Deformation spaces of representations of Galois groups of p-adic fields with p-adic Hodge theoretic conditions play a central role in many arithmetic questions, such as the weight part of Serre's conjecture and modularity lifting theorems, yet they remain poorly understood in most situations, especially in dimension >2. I will describe how (some of) these deformation spaces can be algebraized to nice finite algebraic varieties, and how this allows one to deduce structural properties about them, such as the geometric Breuil-Mezard conjecture. This is joint work in progress with D. Le, B. Levin and S. Morra.