Mathematics Graduate Student Colloquium

MGSC Website: http://math.uci.edu/~mgsc/

Bounding the regularity of a graded module

Roger Dellaca
Tuesday, November 4, 2014
4:00 pm - 4:50 pm
RH440R

Talk Abstract:

Given a set X of d points in the complex plane, every complex-valued function on X is the restriction of a polynomial of degree at most d-1; some configurations of points allow the maximal degree of such polynomials to be strictly smaller than d-1, while others do not. This is an example of the notion of (Castelnuovo-Mumford) regularity of the algebraic set. Regularity measures the complexity of the geometric object, and its associated algebraic object as well. A general definition of regularity can be given in terms of a minimal free resolution of a graded module, which is used in several explicit techniques in Commutative Algebra, including the Hilbert function and Hilbert polynomial. A theorem of Gotzmann takes an odd-looking way of writing a Hilbert polynomial, and uses it to bound the regularity of all ideals with that Hilbert polynomial. This was used to give an explicit construction of the set of all such ideals as a geometric space, called the Hilbert scheme. I will show how this technique can be used to bound the regularity of certain classes of modules with a given Hilbert polynomial. This allows an explicit construction of a generalization of the Hilbert scheme.

About the Speaker:

Roger is a 6th-year student. He has a M.S. in Math from CSULB and a M.S. in Management Science from CSUF. His research is in Commutative Algebra and Algebraic Geometry. He enjoys cycling.

Refreshments:

Pizza will be served after the talk.