Degenerations and applications : polynomial interpolation and secant degree

The polynomial interpolation problem in several variables and higher multiplicities is a subject that has been widely studied, but there is only a little understanding about the question. What is known, so far, is essentially concentrated in the Alexander-Hirschowitz Theorem which says that a general collection of double points in Pr gives independent conditions on the linear system L of the hypersurfaces of degree d, with a well known list of exceptions. In the first part of this thesis we present a new proof of this theorem which consists in performing degenerations of Pr and analyzing how L degenerates. Our construction gives hope for further extensions to greater multiplicities. There is a long tradition within algebraic geometry that studies the dimension and the degree of k -secant varieties. These are problems that are unsolved in general. In the second part of the thesis, we consider any projective toric surface XP associated to a polytope P ⊆ R2 and we perform planar toric degenerations D of XP in order to study the k -secant varieties of XP . In particular we give a lower bound to the secant degree and to the 2-secant degree of XP , taking into account the singularities of the configuration D of non-delightful planar toric degenerations. 1