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?

April 23, 2011 at 12:20 pm

What is k in the question about genus 1 curves? The answer for k = C(t) is yes: for every n there exists a smooth projective geometrically irreducible genus 1 curve over k with no closed points whose residue field has degree < n over k.

April 24, 2011 at 4:03 am

Fantastic!

April 23, 2011 at 3:01 pm

Can also make examples over any number field (with whatever Jacobian we wish).

April 24, 2011 at 4:03 am

Brian Conrad to the rescue! His explanation (copied in with his permission; my earlier cut-and-paste messed something up):

To explain this, let E be the Jacobian of such a genus-1 curve C, so C is a E-torsor and thereby corresponds to a class in

H^1(k,E)

that is trivial iff C(k) is non-empty (and formation is functorial with respect to extension on k). So you’re really asking for an elliptic curve E over k and a class in H^1(k,E) which is not split by any extension k’/k of degree less than n.

Let’s first consider k varying through p-adic fields (finite extensions of Q_p), so there are only finitely many extensions of a given degree. Thus, for such k an equivalent version of the question is whether all classes in H^1(k,E) are killed by restriction to H^1(k’,E) for some finite extension k’/k; equivalently, is there some finite extension k’/k such that the restriction map

H^1(k,E) —-> H^1(k’,E)

vanishes? Well, we can identify this etale cohomology with Galois cohomology (since torsors acquire a point over a finite Galois extn), and there’s a corestriction map H*(k’,—) —-> H*(k,—) as delta functors on discrete G_k-modules (such as E(k_s)) such that its composition with restriction is multiplication by the degree n of k’ over k. Thus, if the above restriction vanishes then H^1(k,E) is killed by n. However, we have an exact sequence of k-groups

1 —> E[n] —> E —-> E —> 1

(where the map E —-> E is multiplication by n) so H^1(k,E[n]) —> H^1(k,E)[n] is surjective. Now H^1(k,E[n]) is finite (as is Galois cohomology of p-adic fields with any finite coefficients), so H^1(k,E)[n] is finite. Hence, under the above vanishing hypothesis we’d have that H^1(k,E) = H^1(k,E)[n] is finite.

But it is a theorem for Tate, for any abelian variety A over a p-adic field k (= finite extension of Q_p), H^1(k,A) is Q/Z-dual to the compact group A*(k) where A* = dual abelian variety. In particular, H^1(k,A) is Q/Z to the uncountable A*(k) when A is nonzero, so H^1(k,E) is huge (certainly not finite!).

If one uses deeper global input then one can make examples over number fields. In fact, for any nonzero abelian variety A over any number field k and any finite set S of places of k that is “sufficiently large” (depending on the rank of A(k) and a few other things), the subgroup Sha^1_S(k,A) of classes in H^1(k,A) that have a local point at all places outside S is infinite. But its n-torsion subgroup is again finite for any specific n > 0 (if you enlarge S to contain all places over primes dividing n, say), so H^1(k,A) contains classes that are not split by any number field k’/k of degree below whatever bound you choose. That gives examples over k = Q as torsors for whatever elliptic curve you like. (By the way, this sort of thing is one reason that for the conjectural finiteness of Tate-Shafarevich of abelian varieties over global fields, it is crucial to take the set S of ignored places to be empty. Ironically, if one considers torsors for smooth connected linear algebraic groups G rather than abelian varieties, then allowing any finite S is fine; the pointed set Sha^1_S(k,G) turns out to always be finite, due to a lot of deep results in the arithmetic theory of semisimple groups and global class field theory.)