# Assymptotic ideal theory and parametrized rational varieties

Monday, September 25, 2006 - 11:15am - 12:15pm

Lind 305

David Eisenbud (Mathematical Sciences Research Institute)

If I is a homogeneous ideal in a polyonomial ring over a field, then in some ways high powers of I are simpler than I itself. This fact is implicitly used whenever one blows up a subvariety of a variety, for example in the process of resolution of singularities. The attendant phenomena appear in commutative algebra in the study of the Rees algebra of a variety. I will explain a new point of view that connects, for example, the parametrization of a rational curve to the singularities of that curve without passing through the implicitization step of finding the equations of the image curve. The work I will describe is joint with Joe Harris and Craig Huneke.