Generic Global Rigidity in ₚ-Space and the Identifiability of the p-Cayley-Menger Varieties

The celebrated result of Gortler-Healy-Thurston (independently, Jackson-Jord\'an for d=2) shows that the global rigidity of graphs realised in the d-dimensional Euclidean space is a generic property. Extending this result to the global rigidity problem in ₚ-spaces remains an open problem. In this paper we affirmatively solve this problem when d=2 and p is an even positive integer. A key tool in our proof is a sufficient condition for the d-tangentially weakly non-defectiveness of projective varieties due to Bocci, Chiantini, Ottaviani, and Vannieuwenhoven. By specialising the condition to the p-Cayley-Menger variety, which is the ₚ-analogue of the Cayley-Menger variety for Euclidean distance, we provide an ₚ-extension of the generic global rigidity theory of Connelly. As a by-product of our proof, we also offer a purely graph-theoretical characterisation of the 2-identifiability of an orthogonal projection of the p-Cayley-Menger variety along a coordinate axis of the ambient affine space.