  We study projective schemes arising from eigenvectors of tensors, called
eigenschemes. After some general results, we give a birational description of
the variety parametrizing eigenschemes of general ternary symmetric tensors and
we compute its dimension. Moreover, we characterize the locus of triples of
homogeneous polynomials defining the eigenscheme of a ternary symmetric tensor.
Our results allow us to implement algorithms to check whether a given set of
points is the eigenscheme of a symmetric tensor, and to reconstruct the tensor.
Finally, we give a geometric characterization of all reduced zero-dimensional
eigenschemes. The techniques we use rely both on classical and modern complex
projective algebraic geometry.
