Interpolation of toric varieties

Abstract

Let ⊂ ℙ be an -dimensional variety in -dimensional complex projective space. Let be a positive integer such that the combinatorial number (+, k) is smaller than or equal to . Consider the following interpolation problem: does there exist a variety ⊂ ℙ of dimension strictly smaller than (m+k,k) with ⊂ , such that the tangent space to at a point ∈ isequal to the th osculating space to at , for almost all points ∈ ? In this paper we consider this question in the toric setting. We prove that if is toric, then there is a unique toric variety solving the above interpolation problem. We identify in the general case and we explicitly compute some of its invariants when is a toric curve.