# Basic Algebraic Geometry with Dr. Bob

## Video Lectures

Displaying all 10 video lectures.
 Lecture 1 Play Video Toric Varieties 1 - Affine Varieties over CBasic Algebraic Geometry: We define affine varieties over the complex numbers, the Zariski topology on C^n, and the Zariski closure of a subset X in C^n. Prerequisites: A first year of graduate algebra, but the course was run with advanced undergraduates in mind. Chapter 0 forthcoming, which will give a better idea of the big picture. Lecture 2 Play Video Toric Varieties 2 - Affine Toric VarietiesBasic Algebraic Geometry: We define affine toric varieties. There are four ways to characterize ATVs and we note three here: as the Zariski closure of a torus in C^s; as the space for a torus group action; and as the variety of a toric ideal. Lecture 3 Play Video Toric Varieties 3 - Coordinate Rings and MorphismsBasic Algebraic Geometry: We introduce the notions of coordinate rings and morphisms, and use properties of tori as examples. Lecture 4 Play Video Toric Varieties 4 - Spec(R) and Affine SemigroupsBasic Algebraic Geometry: In this part, we introduce Spec(R) and affine semigroups. This allows us to give yet another characterization of affine toric varieties in terms of affine semigroups. Lecture 5 Play Video Toric Varieties 5 - Polyhedral Cones for Affine Toric VarietiesBasic Algebraic Geometry: We review the basic properties of convex polyhedral cones and give an application to affine toric varieties. Lecture 6 Play Video Toric Varieties 6 - Faces and LocalizationBasic Algebraic Geometry: We give an overview of convex geometry for polyhedral cones, including Gordan's Lemma and the Separation Lemma. A first connection to toric varieties is given by localization. Lecture 7 Play Video Toric Varieties 7 - Overview of Smoothness and NormalityBasic Algebraic Geometry: In this part, we give a general overview of smoothness and normality for affine varieties. The second part will cover the case of affine toric varieties. (Note: BAG1.6 is in preparation and not needed yet.) Lecture 8 Play Video Projective Toric Varieties - Part 1Basic Algebraic Geometry: We define complex projective space, projective varieties, and projective toric varieties. For PTVs, we identify the character lattice and lattice of one-parameter subgroups. Lecture 9 Play Video Projective Toric Varieties - Part 2Basic Algebraic Geometry: Continuing from the previous video, we give several equivalent conditions for when the cone over the projective toric variety X_A is equal to the affine toric variety Y_A. Lecture 10 Play Video Affine Pieces of Projective Toric VarietiesBasic Algebraic Geometry: This part has three goals: formalizing some notions used in the previous parts; noting a result about tori; and begin study of affine pieces of projective toric varieties.