The Canonical Module of GT-Varieties and the Normal Bundle of RL-Varieties

In this paper, we study the geometry of GT-varieties $$X_{d}$$ X d with group a finite cyclic group $$\Gamma \subset {{\,\mathrm{GL}\,}}(n+1,\mathbb {K})$$ Γ ⊂ GL ( n + 1 , K ) of order d. We prove that the homogeneous ideal $${{\,\mathrm{I}\,}}(X_{d})$$ I ( X d ) of $$X_{d}$$ 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 $$X_{d}$$ X d and we show that it is generated by monomial invariants of $$\Gamma $$ Γ of degree d and 2d. This allows us to characterize the Castelnuovo–Mumford regularity of the homogeneous coordinate ring of $$X_d$$ 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 $$\nu _{d}(\mathbb {P}^{n}) \subset \mathbb {P}^{\left( {\begin{array}{c}n+d\\ n\end{array}}\right) -1}$$ ν d ( P n ) ⊂ P n + d n - 1 which naturally arise from level GT-varieties.