|February 5, 12, 26|
|Prof. S. R. Ghorpade|
|Hilbert Functions of Determinantal Varieties|
|March 5, 12, 19|
|Mr. A. V. Jayanthan|
|Castelnuovo-Mumford Regularity and Blow-Up Algebras|
|Abstract : The Goto-Shimoda theorem about Cohen-Macaulayness of the Rees algebras was generalized to an arbitrary ideal by a number of researchers. We shall present one of the simplest proofs of the generalized Goto-Shimoda theorem due to Johnston and Katz which appeared in the proceedings of the AMS. Effort will be made to present the proof so that it is accessible to people who know only basic theory of CM rings.|
|April 2, 9, 16|
|Prof. Balwant Singh|
Hilbert functions, Lech's Inequality and Blowing Up
Abstract :We state three open questions concerning inequailities
between Hilbert functions of local rings. One of the questions
compares the Hilbert function at a closed point with the one at
the generic point. Another one is the Hironaka-Lech Conjecture
comparing Hilbert functions under a flat local homomorphism.
The third question compares the Hilbert function of a local ring
with the Hilbert function of the fiber of a blowing-up with a
normally flat center.
We shall show that the first two questions are essentially
equivalent and that an affirmative answer to the third question
implies an affirmative answer to the first two. We shall also
describe some cases in which the questions have been settled.