TITLE: flat metrics on the 2-sphere with conical singularities
QUESTION [7 upvotes]: Consider some flat metric on $S^2$ with a fixed finite number of conical singularities $p_1,\ldots,p_n$.
What is the moduli space of such metrics up to isometry? In particular what is its dimension?

REPLY [3 votes]: Another example: any meromorphic quadratic differential on the Riemann sphere (with at most simple poles) gives such a metric. There the cone angles are always positive integer multiples of $\pi$ . 
These moduli spaces have been extensively studied from many different points of view.