An Introduction to Algebraic Geometry : Chapter 1, Section 2 - Projective Varieties

This is the second part of my playlist going over the content in Robin Hartshorne's book Algebraic Geometry - in this part we go over section two (projective varieties) of chapter one (varieties), covering polynomials that have zero sets with algebraic properties and from there developing a visualization for projective space, the Zariski Topology in the projective case, the open cover of projective space by affine spaces, the relation between ideals and zero sets, projective varieties, relation of the homogeneous coordinate ring to dimension-theoretic results, basic dimension theory of the affine cone, and end off with a discussion of the Segre Embedding.

The collection of links this lecture is as follows :

[CHECK PINNED COMMENT FOR UPDATE ON THE PREVIOUS PARAGRAPH]

Second, I mentioned in the chapter I would discuss more of my general philosophy explicitly in writing here, especially since a lot of the comments I got related to that - I am not going to be doing every exercise, to steady expectations. The goal of this, while partially to document everything in this book in google slides lecture form, is also ultimately to teach, and going over every small technical lemma is just not going to cut it (I don't even think I hit a fourth of the exercises in this section and look how long the exercises took, for example.) As well, I didn't mention this vocally during the discussion, but I want to use the exercises as more free-form ways of approaching problems - this'll be a lot of where I add my style in, which also will lead to mistakes (as shown by the example of the homogeneous coordinate ring), which brings up a big statement I want to say for them - I do not intend to be correct 100% in the exercises unlike my intentions in the main text, I intend to showcase methods and can only confirm the results will be correct. If you've come here for a guide that perfectly weaves through all the exercises... sorry, I'm crashing in style.

Chapters :

00:00-01:28 : Introduction
01:28-12:46 : Spaces of Zeroes
12:46-17:55 : Locally Affine
17:55-19:00 : (Discussion about Philosophy of Exercises, check description for more)
19:00-23:44 : The Homogeneous Case
23:44-35:00 : Proof by Locality [CHECK PINNED COMMENT]
35:00-41:36 : Products, Projectively
41:36-43:41 : End(ing Remarks)

[xdd]

Quickly just clarifying an issue that I realized after publishing, at 28:51 - I had made an edit in the video where I had "corrected incorrect notation" based on the blog post in the description my mathmusings. THAT BLOG POST IS INCORRECT, at least when defining the inverse - to see this, notice that if your polynomial in the affine coordinate ring is not homogeneous, then if you try to apply the mapping the blog post says you won't map into the degree 0 elements. What the blog post missed is that you need to homogeneous-ize the polynomial first before doing that, WHICH IS WHAT IS WRITTEN IN THE SLIDES. Tl;dr, The slides are correct at the timecode, the voiceover is wrong. While I could reupload the video with corrections, I don't want to be annoying in subscription boxes, so for now the video will stay, but (playing my hands a bit early) just know that there will eventually be a definitive way to watch all the lectures for each chapter, and when that comes out it will be corrected. I apologize in advanced for any confusion this brought.

[rafisics]

waiting for the next videos of the playlist

[ubqiyek]

Sir keep showing high quality educational content please

[SPVLaboratories]

okay I just came across this video and I'm saving this for later, thank you so much for the effort you've put into this series