Basic Algebraic Geometry with Dr. Bob
Video Lectures
Displaying all 10 video lectures.
Lecture 1![]() Play Video |
Toric Varieties 1 - Affine Varieties over C Basic 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 Varieties Basic 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 Morphisms Basic 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 Semigroups Basic 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 Varieties Basic 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 Localization Basic 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 Normality Basic 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 1 Basic 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 2 Basic 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 Varieties Basic 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. |