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Math 622: Toric varieties, Grassmannians,
Instructor:
Frank Sottile |
Office Hours : |
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11:00–12:00 |
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Among the most accessible classes of algebraic varieties are toric varieties and homogeneous spaces, specifically Grassmannians. This is fortunate for these are also among the most commonly encountered outside of algebraic geometry within mathematics and in the applications of algebraic geometry. Toric varieties in particular are currently widely studied in algebraic geometry and its applications.
In this course we will introduce and develop the elementary theory of these two classes of varieties, emphasizing their fundamental combinatorial nature while focusing on concrete examples and explaining some of their applications.
Because of the elementary nature of these varieties, the prerequisite will be graduate algebra, although courses in commutative algebra and algebraic geometry will be helpful.
I feel that in advanced graduate classes, the students get out of the class what they put in. Consequently, I do not assign regular homework to be collected and marked. (But I will mention exercises, which are intended for you to fill in gaps in the presentation, thereby learning a bit more.) At the end of the term, students will present projects; I am starting a file with some suggestions here.
Schedule Toric Varieties
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Text Book: I do not plan to follow a text book for this class. I will be writing notes throughout the semester, these are for part of a book I am writing with Thorsten Theobald, Applicable Algebraic Geometry. The sections on toric varieties will be written this semester, while the sections on Grassmannians are being revised. These will be made available as they are completed. While there are several attractive introductions to toric varieties (see below), I am afraid that I cannot recommend a single source for Grassmannians, as there has not been a definitive treatment of these, except as chapters in various books. Other Sources:
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