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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 $\eta(S)$.