The following situation showed up this spring in my research, and although it ended up not seeming to lead anywhere, I still think there’s something deeper going on.
Consider an algebraic curve defined over . I should emphasize that this is a complex curve with real structure, that is, a Riemann surface with an action of complex conjugation. The fixed points of this action are the curve’s real points.
There are a handful of interesting topological questions we can ask about algebraic curves defined over . For instance:
- Does have any real points at all? Is smooth? What about ?
- How do the real points of sit inside its complex points?
- If is smooth, then it is a disjoint union of circles. How many circles are there? We’ll call this quantity .
I’m just going to assume is smooth from here on out. (If is smooth, so is .)
The third bullet point has an interesting connection to the algebraic structure of : it turns out that there can be at most real connected components. (This follows from (1) Mayer-Vietoris, and (2) the next thing I’ll discuss.)
The second bullet point is more subtle. One way to put the question is to look at the space . It turns out that there are two broad possibilities about this space, and they are partly governed by some numerology:
Theorem. The space is either connected or has exactly two connected components. (In the latter case, we say “ is disconnected by its real points.”) Furthermore:
- If is disconnected by its real points, we always have
- If , then must be disconnected by its real points.
Roughly speaking, this theorem follows from the Mayer-Vietoris sequence, plus the observation that is disconnected by its real points if and only if the quotient of by complex conjugation is orientable.
The first part is more elementary, though. Here is a proof: let be a real point. Take a small disk around , which is cut into two pieces by complex conjugation. Let be the connected component of containing one half of the disk. Let be the connected component of the other half. (It is possible that .) Then complex conjugation maps to , and their boundaries coincide:
Then it is clear that is closed; but in fact it is also open! Hence it must be all of . Thus there are at most two connected components.
I can now state the particular situation I was in, which seems to have interesting connections to this theorem.
Situation: Let’s assume is smooth, and that we have a finite map such that the preimage of any real point consists entirely of reduced, real points. In this case,
and the corresponding map of real points, , is a covering map. We’ll see in a moment that this covering map, combined with the Riemann-Hurwitz formula, leads to something mysterious.
First of all, with this setup, is automatically disconnected by its real points: take the preimages of the upper and lower hemispheres of . So it is the case that
Let’s pick a base point ; its preimage consists of real points. We have the monodromy action
which is an action of on a set of size , i.e. a permutation . Moreover, the number of orbits of is the number of real connected components of .
Fact. Let be a permutation with cycle type . We have
where the sign of a permutation is 0 or 1.
Finally, we have the Riemann-Hurwitz formula, which in this case says
where is the ramification divisor on , that is, the formal sum of the ramification points (with multiplicity).
Combining equations (1), (2) and (3) gives the following intriguing numerical fact:
The quantity has an important geometrical meaning here: all the ramification points of occur in complex conjugate pairs, so this is the number of complex conjugate pairs of ramification points.
So, my question is: what does this mean? Is there a simple topological phenomenon linking pairs of ramification points to (sizes of) monodromy orbits? Ideally, I would like a phenomenon that isn’t just mod 2.