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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.