Interpolation of toric varieties
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909313812922368 |
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| author | Dickenstein, Alicia Di Rocco, Sandra Piene, Ragni |
| author_facet | Dickenstein, Alicia Di Rocco, Sandra Piene, Ragni |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_08965 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Interpolation of toric varieties Dickenstein, Alicia Di Rocco, Sandra Piene, Ragni Algebraic Geometry 14M25 (Primary) 14H45, 14H81, 52B20 (Secondary) 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. |
| title | Interpolation of toric varieties |
| topic | Algebraic Geometry 14M25 (Primary) 14H45, 14H81, 52B20 (Secondary) |
| url | https://arxiv.org/abs/2308.08965 |