Title: Introduction to Toric Varieties

Group actions are essential tools in many areas of mathematics, and sometimes they help reveal deep and hidden properties. Roughly speaking, toric varieties are algebraic varieties with torus actions. Algebraic varieties are the sets of common roots of polynomials. Indeed, algebraic tori are group objects among algebraic varieties.

Torus actions provide several characterizations of toric varieties. In particular, toric varieties can be constructed from a fan, a combinatorial object. Sometimes this combinatorial approach helps understand deep theorems in algebraic geometry concretely. For instance, the resolution of singularities of two-dimensional toric varieties can be understood via a version of continued fractions.

Toric varieties are often conceived sitting in the sweet spot. They are concrete enough to carry out explicit calculations and sophisticated enough to demonstrate general properties. The course aims to study toric varieties with special attention to the dimension two case. If time permits, we will cover computer algebra software such as Sage and Macaulay2. There is no set textbook, and the software and computing resources will be available for free.

The course requires the instructor’s permission to enroll.

Required or strongly suggested courses: Math 6016, Math 2320

Suggested courses and topics: Math 5310, algebraic geometry (algebraic varieties), convex sets

For students having a strong background in Math 4600 or a basic understanding of algebraic geometry, contact the instructor with a brief explanation of your background.

For questions, contact Youngsu Kim at youngsu.kim@csusb.edu.

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