  In this paper we study the identifiability of specific forms (symmetric
tensors), with the target of extending recent methods for the case of $3$
variables to more general cases. In particular, we focus on forms of degree $4$
in $5$ variables. By means of tools coming from classical algebraic geometry,
such as Hilbert function, liaison procedure and Serre's construction, we give a
complete geometric description and criteria of identifiability for ranks $\geq
9$, filling the gap between rank $\leq 8$, covered by Kruskal's criterion, and
$15$, the rank of a general quartic in $5$ variables. For the case $r=12$, we
construct an effective algorithm that guarantees that a given decomposition is
unique.
