Given any diagonal cyclic subgroup \(\Lambda \subset \text {GL}(n+1,k)\) of order d, let \(I_d\subset k[x_0,\ldots , x_n]\) be the ideal generated by all monomials \(\{m_{1},\ldots , m_{r}\}\) of degree d which are invariants of \(\Lambda\). \(I_d\) is a monomial Togliatti system, provided \(r \le \left( {\begin{array}{c}d+n-1\\ n-1\end{array}}\right)\), and in this case the projective toric variety \(X_d\) parameterized by \((m_{1},\ldots , m_{r})\) is called a GT-variety with group \(\Lambda\). We prove that all these GT-varieties are arithmetically Cohen–Macaulay and we give a combinatorial expression of their Hilbert functions. In the case \(n=2\), we compute explicitly the Hilbert function, polynomial and series of \(X_d\). We determine a minimal free resolution of its homogeneous ideal and we show that it is a binomial prime ideal generated by quadrics and cubics. We also provide the exact number of both types of generators. Finally, we pose the problem of determining whether a surface parameterized by a Togliatti system is aCM. We construct examples that are aCM and examples that are not.

给定阶数为d的任何对角线循环子组\（\ Lambda \ subset \ text {GL}（n + 1，k）\），令\（I_d \ subset k [x_0，\ ldots，x_n] \）是理想生成的通过所有度为d的单项式\（\ {m_ {1}，\ ldots，m_ {r} \} \）都是\（\ Lambda \）的不变式。\（I_d \）是单项式Togliatti系统，提供\（r \ le \ left（{\ begin {array} {c} d + n-1 \\ n-1 \ end {array}} \ right）\），在这种情况下，由\（（m_ {1}，\ ldots，m_ {r}）\）参数化的投影复曲面变体\（X_d \）被称为GT变种，其组为\（\ Lambda \）。我们证明了所有这些GT变量在算术上都是Cohen–Macaulay，并且给出了它们的希尔伯特函数的组合表示。在\（n = 2 \）的情况下，我们显式计算Hilbert函数，多项式和\（X_d \）的级数。我们确定其齐次理想的最小自由分辨率，并证明它是由二次曲面和三次方生成的二项式素理想。我们还提供两种类型发电机的确切数量。最后，我们提出确定由Togliatti系统参数化的曲面是否为aCM的问题。我们构造的示例是aCM，而并非示例。