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. |