Commutative Algebra Seminar-- Spring 2010

• Wednesday, January 27, at 10:10AM, room 312, A. Kustin, "The branches of a curve singularity. I." We prove the following Theorem. Let R be a local one dimensional ring. In the typical case, there is a one-to-one correspondence between the minimal prime ideals of the completion of R and the maximal ideals of the integral closure of R. This result, due to Zariski, has geometric significance. The ring R could be the local ring of rational functions defined at a point p on a curve C. The minimal prime ideals of the completion of R correspond to the branches of C at p. The maximal ideals of the integral closure of R correspond to points on the desingularization of C which lie over p.
  
  
• Wednesday, February 3, at 10:10AM, room 312, A. Kustin, "The branches of a curve singularity. II."
  
  
• Wednesday, February 10, at 10:10AM, room 312, Speaker: Adela Vraciu, Title: "Homological properties of almost Gorenstein rings"
  
  
• Wednesday, February 17, at 10:10AM, room 312, Speaker: Adela Vraciu, Title: "Homological properties of almost Gorenstein rings, II."
  
  
• Wednesday, February 24, at 10:10AM, room 312, Brett Barwick, "Existence of Pure Filtrations of Modules". Recent work by Eisenbud-Schreyer, Boij-Soederberg, and others has shown that the betti diagram of any Cohen-Macaulay module can be decomposed into a positive rational linear combination of pure diagrams, called the Boij-Soederberg decomposition. A recent paper of Eisenbud, Erman, and Schreyer investigates the question of whether these numerical decompositions may correspond in some way to decompositions of the algebraic structures involved. This talk will give some review of the basic definitions and theorems involved in Boij-Soederberg theory and we will go through an example of how one may compute the submodules involved in a pure filtration, if such a filtration exists.
  
  
• Wednesday, March 3, at 10:10AM, room 312, Brett Barwick, "Existence of Pure Filtrations of Modules, II." In this talk we will see examples of some tools that Boij-Soederberg theory provides for computing a pure filtration of a graded module with first syzygies in more than one degree whose Betti diagram satisfies certain numerical requirements.
  
  
• Wednesday, March 17, at 10:10AM, room 312, Brett Barwick, "Existence of Pure Filtrations of Modules, III."
  
  
• Wednesday, April 14, 10:10-11 AM, room 312. Andrew Kustin "Singularities of parameterized plane curves." We focus on singularities of multiplicity d/2 on curves of degree d. There can be at most three such singularities counting both points on the curve and infinitely near points. We describe the Hilbert-Burch matrix for each of the six possible configurations.

Previous semesters

• Fall 2009.
• Spring 2009.
• Fall 2008.
• Spring 2008.
• Fall 2007.
• Spring 2007.
• Fall 2006.
• Spring 2006.
• Fall, 2005.
• Spring, 2005.