Degenerations of the Veronese and Applications

The classical Hermite interpolation technique approximates a given differentiable function defined in an open subset of R with a polynomial of a given degree which takes at an assigned set points p1, . . . , pn, usually assumed to be sufficiently general, the same values as the function and of its derivatives up to orderm1, . . . ,mn respectively. A geometric counterpart of Hermite interpolation is the following. Fix general points p1, . . . , pn in the complex projective space P, and multiplicities m1, . . . ,mn. We will denote by L = Lr,d(m1, . . . ,mn) the linear system of hypersurfaces of degree d in P having multiplicity at least mi at pi for each i = 1, . . . , n, and we will employ exponential notation Lr,d(m 1 , . . . ,m eh h ) for repeated multiplicities. A basic question is to determine the dimension of the system L. In the 1–dimensional case r = 1 this is easy. Ruffini’s theorem says that