  We consider the question of how the eigenvarieties of a hypergraph relate to
the algebraic multiplicities of their corresponding eigenvalues. Specifically,
we (1) fully describe the irreducible components of the zero-eigenvariety of a
loose $3$-hyperpath (its "nullvariety"), (2) use recent results of
Bao-Fan-Wang-Zhu to compute the corresponding algebraic multiplicity of zero
(its "nullity"), and then (3) for this special class of hypergraphs, verify a
conjecture of Hu-Ye about the relationship between the geometric
(multi-)dimension of the nullvariety and the nullity.
