Karl Schwede
Assistant Professor / Research Fellow
Email: kschwede{at}umich{dot}edu
Office: East Hall 3828
Department of Mathematics
University of Michigan.
Ann Arbor, Michigan 48109-1109

About me:

I am a NSF post-doc in Mathematics at the University of Michigan (last year I was on the RTG grant). I recently finished my Ph.D., titled On F-injective and Du Bois singularities, at the University of Washington under the direction of Sándor Kovács. I am now working under Karen Smith here at the University of Michigan, although I also talk/meet with Mel Hochster and Mircea Mustaţă fairly regularly. I am currently working on questions related to singularities in birational geometry and singularities defined by the action of Frobenius in positive characteristic. I live on the boundary between commutative algebra and algebraic geometry.

Click HERE for a copy of my CV.

Courses and Seminars:

Winter 2008:  Math 632
Fall 2007:  Math 185
Winter 2007:  Math 217

I'm currently organizing the Commutative Algebra Seminar with Neil Epstein.

In the past I helped organize the Fall 2007 Topics in Algebraic Geometry Seminar and the Winter 2008 Topics in Algebraic Geometry Seminar. (aka the post-doc AG seminar)

Pre-prints, Re-prints & Other stuff:

Papers

  • A geometric algorithm for computing seminormalization, based off my thesis, in preparation. In principal, this algorithm could be implemented in Macaulay2 or a similar program. I described this algorithm in my Thesis, and if you CLICK HERE you can view the relevant portion of my thesis on the topic.
  • Globally F-regular and log Fano varieties, with Karen Smith, in preparation.
  • Centers of F-purity, submitted. We discuss a positive characteristic analogue of a notion from characteristic zero, centers of log canonicity. Click HERE for my slides on this topic from the conference in honor of Mel Hochster's 65th birthday..
  • Cohen-Macaulay semi-log canonical singularities are Du Bois  With Sándor Kovács and Karen Smith. Submitted. You can also view Karen Smith giving a talk on these results HERE.
  • Generalized test ideals, sharp F-purity, and sharp test elements    To appear in to Mathematical Research Letters.
  • Rational singularities associated to pairs    With Shunsuke Takagi. It has now appeared in the Michigan Mathematical Journal.
  • F-injective singularities are Du Bois    Based off of a part of my thesis. To appear in The American Journal of Mathematics.
  • A simple characterization of Du Bois singularities    Also based on my thesis. It has now appeared in Composito Mathematica.
  • Gluing schemes and a scheme without closed points    This provides an explicit example that studying algebraic geometry can in fact be pointless (that joke was originally due to Paul Smith). I came up with this as a second year graduate student and managed to get it published. See
      Proceedings of the 2002 John H. Barrett Memorial Lectures Conference on Algebraic/Arithmetic Geometry, (eds Y. Kachi, S. Mulay, P. Tzermias). 2005.

    Other stuff

  • Generalized divisors and reflexive sheaves on normal varieties    These are some notes I wrote for a class I taught on "Introduction to schemes and cohomology". These are largely based on several papers of Hartshorne on the same topic (although he did it in greater generality, that is, without the normality hypothesis).
  • Examples of odd behavior with seminormality    This is just a small document which brings together counter-examples to easily believed (but false) questions about seminormality. The examples included are not original, I just put them all in one place (half for my own convenience).
  • A characterization of finitely generated modules     This was my undergraduate thesis which I did at Whitman College. It gives an algorithm for expressing finitely generated modules over a Euclidean domain as a sum of cyclic modules. It then gives some somewhat standard results.

    At one point in my life, I solved all the problems in Hartshorne's algebraic geometry. I typed up a few of the solutions so I may post some of them here in a little while.

    Links:

    Situs Geometriae Algebraicae     A website which helps people new to algebraic geometry find references.

    Macaulay2     A nice tool for doing computations in commutative algebra and algebraic geometry.

    Commalg     The trusted source for commutative algebra news on the web.

    Mathematics Genealogy Project

    The following people are related to me (mathematically) in some way.
    Sándor Kovács Graduate Advisor.
    Karen Smith Postdoctoral Advisor.
    Michael Van Opstall Sándor's, first student. He's currently at the University of Utah.
    Davis Doherty A recent graduate of Sándor's (my year), he's now at Seattle University.
    Zachary Treisman Another recent graduate of Sándor's (my year). I believe he is now visiting the University of British Columbia
    Kelly Jabbusch A student of Sándor's (my year). She now has a position at the University of Cologne
    Travis Kopp A student of Sándor's a couple years behind me.
    Kevin Tucker A student of Karen's who I chat with regularly
    Shunsuke Takagi A collaborator with similar interests.