Title: Toric Varieties
Prerequisites: 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.
Course Description: Among the most accessible classes of algebraic varieties
are toric varieties. 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.
This course would introduce the students to toric varieties,
emphasizing their fundamental combinatorial nature while focusing
on concrete examples and explaining some of their applications.
Here is an Outline:
-Toric ideals
Generation of toric ideals
Groebner bases
Computation
-Affine toric varieties
-Projective toric varieties
Orbit decomposition and Limiting behavior
-Lattice Polytopes
Algebraic-Combinatorial Geometric Dictionary
Normality of Sturmfellian Toric Varieties
-Real projective toric varieties
Irrational toric varieties
Algebraic moment map
-Geometric combinatorics, lattices and fans
-Abstract Construction of Toric Varieties
Arithmetic toric varieties
-Cox/Delzant quotient construction
Cox Coordinate Ring
Line bundles on toric varieties
Toric Degenerations
Avg Amount of Time Dedicated Per Week: