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Monthly Archives: April 2015

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).