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

Abstract

In this paper, we study the geometry of GT-varieties with group a finite cyclic group of order d. We prove that the homogeneous ideal of 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 and we show that it is generated by monomial invariants of of degree d and 2d. This allows us to characterize the Castelnuovo–Mumford regularity of the homogeneous coordinate ring of . Finally, we compute the cohomology table of the normal bundle of the so-called RL-varieties. They are projections of the Veronese variety which naturally arise from level GT-varieties.