INdAM-COFUND Incoming fellow between 2012-01-10 and 2013-02-15

Project information

Title

Castelnuovo-Mumford Regularity: Homological and Computational Aspects

Abstract

The research training project is concentrated around an important topic in the international research: the study of the Castelnuovo-Mumford regularity of a finitely generated module over a Noetherian commutative ring.

The Castelnuovo-Mumford regularity is a kind of universal bound for important invariants of graded k-algebras such as the maximum degree of the syzygies and the maximum non-vanishing degree of the local cohomology modules.

A central problem in Commutative Algebra and Algebraic Geometry is the Eisenbud-Goto conjecture on regularity, multiplicity and codimension of domains. We can try to prove the conjecture for specific classes of graded domains. In particular we are interested to characterize the initial ideals (or the Groebner bases) of prime ideals. This problem has a strong interest in our project because there is a clear connection between the Castelnuovo-Mumford regularity of an ideal and those of its initial ideal. We try to find suitable extension of Eisenbud-Goto Conjecture for local domains.

We try to see whether the regularity of the associated graded ring can be bounded by a polynomial function (possibly linear) of the multiplicity and the codimension.

Another problem is to find some uniform bounds for all the coefficients of the Hilbert polynomial using regularity bounds. We try to give some optimal uniform bounds for all the coefficients of the Hilbert polynomial corresponding to m-primary
ideals.

Fellow information

Research interests

• Castelnuovo-Mumford Regularity
• Hilbert Coefficients and Blowup Algebras
• Combinatorial Commutative Algebra

Previous positions, awards

• September 2011-January 2012: Visiting Fellow, Tata Institute of Fundamental
  Research, Mumbai, India
• April 2011-August 2011: Research Associate, IIT Bombay, Mumbai, India
• January 2006-March 2011: Research Scholar, IIT Bombay, Mumbai, India

Publications, preprints, other works

1. M. Mandal, J. K. Verma, On the Chern number of I-admissible filtrations of
   ideals, to appear in Journal of Commutative Algebra (2012)
2. M. Mandal, S. Masuti, J. K. Verma, Normal Hilbert polynomials: a survey, to appear in Proceedings of the International
   Conference in Commutative Algebra and Algebraic Geometry (2012)
3. S. Goto, J. Hong, M. Mandal, The positivity of the first coefficients of normal
   Hilbert polynomials, Proc. Amer. Math. Soc. 139 (2011), 2399-2406
4. S. Goto, M. Mandal, J. K. Verma, Negativity of the Chern number of parameter ideals, Proceedings of the International Conference in
   Algebra at Aligarh Muslim University
5. M. Mandal, B. Singh, J. K. Verma, On some conjectures about the Chern numbers of
   filtrations, Journal of Algebra. 325 (2011), 147-162
6. M. Mandal, J. K. Verma, On the Chern number of an ideal, Proc. Amer. Math. Soc. 138 (2010), 1995-1999

Conferences, schools, other events

• Chern number formulas and consequences
  Poster presented at Midwest Commutative Algebra and Geometry
  Conference
  Purdue University, Indiana, US, May 2011
• The positivity of the first coefficients of normal
  Hilbert polynomials
  Talk delivered at 32nd Symposium on Commutative Algebra and the 6th
  Japan-Vietnam Joint Seminar on Commutative Algebra
  Japan, December 2010
• The positivity of the first coefficients of normal
  Hilbert polynomials
  Talk delivered at International Conference on Commutative Algebra and
  Algebraic Geometry
  IISc. Bangalore, India, December 2010
• On the Chern number of an ideal
  Talk delivered at CIMPA School and International Conference in
  Commutative Algebra and its application to Algebraic Geometry and
  Combinatorics
  Galatasaray University, Istanbul,
  Turkey, September 2010
• International Congress of Mathematics
  Hyderabad, India, August 2010
• School and Workshop on Local Rings and Local Study
  of Algebraic Varieties
  ICTP, Trieste, Italy, June 2010