Big polygon spaces are compact orientable manifolds defined as the intersection of a product of spheres with a complex hyperplane given by a so-called length vector. They are related to polygon spaces, which appear as their fixed point set under a canonical torus action. What makes big polygon spaces interesting is that they exhibit remarkable new features in equivariant cohomology: The Chang-Skjelbred sequence can be exact for them and the equivariant Poincaré pairing perfect although these spaces fail to be equivariantly formal. We will in particular present a combinatorial formula that expresses the precise syzygy order of the equivariant cohomology of a big polygon space in terms of the length vector. This is in part joint work with Jianing Huang.