Math 732: Prime Characteristic Methods in Commutative Algebra and Algebraic Geometry
Winter 2019


Professor Karen E. Smith

MWF 2-3 pm in 3096 EH

The goal of this course is to introduce the Frobenius morphism and its uses in commutative algebra and algebraic geometry.

These "characteristic p techniques" have been used in commutative algebra, for example, to establish that certain rings are Cohen-Macaulay, as in the famous Hochster-Roberts theorem for rings of invariants (over fields of arbitrary characteristic). In more geometric settings, we can analyze or quantify how singular---that is, how far from being smooth---a particular variety may be, or to establish that the singularities are suitably mild.

Prerequisites are a solid understanding of introductory commutative algebra (local rings) and algebraic geometry (classical varieties and schemes), such as Math 614, 631 and 632.

Course Notes, updated frequently, so check back as we go.

DAILY UPDATE, a brief description of what was covered and any announcements.

Worksheet to Explore Hilbert-Kunz Multiplicity, best to work together with others on it

Worksheet to Learn Macaulay 2, best to work together with others on it

Second Worksheet to Learn Macaulay 2, best to work together with others on it

GRADES AND EXPECTATIONS:
Students are expected to attend lectures, and try to keep up with the reading, which will mostly be the notes posted here. Students are expected to ask questions.

Students each expected to give a lecture in the course. There is flexibility on the timing and topic: students can chose to present a theorem or overview a paper close to their own interests and relevant to the course. I will also be supplying a list of suggested topics/papers.

As an alternative, students may instead run some computer experiments or calculations, and either present this to the class, or write a brief (at most 10 page) summary of what they have done. Some suggested computer exercises will be distributed.

By Friday March 1 (the day before Spring break), a one-page description of your talk (typed in tex) is due. Of course, I am happy to discuss ideas (much!) earlier than this, and if you would like to do something early in the semester, just let me know!

Note that the course material touches on issues studied by many professors in our department, most notably (besides me) Mel Hochster, Mircea Mustata, Bhargav Bhatt, and David Speyer. You might consider studying one of their papers or asking them for ideas on lecture topics as well.

In addition, students who are not yet candidates are expected to keep a notebook (or latex file) in which they solve or attempt some of the exercises that are assigned periodically, or check details omited from lecture. Computer exercises, related readings, or any other related topics can freely be included. Please come discuss questions/confusions/ideas in office hours!

Anyone who attends all (or nearly all) the lectures is guaranteed the grade of B. For some form of A, students need to do some work, but there is considerable freedom and flexibility. I certainly do not expect students to solve every single exercise, though of course this is great if you do. You should follow your own interests and explore!

Testing and Disability: If you think you need an accommodation for a disability, please let me know as soon as possible.