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

TL;DR

This paper investigates the algebraic and geometric properties of GT-varieties with cyclic symmetry, including their defining ideals, canonical modules, and the cohomology of their associated RL-varieties, revealing bounds and structural descriptions.

Contribution

It provides a bound on the generators of the ideal of GT-varieties, describes the canonical module combinatorially, and computes the cohomology of RL-varieties.

Findings

Ideal generated by binomials of degree at most 3

Canonical module generated by monomials of degrees d and 2d

Cohomology table of the normal bundle computed

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 $\nu_{d}(\mathbb{P}^{n}) \subset \mathbb{P}^{\binom{n+d}{n}-1}$ which naturally arise from level $GT-$varieties.