a contact of order n with a
specified curve (this should be counted as n-1 conditions).
A contact formula expresses this finite number in terms of
simpler data for the family and for the specified individual
curves, called characteristic numbers. One is then faced
with the problem of determining the characteristic numbers.
We will look in particular at the case of rational plane
curves and where the specified order of contact is no more
than 3.