  Many-body wavefunctions usually lie in high-dimensional Hilbert spaces.
However, physically relevant states, i.e, the eigenstates of the Schr\"odinger
equation are rare. For many-body systems involving only pairwise interactions,
these eigenstates form a low-dimensional subspace of the entire Hilbert space.
The geometry of this subspace, which we call the eigenstate moduli problem is
parameterized by a set of 2-particle Hamiltonian. This problem is closely
related to the $N$-representability conditions for 2-reduced density matrices,
a long-standing challenge for quantum many-body systems. Despite its
importance, the eigenstate moduli problem remains largely unexplored in the
literature. In this Letter, we propose a comprehensive approach to this
problem. We discover an explicit set of algebraic equations that fully
determine the eigenstate spaces of $m$-interaction systems as projective
varieties, which in turn determine the geometry of the spaces for representable
reduced density matrices. We investigate the geometrical structure of these
spaces, and validate our results numerically using the exact diagonalization
method. Finally, we generalize our approach to the moduli problem of the
arbitrary family of Hamiltonians parameterized by a set of real variables.
