The canonical module of GT-varieties and the normal bundle of RL-varieties

Abstract

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 ${\nu }_d({\mathbb{P}}^n)\subset {\mathbb{P}}^{\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)-1}$ which naturally arise from level $GT-$varieties.

To appear in Mediterranean Journal of Mathematics