Dedicated to the memory of Wolfgang Classical Intersection Theory (see
for example Wei! [Wei]) treats the case of proper intersections, where
geometrical objects (usually subvarieties of a non- singular variety)
intersect with the expected dimension. In 1984, two books appeared which
surveyed and developed work by the individual authors, co- workers and
others on a refined version of Intersection Theory, treating the case of
possibly improper intersections, where the intersection could have ex-
cess dimension. The first, by W. Fulton [Full] (recently revised in
updated form), used a geometrical theory of deformation to the normal
cone, more specifically, deformation to the normal bundle followed by
moving the zero section to make the intersection proper; this theory was
due to the author together with R. MacPherson and worked generally for
intersections on algeb- raic manifolds. It represents nowadays the
standard approach to Intersection Theory. The second, by W. Vogel
[Vogl], employed an algebraic approach to inter- sections; although
restricted to intersections in projective space it produced an
intersection cycle by a simple and natural algorithm, thus leading to a
Bezout theorem for improper intersections. It was developed together
with J. Stiickrad and involved a refined version of the classical
technique ofreduc- tion to the diagonal: here one starts with the join
variety and intersects with successive hyperplanes in general position,
laying aside components which fall into the diagonal and intersecting
the residual scheme with the next hyperplane; since all the hyperplanes
intersect in the diagonal, the process terminates.
