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 .