Polynomial Interpolation Problems in Projective Spaces and Products of Projective Lines

Abstract

These notes summarize part of my research work as a SAGA post-doctoral fellow. We study a class of polynomial interpolation problems which consists of determining the dimension of the vector space of homogeneous or multi-homogeneous 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 $\mathbb{P}^{n}$ in a family of cases for which the base locus is only linear (Brambilla et al., Trans Am Math Soc, 2014, to appear). Moreover we completely classify linear systems with double points in general position in products of projective lines $(\mathbb{P}^{1})^{n}$ (Laface and Postinghel, Math. Ann. 356(4):1455-1470, 2013) and we relate this to the study of secant varieties of Segre-Veronese varieties.