Normal cyclic polytopes and cyclic polytopes that are not very ample

TL;DR

This paper investigates the conditions under which cyclic polytopes are normal or very ample, establishing new bounds on the parameters that guarantee these properties and identifying cases where they do not hold.

Contribution

The paper introduces a new inequality $oldsymbol{oldsymbol{ ext{γ}_d extless d^2 - 1}}$ for the normality of cyclic polytopes and shows certain polytopes are not very ample under specific conditions.

Findings

Established $ ext{γ}_d extless d^2 - 1$ as a bound for normality.

Proved that certain cyclic polytopes are not very ample when $ au_3 - au_2 = 1$ for $d extgreater 3$.

Improved upon previous bounds for the normality of cyclic polytopes.

Abstract

Let $d$ and $n$ be positive integers with $n \geq d + 1$ and $\tau_{1}, ..., \tau_{n}$ integers with $\tau_{1} < ... < \tau_{n}$. Let $C_{d}(\tau_{1}, ..., \tau_{n}) \subset \RR^{d}$ 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 $\tau_{i+1} - \tau_{i} \geq \gamma_{d}$ for $1 \leq i < n$, then $C_{d}(\tau_{1}, ..., \tau_{n})$ is normal. One of the known results is $\gamma_{d} \leq d (d + 1)$. In the present paper a new inequality $\gamma_{d} \leq d^{2} - 1$ is proved. Moreover, it is shown that if $d \geq 4$ with $\tau_{3} - \tau_{2} = 1$, then $C_{d}(\tau_{1}, ..., \tau_{n})$ is not very ample.