  In this paper, we study the geometry of $GT-$varieties $X_{d}$ with group a
finite cyclic group $\Gamma \subset \mathrm{GL}(n+1,\mathbb{K})$ of order $d$.
We prove that the homogeneous ideal $\mathrm{I}(X_{d})$ of $X_{d}$ is generated
by binomials of degree at most $3$ and we provide examples reaching this bound.
We give a combinatorial description of the canonical module of the homogeneous
coordinate ring of $X_{d}$ and we show that it is generated by monomial
invariants of $\Gamma$ of degree $d$ and $2d$. This allows us to characterize
the Castelnuovo-Mumford regularity of the homogeneous coordinate ring of $X_d$.
Finally, we compute the cohomology table of the normal bundle of the so called
$RL-$varieties. They are projections of the Veronese variety
$\nu_{d}(\mathbb{P}^{n}) \subset \mathbb{P}^{\binom{n+d}{n}-1}$ which naturally
arise from level $GT-$varieties.
