The Geometry of \(\overline {G}\) -Varieties

Any \(\overline {G}\) -variety \(X_d\) with a finite abelian group \(G \subset \mathrm {GL}(n+1,\mathbb {K})\) of order d is an aCM monomial projection of the Veronese variety \(X_{n,d} \subset \mathbb {P}^{N_{n,d}-1}\) related to invariant theory of finite groups and the theory of semigroup rings. The homogeneous coordinate ring \(A(X_d)\) of \(X_d\) is a graded CM ring isomorphic to the ring \(R^{\overline {G}}\) . Combinatorially, \(A(X_d)\) is isomorphic to the semigroup ring of the normal affine semigroup \(H_{\mathcal {A}} \subset \mathbb {Z}_{\geq 0}^{n+1}\) associated to \(R^{\overline {G}}\) . These features endow the homogeneous coordinate ring \(A(X_d)\) with a rich structure. In this chapter, we use these connections to study the geometry behind a \(\overline {G}\) -variety \(X_d\) with group \(G \subset \mathrm {GL}(n+1,\mathbb {K})\) . We pursue to determine the Hilbert function and Hilbert series of \(A(X_d)\) , to understand the structure of a minimal set of binomial generators of the homogeneous ideal \(\mathrm {I}(X_{d})\) of \(X_d\) , to investigate the canonical module of \(A(X_d)\) , to characterize the Castelnuovo–Mumford regularity of \(A(X_d)\) and to describe the Betti diagram of a minimal graded free resolution of \(A(X_d)\) .