Algebraic geometry is often presented as the study of zeroes of polynomial equations. But it's really about something much deeper: the duality between abstract algebra and geometry.

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A HUGE HUGE thank you to Faisal Al-Faisal for working with me on the script and storyboard for this video!

And another thank you to Davide Radaelli for helpful conversations when making this video.

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CORRECTIONS:

At 4:26, I mistakenly wrote that g(1,1)=-2. This is a typo! The corrected version is g(1,-1)=-2.

SOURCES and REFERENCES for Further Reading!

(a) “A guide to plane algebraic curves” by Keith Kendig. It’s written in a very elementary style and has lots of really captivating diagrams throughout. If you look at the table of contents, it starts off with lots of examples that only require elementary algebra. And by the end, it actually gets to some pretty deep theorems in algebraic geometry.

(b) "Ideals, Varieties, and Algorithms” by Cox, Little, O’ Shea. This book does not assume any knowledge of abstract algebra and teaches everything from the ground up. It is a very nice book with plenty of computational examples and exercises.

(c) “Algebraic Geometry and Arithmetic Curves” by Qing Liu. This books is all about schemes and Spec. It's a rather terse theorem-proof style book, but it is beautifully written and has lots of exercises.

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MUSIC CREDITS:
The song is “Taking Flight”, by Vince Rubinetti.
https://www.vincentrubinetti.com/

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What is algebraic geometry?: (0:00)
Coordinate Ring: (3:04)
How algebra detects reducibility: (3:54)
How algebra detects a node: (5:15)
Schemes!: (8:00)