The fourteenth version of the notes is the April 22 version in the usual place. This post covers up to 23.3 (although more is included). The next post should appear around May 14. I continue to be far behind on responding to comments, including a number from the fall, but I am steadily catching up.

This installment includes an extended discussion of genus 1 curves, a starred section on intersection theory, and the start of the chapter on differentials.

The intersection theory section is suitable for public viewing, but will be edited further in response to some comments of Sándor Kovács. Also, Christian Liedtke convinced me that the Hodge Index Theorem deserves inclusion (and proof).

Other news

In 17.4.8, I’ve added the simplest example of a proper nonprojective k-scheme that I know. “Simplest” means something that can be done with as little work as possible, as early as possible. It’s obtained by gluing to P^3’s together along curves, so I need to describe the theory of gluing two schemes together along isomorphic closed subschemes. Surprisingly, it was most
annoying to show that gluing together two finite type things yields something finite type — Jack Hall and Karl Schwede explained how to do this the right way. The argument in the notes is from Karl, and is better than mine. I also have some additional worthwhile comments to add, proposed by Andrew Critch [now done – Apr. 24].

I’d like to write a short section in the curves chapter proving Pappus’ Theorem, Pascal’s “Mystical Hexagon” Theorem, and Poncelet’s Theorem. I haven’t managed to do it yet.

For learners

I realize some of you are writing apologetically that you are still on earlier sections. Please don’t apologize! These notes are not being posted at a reasonable rate at which to work through; their rate and timing is purely a function of when I get them done. Plus you should certainly be thinking about earlier things even as you read new things. So please keep responding/posting/emailing.

If you are reading along, please read from 21.8-21.9 and 23.1-23.3.

21.8 is on curves of genus 1. It is very rich. Showing that elliptic curves (genus 1 curves with k-points) form a group variety without doing any explicit algebra was fun.

21.9 uses elliptic curves to exhibit some long-promised counterxamples (a scheme that is factorial but such that no affine open neighborhood is Spec of a UFD; an affine open subset of an affine scheme that is not distinguished; a Picard group that has no chance of being a scheme; and a variety with non-finitely-generated ring of global sections.

The optional (starred) chapter 22 gives an introduction to intersection theory, including intersection theory on a surface, and Nakai and Kleiman’s criteria for ampleness.

Chapter 23 is on differentials. I try to motivate it geometrically, then develop the theory, then see it in practice. 23.1 is for motivation; 23.2 gives the definitions and first properties, starting with the affine case from three viewpoints; and 23.3 has a number of examples.

Exercises to do: There are now enough explicit examples that I won’t separate out the “explicit example” exercises. I worked hard to prune this down to ten, but as usual, please try others that catch your eye. Here they are: 21.8.B, 21.8.I, 21.8.K. 21.9.A, 23.2.E, 23.2.F, 23.2.H, 23.2.I, 23.3.A, 23.3.D.

For experts

I don’t have much to ask experts here. I’m toying with various terminological experiments, but that’s discussed on the Notation page.

Small question: is it true that for each positive integer n, there is a genus 1 curve over k with no points of degree less than n?

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