Toric Geometry Reading Course (Spring 2016)

Wednesdays 10am-12pm in K9509

Lecture topics and tentative dates are listed below. Tentative speakers are in italics. You are not expected to present every detail from your assigned section, but instead to paint the big picture. If choosing between an example illustrating a proof idea, or the actual proof, always present the example!

Tentative Date Topic Speaker
16.01.06 Introduction and Motivation

Bernstein-Kushnirenko Theorem. The algebraic torus. Stanley's g-theorem. Examples of toric varieties.

 Nathan
16.01.13 Affine Varieties I

Explain the contents of [CLS] pp 3-5 up to normalization, including a discussion of irreducibility. Sections 3.1-3.3, 6.1-6.2, and 7.1 of [Hassett] provide good background. See also [Hart] I.1.

Matthew
16.01.20 Affine Toric Varieties

[CLS] section 1.1.

Avi
16.01.27 Polyhedral Cones

[CLS] section 1.2. See also [Fulton] 1.2 and 1.3.

 Marni
16.02.03 Properties of Affine Varieties

[CLS] pp 5 (Normal Affine Varieties) to end of section. Now is a good time to define dimension, see [Hart] pp 5-6. See also e.g. 7.4 and 7.7 of [Hassett].

 Sasha
16.02.17 Properties of Affine Toric Varieties

[CLS] 1.3 through pp 41. See also [Fulton] 2.1.

 Charles
16.02.24 Equivariant Maps

[CLS] 1.3 pp 41 to end of section.

 Jens
16.03.02 Abstract Varieties

[CLS] 3.0.

 Brett
16.03.09 Abstract Toric Varieties

[CLS] 3.1. See also [Fulton] 1.4.

 Nathan
16.03.16 The Orbit-Cone Correspondence

[CLS] 3.2. See also [Fulton] 3.1.

 Matthew
16.03.23 Basics on Divisors; Toric Divisors

[Fulton] 3.3 and [CLS] pp 155-160, 170-171. Weil divisors, discrete valuations, principal divisors, Cartier divisors, invariant Weil and Cartier divisors on toric varieties, the multiplicity formula for a character.

 Sasha
16.03.30 Properties of Divisors

Global sections of a divisor (don't use the word sheaf!). Piecewise linear functions and the global sections of a toric divisor [Fulton] pp 65-66, see also [CLS] pp 183-184, 189-190. Global generation of a (toric) Cartier divisor [Fulton] 67-68 or [CLS] 262-267. Define the degree of a (nef) divisor to be the normalized leading coefficient of its Hilbert polynomial. Volume formula for degree of toric divisors (talk to Nathan).

 Avi
16.04.06 Intersection Theory and Proof of Bernstein-Kushnirenko

Basics of intersection theory for divisors (talk to Nathan). For proof of BK, see [Fulton] pp 114-116 and section 5.5.

 Nathan

References