Algebraic Geometry 3: Toric Varieties

Toric varieties are a class of algebraic varieties, which can be constructed and studied via combinatorial methods. They serve as concrete examples to study and to test conjectures.

Prerequisites:

Algebraic Geometry 2 (SS 2018) by Prof. Jan Kohlhaase, see: AlgGeo2

Content:

We will mostly follow the rst three chapters of Fulton's book, which give the basics of toric varieties. If time permits, we will also study a few specific topics.

Time and place:

Vorlesung: Monday 10-12 WSC-S-3.14

Vorlesung: Wednesday 10-12 WSC-S-3.14

Übung: Tuesday 8-10 WSC-N-U-4.03

References:

Fulton William. Introduction to Toric Varieties. (Annals of Mathematics Studies 131). Princeton University Press, 1993. See: due-library e-book

Danilov Vladimir. The geometry of toric varieties. See: danilov-Toric

Further readings:

Cox David, Little John, Schenck Henry. Toric Varieties. (Graduate Studies in Mathematics 124). American Mathematical Soc., 2011.

Comment: This is a comprehensive book on Toric varieties.

 

Kempf George, Knudsen Finn Faye, Mumford David, Bernard Saint-Donat. Toroidal embeddings I. (Lecture notes in mathematics 339). Springer-Verlag, 1973.

Comment: Toroidal embeddings are varieties which look like a toric variety locally.

 

Ogus, Arthur. Lectures on logarithmic algebraic geometry. (Cambridge Studies in Advanced Mathematics 178). Cambridge University Press, 2018. 

Comment: Toric varieties and toroidal embeddings are important examples of log schemes. In fact, log schemes are closely related to toric schemes locally. More precisely, the very important notion of chart of log scheme is modeled on toric scheme.