Normal Bundle of RL-Varieties

Our purpose in this chapter is to study the geometry and the normal bundle of a family of smooth rational monomial projections \(\mathcal {X}_{d} \subset \mathbb {P}^{N_{d}}\) of Veronese varieties \(X_{n,d}\subset \mathbb {P}^{N_{n,d}-1}\) , we called them RL-varieties. This family of monomial projections is naturally related to \(\overline {G}\) -varieties \(X_{d} \subset \mathbb {P}^{\mu _{d}-1}\) with a finite abelian group \(G \subset \mathrm {GL}(n+1,\mathbb {K})\) of order d and whose coordinate rings \(A(X_d)\) are level rings with \(\mathrm {reg}(A(X_{d})) = n+1\) .