An Introduction to Algebraic Geometry : Chapter 1, Section 2 - Projective Varieties
Channel: xdd1
Date: 2022-12-05

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 :

https://docs.google.com/document/d/1ZhC3bVycv8qc075CPYA3saRAJTSlGiedPGiNbpevxP4/edit?usp=sharing - Video Script
https://docs.google.com/presentation/d/1TUqybnZWcNTE1Cn3110UViICreetjQUdV9skc8b0bGo/edit?usp=sharing - Slides
http://abstract.ups.edu/download.html - Abstract Algebra, Theory and applications (Tom Judson)
https://www.math.ru.nl/~mueger/topology.pdf - Topology for the Working Mathematician (Michael Müger)
https://aareyanmanzoor.github.io/assets/matsumura-CA.pdf - Commutative Algebra (Hideyuki Matsumura, re-texed by the TEXromancers)
https://www.youtube.com/c/OliverLugg - Oliver Lugg's channel, go learn category theory there if you haven't already

Before the chapters, two things : first, I want to elaborate on an issue that came up in the video. During the discussion of dimensionality results for the homogeneous coordinate ring, I had made multiple blunders when setting up a map that would become an isomorphism - you may ask why I don't just fix it and instead kept the blunders in the video. My current view on that is to keep it as a record of where I went wrong in the solving of this problem, since I make it policy to solo solve these problems so it can be insightful to see where one can go wrong in such an isolated environment, while also pointing out my mistake to show one possible pitfall to not fall into (even if it should have been an easy one to avoid), and hopefully the signaling explicitly in the voiceover makes that clear. However, I do understand disagreement on that, especially since this is in part documentation, so having... well, incorrect documentation of the intended solution on my end is a strictly bad thing in that regard. Towards that end, a complete solution to that problem and the explanation behind it is given in this post by mathmusings : http://hartshornesolutions.blogspot.com/2015/06/chapter-1-exercise-26-dimension-of.html . 

[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)