Explaining genus to students

Question

I need to do a presentation on my thesis, which is in arithmetic geometry. This presentation is meant for all students of mathematics, but I will assume some knowledge of abstract algebra (i.e. groups, rings, fields).

In the first part of my presentation, I want to explain what Faltings' Theorem is and why it is important (and nice). Defining a curve $C$ is not too hard to do (although I skip over some details) and defining $C(k)$ (or I might just stick with $C(\mathbb{Q})$) for $k$-rational points on the curve. I then introduce some examples of interesting curves, such as the circle, an elliptic curve, and hyperelliptic curves.

I'm having difficulty however, to explain in simple terms the concept of the genus $g(C)$ of the curve $C$. Most intuitive might be too think of it topologically "as the number of holes" and use the Weierstrass $\wp$ function to explain how we can think of the elliptic curve $\mathcal{E}$ as some torus $\mathbb{C}/\Lambda$. This takes up quite a lot of time, though.

Is there an easier, shorter, way to explain the genus $g(C)$ or might I be able to skip the whole genus and explain Faltings' in some other way?

Answer 1

As a fellow arithmetic geometer working on some problems related to Faltings' theorem, I've given a fair number of explanations of algebraic curves to non-specialists, so I can tell you how I usually do it.

One thing I never do unless I'm talking to number theorists or algebraic geometers is get into the details of complex tori, the Weierstrass $\wp$-function, or anything like that. What is usually do for explaining the genus is something like this:

1. Describe the genus as the "number of holes" without giving a precise definition.
2. Draw a few examples on the board and say what their genus is. I usually do a sphere, a torus, and a genus two or three curve. (Drawing tip: draw the holes first, then the visual boundary.)
3. Briefly mention that what I'm drawing here is the set of complex solutions to the defining equations of the curve, not just the real solutions, which is why it looks like a surface rather than a curve. (For a non-mathematician audience, I might leave this out entirely and only bring it up if someone asks why my drawings look 2-dimensional. If the audience knows complex analysis, I'll mention these are just compact Riemann surfaces.)

Most people can visualize "number of holes" just fine once you've drawn a few examples. If you wanted to spend more time on the concept of genus, you could give examples of how to construct a genus two curve by gluing sides of an octagon, like here: https://plus.maths.org/content/making-double-torus-octagon (I've never taken this approach in a talk, but it seems like a reasonable thing to do.)

As for actually motivating why higher genus curves behave differently, I like to handwave a bit about the fundamental group: I'll note how any loop on a sphere can be contracted to a point, loops on a torus can't always be contracted but still have a simple description, and loops on a higher genus curve have a much more complicated structure (I'll say "nonabelian" here if they know groups). Since my research is on Kim's non-abelian Chabauty method, which is motivated by anabelian geometry, this is a natural perspective for me, but may be less natural if you're viewing it from Faltings' or Vojta's perspectives (which I don't think directly use the fundamental group).