  We show how methods of algebraic geometry can produce criteria for the
identifiability of specific tensors that reach beyond the range of
applicability of the celebrated Kruskal criterion. More specifically, we deal
with the symmetric identifiability of symmetric tensors in
Sym$^4(\mathbb{C}^{n+1})$, i.e., quartic hypersurfaces in a projective space
$\mathbb{P}^n$, that have a decomposition in 2n+1 summands of rank 1. This is
the first case where the reshaped Kruskal criterion no longer applies. We
present an effective algorithm, based on efficient linear algebra computations,
that checks if the given decomposition is minimal and unique. The criterion is
based on the application of advanced geometric tools, like Castelnuovo's lemma
for the existence of rational normal curves passing through a finite set of
points, and the Cayley-Bacharach condition on the postulation of finite sets.
In order to apply these tools to our situation, we prove a reformulation of
these results, hereby extending classical results such as Castelnuovo's lemma
and the analysis of Geramita, Kreuzer, and Robbiano, "Cayley-Bacharach schemes
and their canonical modules", Trans. Amer. Math. Soc. 339:443-452, 1993.
