Polynomial interpolation problems in projective spaces and products of projective lines

These notes summarize part of my research work as a SAGA postdoctoral fellow. We study a class of polynomial interpolation problems which consists of determining the dimension of the vector space of homogeneous or multihomogeneous polynomials vanishing together with their partial derivatives at a finite set of general points. After translating the problem into the setting of linear systems in projective spaces or products of projective lines, we employ algebro-geometric techniques such as blowing-up and degenerations to calculate the dimension of such vector spaces. We compute the dimensions of linear systems with general points of any multiplicity in Pn in a family of cases for which the base locus is only linear [8]. Moreover we completely classify linear systems with double points in general position in products of projective lines (P1)n [26] and we relate this to the study of secant varieties of Segre-Veronese varieties.