Six points and a surface

Every compact Riemann surface of genus two is hyperelliptic. It can be written as an algebraic curve

$$y^2 = (x - a_1)(x - a_2)(x - a_3)(x - a_4)(x - a_5)(x - a_6),$$

with the $a_i$ distinct, which exhibits it as a double cover of the Riemann sphere branched over six points. Those six branch points are exactly the Weierstrass points of the surface: the fixed points of the hyperelliptic involution $(x, y) \mapsto (x, -y)$.

Two such surfaces are isomorphic precisely when their sets of branch points differ by a Möbius transformation. So the moduli space $\mathcal{M}_2$ can be identified with unordered configurations of six distinct points on $\mathbb{CP}^1$, modulo $\mathrm{PSL}(2, \mathbb{C})$. Counting dimensions gives $6 - 3 = 3$, the complex dimension of $\mathcal{M}_2$, which agrees with the general formula $3g - 3$.

This is why six points open the home page. Much of the geometry of a genus-two surface is carried by its Weierstrass points, and the Delaunay and Voronoi decompositions they define are a natural route to decomposing moduli space itself.