Secant varieties in particular of Segre and Segre-Veronese varieties; rank and border rank of tensors. Linear systems of hypersurfaces of P KU Leuven
In the direction of solving the problem one has to describe the base locus of L and to understand the contribution given by each component of the base locus to the speciality of L. This is related to the algebraic conjectures by Froberg and Iarrobino that give a predicted value for the Hilbert series of an ideal generated by general forms of given degree