Basic Algebraic Geometry with Dr. Bob

Video Lectures

Displaying all 10 video lectures.
Lecture 1
Toric Varieties 1 - Affine Varieties over C
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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
Toric Varieties 2 - Affine Toric Varieties
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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
Toric Varieties 3 - Coordinate Rings and Morphisms
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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
Toric Varieties 4 - Spec(R) and Affine Semigroups
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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
Toric Varieties 5 - Polyhedral Cones for Affine Toric Varieties
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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
Toric Varieties 6 - Faces and Localization
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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
Toric Varieties 7 - Overview of Smoothness and Normality
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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
Projective Toric Varieties - Part 1
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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
Projective Toric Varieties - Part 2
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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
Affine Pieces of Projective Toric Varieties
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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.