## 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 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 .

Advertisements