Interpolation of toric varieties (Q6635594)

In the present paper the authors study the interpolation problem in the toric setting. Recall that by an interpolation problem we understand the study of a class of algebraic varieties together with a collection of incidences, for instance, between linear subspaces and possible tangencies or higher order osculating conditions and ask for those varieties in this class that satisfy the given conditions. By \(k\)-th osculating space to an \(m\)-dimensional variety \(X \subset \mathbb{P}^{d}\) at a point \(p\in X\) we understand a linear space of dimension \(\leq \binom{m+k}{k}-1\) in \(\mathbb{P}^{d}\) that is tangent to \(X\) at \(p\) to the order \(k\). The kind of interpolation that the authors consider here is the following. Fix a variety \(X \subset \mathbb{P}^{d}\) of dimension \(m\) in the complex projective space of dimension \(d\). Let \(k\) be a positive integer satisfying \(\binom{m+k}{k}\leq d\) and consider the set of all varieties \(Y \subset \mathbb{P}^{d}\) of dimension \(\leq \binom{m+k}{k}-1\) such that \(X \subset Y\). We say that \(Y\) satisfies the \(k\)-th interpolation condition with respect to \(X\) if the embedded tangent space to \(Y\) at almost all points of \(X\) is equal to the \(k\)-th osculating space to \(X\) at that point. A natural question is to determine the existence and the uniqueness of a \(k\)-th interpolant and, if exists, explore the methods for its construction. The main result of the paper under review tells us that if \(X\) is a toric variety, then there is a unique toric variety \(Y\) solving the above interpolation problem.