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Fenchel–Nielsen coordinates vs Fock–Goncharov coordinates

This is done, for shearing coordinates (both for pants decompositions and triangulations), by Theorem 1.3 of the paper "Shearing coordinates and convexity of length functions on Teichmüller space&...
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4 votes
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Curves having only one linear system realizing its gonality

A generic $d$-gonal curve of genus $g$ satisfies this property unless $g \leq 2d-2$. So the only possible restriction for curves with this property is $g \geq 2d-1$, which I believe follows from Brill-...
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7 votes

Explicit triples of isomorphic Riemann surfaces

There are indeed very few pairs (except spheres with 3 or 4 singularities, or tori, and what can be obtained from them by finite coverings, where correspondence 2)-3) is completely explicit. See: H. P....
6 votes

Explicit triples of isomorphic Riemann surfaces

A classical, wonderful example in which is possible to explicitly see all the three descriptions is the Klein quartic.
8 votes

Explicit triples of isomorphic Riemann surfaces

A particular example is that of the Lawson surface $\xi_{g,1}$ of genus $g$. As defined here, it is a compact minimal surface in the 3-sphere, obtained by reflecting the solution of the Plateau ...

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