Interpolation of toric varieties

TL;DR

This paper investigates a specific interpolation problem for toric varieties, proving the existence and uniqueness of solutions in the toric setting and explicitly computing invariants for toric curves.

Contribution

It establishes the existence and uniqueness of a toric variety solving the interpolation problem and provides explicit computations for toric curves.

Findings

Unique toric variety Y exists for the interpolation problem.

Y can be explicitly identified in the general case.

Invariants of Y are computed for toric curves.

Abstract

Let $X\subset \mathbb P^d$ be a $m$-dimensional variety in $d$-dimensional projective space. Let $k$ be a positive integer such that $\binom{m+k}k \le d$. Consider the following interpolation problem: does there exist a variety $Y\subset \mathbb P^d$ of dimension $\le \binom{m+k}k -1$, with $X\subset Y$, 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.