Computational Algebra Seminar: Brown -- Introduction to toric geometry

Speaker: Gavin Brown (Kent, Sydney)
Title: Introduction to toric geometry: Part 1: cones and fans
Time & Place: 4-5pm, Thursday 22 Jan, Carslaw 535.

Abstract: Part 1: cones and fans.
Affine 2-space k^2 (over a field k) has the polynomial ring
k[x,y] as its ring of (polynomial) functions.
If sigma in R^2 is the first quadrant and M = Z^2 is the
integral lattice in R^2, then I can regard the set of all monomials
x^i*y^j (and their multiplicative structure) as $sigma  \cap M$.
Thus $k[x,y] = k[  sigma  \cap M ]$, and so I think of sigma
(sitting in $M \otimes \Q$) as determining k^2.

Toric geometry makes a lot out of this by allowing other
choices of cones than sigma, allowing collections of cones,
and considering maps between cones. Many features of
geometry can be interpreted as questions about cones in
lattices. I’ll go through the famous first examples.