POSTINGHEL, E., 2014. Polynomial interpolation problems in projective spaces and products of projective lines. IN: Dokken, T. and Muntingh, G. (eds.) SAGA – Advances in ShApes, Geometry, and Algebra: Results from the Marie Curie Initial Training Network, Cham: Springer, pp. 199-216.
Geometry and Computing;10
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
. Moreover we completely classify linear systems with double points in general
position in products of projective lines (P1)n  and we relate this to the study of
secant varieties of Segre-Veronese varieties.