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Ramification and Monodromy

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 S defined over \mathbb{R}. 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 \mathbb{R}. For instance:

  • Does S have any real points at all? Is S(\mathbb{R}) smooth? What about S(\mathbb{C})?
  • How do the real points of S sit inside its complex points?
  • If S(\mathbb{R}) is smooth, then it is a disjoint union of circles. How many circles are there? We’ll call this quantity \eta(S).

I’m just going to assume S is smooth from here on out. (If S(\mathbb{C}) is smooth, so is S(\mathbb{R}).)

The third bullet point has an interesting connection to the algebraic structure of S: it turns out that there can be at most g(S) + 1 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 S(\mathbb{C}) - S(\mathbb{R}). It turns out that there are two broad possibilities about this space, and they are partly governed by some numerology:

Theorem. The space S(\mathbb{C}) - S(\mathbb{R}) is either connected or has exactly two connected components. (In the latter case, we say “S is disconnected by its real points.”) Furthermore:

  • If S is disconnected by its real points, we always have \eta(S) = g(S) + 1 \ (\mathrm{mod} \ 2).
  • If \eta(S) = g(S) + 1, then S must be disconnected by its real points.

Roughly speaking, this theorem follows from the Mayer-Vietoris sequence, plus the observation that S is disconnected by its real points if and only if the quotient of S by complex conjugation is orientable.

The first part is more elementary, though. Here is a proof: let s \in S be a real point. Take a small disk around s, which is cut into two pieces by complex conjugation. Let U_1 be the connected component of S(\mathbb{C}) - S(\mathbb{R}) containing one half of the disk. Let U_2 be the connected component of the other half. (It is possible that U_1 = U_2.) Then complex conjugation maps U_1 to U_2, and their boundaries coincide:

B = \partial U_1 = \partial U_2.

Then it is clear that U_1 \cup B \cup U_2 is closed; but in fact it is also open! Hence it must be all of S(\mathbb{C}). 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 S is smooth, and that we have a finite map f : S \to \mathbb{P}^1 such that the preimage of any real point consists entirely of reduced, real points. In this case,

f^{-1}(\mathbb{RP}^1) = S(\mathbb{R}),

and the corresponding map of real points, f : S(\mathbb{R}) \to \mathbb{RP}^1, 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, S is automatically disconnected by its real points: take the preimages of the upper and lower hemispheres of \mathbb{CP}^1. So it is the case that

(1)\ \ \eta(S) = g(S) + 1 \ (\mathrm{mod} \ 2).

Let’s pick a base point z \in \mathbb{RP}^1; its preimage consists of d = \deg(f) real points. We have the monodromy action

\pi_1(\mathbb{RP}^1) \times f^{-1}(z) \to f^{-1}(z),

which is an action of \mathbb{Z} on a set of size d, i.e. a permutation \omega \in S_d. Moreover, the number of orbits of \omega is the number of real connected components of S(\mathbb{R}).

Fact. Let \omega \in S_d be a permutation with cycle type \lambda_1, \ldots, \lambda_k. We have

(2)\ \ \#\mathrm{orbits}(\omega) = d - \sum (\lambda_i - 1) \equiv d - \mathrm{sign}(\omega)\ (\mathrm{mod}\ 2),

where the sign of a permutation is 0 or 1.

Finally, we have the Riemann-Hurwitz formula, which in this case says

(3)\ \ g(S) - 1 = -d + \frac{1}{2}deg(R),

where R is the ramification divisor on S, that is, the formal sum of the ramification points (with multiplicity).

Combining equations (1), (2) and (3) gives the following intriguing numerical fact:

\frac{1}{2} \deg(R) \equiv \sum (\lambda_i - 1) \equiv \mathrm{sign}(\omega) \ (\mathrm{mod}\ 2).

The quantity \frac{1}{2} \deg(R) has an important geometrical meaning here: all the ramification points of f 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.

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