Periodic Points in Genus Two: Holomorphic Sections Over Hilbert Modular Varieties, Teichmüller Dynamics, and Billiards

We show that all GL(2, R) equivariant point markings over orbit closures of primitive genus two translation surfaces arise from pairs of points exchanged by the hyperelliptic involution, Weierstrass points, or the golden points on Ω E₅. As corollaries, we classify holomorphically varying families of points over orbifold covers of genus two Hilbert modular surfaces and solve the finite blocking problem on genus two translation surfaces.