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

This work was partially supported by the project Pure Mathematics in Norway, funded by Trond Mohn Foundation and Tromso Research Foundation. AD was partially supported by UBACYT 20020220200166BA and CONICET PIP 20110100580, Argentina. We would like to thank the referees for their valuable comments and questions that have helped us improve the exposition.