Toric Varieties

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Time and Place:

Lectures: Monday, 10-12 c.t., Room B 045.

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Abstract:

Toric varieties are algebraic varieties with an algebraic torus acting on with a Zariski open dense orbit. These geometric objects may be defined using purely combinatorial data (lattices, cones, fans, etc.) and provide a rich class of explicitly computable examples in algebraic geometry. In these lectures we will explore various aspects of this subject, starting with the combinatorial description of toric varieties, and see how various algebro-geometric notions and constructions, such as smoothness, completeness, Weil divisors, blow-ups, resolution of singularieties, etc., are reflected in the respective combinatorics.
If time permits we will also discuss the presentation of the Chow ring of a smooth toric variety and also the structure of the derived category of a smooth complete toric variety.

Prerequisites:

This course will be of interest to Master students majoring in Algebra. I will assume that the students have some familiarity with Algebraic Geometry and Commutative Algebra. Some knowledge of Algebraic Topology will be also helpful.
The course will be taught in English.

Exam:

The exam will take place on Monday 15 July from 10:15 until 12:00 in room B 045. No materials are permitted in the exam. Please ensure you bring with you an appropriate identity document with photo. If you are considering taking the exam, please register by sending me (ananyevs@math.lmu.de) a short email with your name and Matrikelnr.

Notes:

Handwritten notes for the course

References:

[1] David A. Cox, John B. Little, and Henry K. Schenck. Toric varieties. American Mathematical Soc., 2011
[2] William Fulton. Introduction to toric varieties. Number 131. Princeton University Press, 1993
[3] Vladimir Danilov. The Geometry of Toric Varieties. In: Russian Mathematical Surveys 33:2 (1978), pp. 97–154

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Ph.D. Alexey Ananyevskiy