15:00 – 17:00, Friday, September 14, 2012 : The Hoskin-Deligne Formula
Abstract:
I will sketch a proof of Hoskin-Deligne formula and show that it gives a complete description of the Hilbert-Samuel polynomial of a complete ideal. It also shows that a complete m-primary ideal of a two-dimensional regular local ring has reduction number one.
I will also discuss results and examples in higher dimensions and in two-dimensional local rings and point out some open problems.
09:00 – 11:30, Friday, September 7, 2012 : Transforms of ideals
Abstract:
We will introduce transform of an ideal in a two-dimensional regular local ring R and show that transform of a complete ideal is complete. We will also show that the colength of a zero-dimensional ideal of R is bigger than the colength of its transform in a local quadratic transform. This gives an inductive tool to begin the proofs of two main theorems of Zariski.
15:00 – 17:00, Friday, August 31, 2012: Integral closure, valuations and quadratic transforms
Abstract:
In this lecture, I will recall the basics of reductions and integral closures of ideals.
It will be shown that any complete ideal is an intersection of valuation ideals in a Noetherian domain. Construction of quadratic transform is central in Zariski’s approach. These will be discussed and Lipman-Rees characterization of ideals contracted from quadratic transforms will be presented. I will present a proof of this due to Huneke and Sally in which Hilbert-Burch theorem is used for the structure of ideals of projective dimension one.