Degenerations of the Veronese and applications (Q1049895)

The authors describe some degeneration process, for the images of the Veronese embeddings of the projective plane, which can be applied to the Hermite interpolation problem. The problem consists in determining the dimension of linear systems of plane curves, having prescribed multiplicities at some general points. It is known that a degeneration of the plane can reduce an interpolation problem to two or more similar problems, involving a smaller number of points. So one can play induction, and often get an answer. The authors explore a great number of degenerations of the plane. Using the correspondence between toric varieties \(X\) and compact polytopes \(P\), the authors describe how to obtain surface degenerations by regular polytopal subdivison of \(P\), starting with the polytope (a triangle) associated to the plane. Specializing the points appropriately, the initial interpolation problem reduces to interpolation problems on many copies of \(\mathbb P^2\), where the number of points and the degree have diminished. The authors show how one can reconstruct many known results, in a unified way, by these techniques. Moreover, they present several applications to the study of secant varieties.