Commutative Algebra Seminar

Spring 2000-2001

Hilbert functions and Blow-up Algebras

Venue : Room 105 : On Mondays at 4.00 pm

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.

Information on last semester's commutative algebra available here