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Joins and Intersections (Hardcover, 1999 ed.)
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Joins and Intersections (Hardcover, 1999 ed.)
Series: Springer Monographs in Mathematics
Expected to ship within 10 - 15 working days
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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.
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