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 ≥9, filling the gap between rank ≤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