REU 09

REU 09: Algebraic Geometry Group

Goals for the first week:

Understand how to construct the secant ideal of an edge ideal.
Understand the definition of a minimal free resolution and how to compute one using Macaulay 2. In particular, you should know how to interpret a graded Betti diagram.
Start to gather together examples of graphs G so that the secant ideals of I(G) have "good" Betti diagrams.
Start to read the "Combinatorial secant varieties" paper. By the end of the REU you should have a good working knowledge of it. It contains the background and motivation for our investigations this summer.

Here are some links to things you might want to be reading:

Combinatorial secant varieties, Bernd Sturmfels and Seth Sullivant, Quarterly Journal of Pure and Applied Mathematics 2 (2006) 285-309. Start with Section 3.
Computations in Algebraic Geometry with Macaulay 2, edited by David Eisenbud, Daniel R. Grayson, Michael E. Stillman, and Bernd Sturmfels, Springer-Verlag, no. 8, Algorithms and Computations in Mathematics, 2001.
Start with pages 21-24.
Lectures on the Geometry of Syzygies, edited by Luchezar L. Avramov, Mark Green, Craig Huneke, Karen E. Smith, and Bernd Sturmfels, Trends in Commutative Algebra, MSRI Publications Volume 51, 2004.
Start with pages 121-123.
What can be computed in algebraic geometry?, Dave Bayer and David Mumford, Computational algebraic geometry and commutative algebra (Cortona, 1991), Cambridge Univ. Press, Cambridge, 1993, pp. 1-48.
An introduction to toric surfaces, Jessica Sidman, rough draft.

Here is a document describing how you might want to deal with file storage.

Here is a LaTex template file that you might want to use as you are getting started writing up your thoughts and examples. If you have your own computer, you may download LaTex yourself. If you have a PC running Windows, you can download LEd. You also need to install MikTex. Those of you with Macs running OS X can download MacTex. MacTex is TexShop plus TexLive, and it is all you need to install to get started in one step.

Goals for the second week:

Write down what you can about the minimal free resolution of one edge ideal and its secant ideal. If you think you have any conjectures, try to write them down as precisely as you can.
Understand how to use Buchberger's algorithm to compute syzygies.
Be able to use weight vectors to compute initial ideals in Macaulay 2. Try to understand what equivalence classes of weight vectors look like.
Start getting acquainted with the gfan software.

Some Macaulay 2 code that you might find useful:

Code for computing the secant ideal of an arbitrary monomial ideal secantMonomialIdeal.m2.
Check out the package EdgeIdeals written by Chris Francisco, Andrew Hoefel and Adam Van Tuyl.

Goals for the third week:

Be able to describe the variety of some edge ideals in terms of unions of linear subspaces.
Start familiarizing yourself with varieties in projective space. Understand why projective varieties are defined by homogeneous polynomials.
Start to sort out what you've learned about the ideals of 2-minors and 3-minors of the matrix of linear forms that you chose to work with last week. Have you computed all of the Groebner bases of both? If it seems too hard, what should you do? Are there any "wonderful" term orderings? (See Sturmfels and Sullivant for a definition.) What did betti diagrams of the initial ideals look like? Do you have questions? Conjectures?
You should be reading Chapters 1,4,8 of Cox, Little, O'Shea. If you have a good handle on that material, try Chapters 9 and 5, with that priority.
By the end of the week you should be able to understand the statements of all of the questions our group will look at this summer. Pick a question and an example, and start playing around. You should feel free to bounce around between questions and examples for the next few weeks or so. Playing around will help you to see patterns and make conjectures. Don't feel pressured to settled down and get results.

Here is a beamer presentation that you can download. What you are downloading is a directory containing all of the source images and LaTex files for a talk. To unpack it (on a mac or in Linux), download the file to a directory. Inside the directory containing the file ima.tar.gz, type

gunzip ima.tar.gz

to unzip it and then

tar -xf ima.tar

to unpack it. This creates a directory called "ima". The files that you want to look at will be there. <

Plan for week four:

Adam Van Tuyl and Tai Ha are visiting us this week. They will talk about free resolutions of edge ideals and their talks will be our lecture time Tuesday and Wednesday.
I would like to meet with each of you individually this week to talk about which questions you are most interested in pursuing. Please do talk to me frequently!
Here is some code, written by Greg Smith, for computing the secant ideal of an arbitrary ideal.
Here is some code that computes all of the forms of degree k+2 in the k-th secant ideal of an ideal generated by quadrics. You'll also need intersection.m2, written by Mike Stillman.

Looking ahead: You've all learned a lot of commutative algebra and algebraic geometry. As you think about the second half of the REU, I think that a reasonable goal is that you pick a specific question and then really analyze it in the context of a concrete famiily of examples. You should be thinking of creating a talk/poster/paper that sets the stage for your question, describes your family of examples, and then states all conjectures and proofs. As you get to know your example well, you may start to gain intuition for the general picture.

For now, focus on your examples, and please do keep me in the loop! You will probably be dancing back and forth between what is known and what is not, and I can help to let you know which side you're likely to be on at any given point in time.

Plan for week six:

On Monday (or as soon as possible) I would like to see the following from each of you:

A list of what you have been reading. Please let me know what you have understood and liked best. I would also like you to write down at least one question about your reading.
A list of all of the examples that you have looked at so far. If there were examples that were particularly interesting to you, please make a note and let me know what interested you. If there were examples that didn't seem to go anywhere, then let me know about them too. You should write down the specific packages that you used for each example in Macaulay 2.
A list of any Macaulay 2 programs or functions that you've written so far. (I'm thinking of any code that goes beyond just using already available functions more or less interactively.)
A list of conjectures/ideas/proofs that you have generated so far.
A to-do list including further reading, examples that you want to check, code that you want to write, ideas that you hope will turn into proofs.

If you don't have this ready by Monday, it's not the end of the world. The point of the exercise is that you sit down and think reflectively about what you've done so far. This will help both of us to see where you should go next. It will also help to get you in a good place for starting your final write-up of the work that you've done this summer. You're really starting to put together a bibliography, sort through examples, and think through what else you'll need to make a nice write-up.
I'll meet with each of you individually after I receive the information above and we'll formulate a plan for the remainder of the REU (weeks 6-8).
The grant number for the REU is DMS-0849637. Please include it in everything you LaTex up from now on.