Department of Mathematics
Colloquium
by
Prof. Paul Roberts

The question of defining intersection multiplicities in Algebraic Geometry has been a central problem in that subject and in Commutative Algebra for many years. About forty years ago Serre proposed a definition of using homological algebra and showed that it fulfilled many of the required conditions. However, there were certain questions which were left open; these questions concerned whether the intersection multiplicities could be negative and the precise conditions for them to be positive, and they were formulated as conjectures. This talk will discuss the origin and history of these conjectures and the recent proof of the Nonnegativity Conjecture by O. Gabber It will conclude with a brief outline of the implications of the new methods for the positivity conjecture, which remains open, and connections with other topics in Commutative Algebra.

References:

1. J.-P. Serre, Algebre Locale-Multiplicites, Springer Lecture Notes 11.

2. W. Fulton, Intersection Theory, Springer-Verlag.