On the description of identifiable quartics

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 > 8, filling the gap between rank < 9, 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.