Integrating a holomorphic 1-form ω on a Riemann surface X induces a flat structure on the surface. The group SL(2, R) acts by post-composition; the standard hyperbolic metric on SL(2,R) modulo SO(2) can be used to show that the SL(2,R)-orbit of a pair (X,ω) descends to give the "complexification" of a Teichmüller geodesic of Riemann moduli space. When the SL(2,R)-stabilizer of (X, ω) is a lattice, this complex Teichmüller geodesic is an algebraic curve. A lattice is in particular finitely generated; answering a question of W. Veech, with P. Hubert we gave examples where the SL(2, R)-stabilizer of (X, ω) is not finitely generated. C. McMullen followed with further examples, in particular completing his classification of the images of all SL(2,R)-orbits with X of genus 2. We discuss some of the geometry --- flat, hyperbolic and algebraic --- of the various examples.