P¹ is simply connected

This is a cute proof that I ran into of the simple connectedness of . It does not use Riemann–Hurwitz or differentials, and instead relies on a purely geometric argument.

Lemma. Let be an algebraically closed field. Then is simply connected.

Proof. Let be a finite étale Galois cover with Galois group . We have to show that is an isomorphism. The diagonal is ample, so the same goes for the pullback to [Hart, Exc. III.5.7(d)]. In particular, is connected [Hart, Cor. III.7.9].

But is isomorphic to copies of because the action

is an isomorphism. If is connected, this forces , so is an isomorphism.

The proof actually shows that if is a smooth projective variety such that is a set-theoretic complete intersection of ample divisors, then is simply connected.

Example. For a smooth projective curve of genus , the diagonal cannot be ample, as . We already knew this by computing the self-intersection , but the argument above is more elementary.

References.

[Hart] Hartshorne, Algebraic geometry. GTM 52, Springer, 1977.

Number theory is heavy metal

As some of you may be aware, I am a musician as well as a mathematician. I often like to compare my experiences between the two. For example, I found that my approach to the creative process is not dissimilar (in both, I work on the more technical side, with a particular interest in the larger structure), and I face the same problems of excessive perfectionism (never able to finish anything).

This post is about some observations about the material itself, as opposed to my interaction with it.

Universal donor.

A look at the arXiv listing for algebraic geometry shows the breadth of the subject. Ideas from algebraic geometry find their way into many different areas of mathematics; from representation theory and abstract algebra to combinatorics and from number theory to mathematical physics. But when I say algebraic geometry is a universal donor (of ideas, techniques, etc.), I also mean that its applications to other fields far outnumber the applications of other fields to algebraic geometry, and that the field of algebraic geometry is largely self-contained.

Much the same role is played by classical music inside musical composition. Common practise theory is used throughout Western music, whether you’re listening to hip hop, trance, blues, ambient electronic, bluegrass, or hard rock. Conversely, the influence of popular genres music on [contemporary] classical is comparatively little. One could therefore argue that classical music is a universal donor in the field of musical composition.

Universal receptor.

The opposite is the case for number theory. Another vast area, number theory often uses ingenious arguments combining ideas from algebra, combinatorics, analysis, geometry, and many other areas. In practical terms, the amount of material that a number theorist needs to master is immense: whatever solves your particular problem.

So is there any musical genre that plays a similar role? I claim that metal fits the bill. With a vast list of subgenres including thrash metal, black metal, doom metal, progressive metal, death metal, symphonic metal, nu metal, grindcore, hair metal, power metal, and deathcore, metal writing often contains creative combinations of other genres, from classical music to ambient electronic, and from free jazz to hip hop. As Steven Wilson of Porcupine Tree said about his rediscovery of metal:

I said to myself, this is where all the interesting musicians are working! Because for a long time I couldn’t find where all these creative musicians were going… You know in the 70’s they had a lot of creative musicians like Carlos Santana, Jimmy Page, Frank Zappa, Neil Young, I was thinking “where are all these people now?” and I found them, they were working in extreme metal.

I sometimes think of metal as a small microcosmos reflecting the full range of Western (and some non-Western) music, tied together by distorted guitars and fast, technical drumming.

In summary, algebraic geometry is classical music, and number theory is heavy metal.