Normal cyclic polytopes and cyclic polytopes that are not very ample

Abstract

Let $d$ and $n$ be positive integers with $n\ge d+1$ and ${\tau }_1,...,{\tau }_n$ integers with ${\tau }_1<...<{\tau }_n$. Let $C_{d}(τ_{1}, ..., τ_{n}) \subset \RR^{d}$ denote the cyclic polytope of dimension $d$ with $n$ vertices $({\tau }_1,{\tau }_1^2,...,{\tau }_1^d),...,({\tau }_n,{\tau }_n^2,...,{\tau }_n^d)$. We are interested in finding the smallest integer ${\gamma }_d$ such that if ${\tau }_{i+1}-{\tau }_i\ge {\gamma }_d$ for $1\le i<n$, then $C_d({\tau }_1,...,{\tau }_n)$ is normal. One of the known results is ${\gamma }_d\le d(d+1)$. In the present paper a new inequality ${\gamma }_d\le d^2-1$ is proved. Moreover, it is shown that if $d\ge 4$ with ${\tau }_3-{\tau }_2=1$, then $C_d({\tau }_1,...,{\tau }_n)$ is not very ample.

17 pages

References in corpus (2)

• Non-very ample configurations arising from contingency tables
• Toric rings arising from cyclic polytopes

Cited by in corpus (2)

• Toric rings arising from cyclic polytopes
• Mixed Integer Reformulations of Integer Programs and the Affine TU-dimension of a Matrix