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The celebrated result of Gortler–Healy–Thurston (independently, Jackson–Jordán 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   $\ell _{p}$  -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   $d$  -tangential weak non-defectivity of projective varieties due to Bocci, Chiantini, Ottaviani, and Vannieuwenhoven. By specialising the condition to the   $p$  -Cayley–Menger variety, which is the   $\ell _{p}$  -analogue of the Cayley–Menger variety for Euclidean distance, we provide an   $\ell _{p}$  -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.

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