On the arithmetic Cohen–Macaulayness of varieties parameterized by Togliatti systems

Given any diagonal cyclic subgroup Λ⊂GL(n+1,k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \subset \text {GL}(n+1,k)$$\end{document} of order d, let Id⊂k[x0,…,xn]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_d\subset k[x_0,\ldots , x_n]$$\end{document} be the ideal generated by all monomials {m1,…,mr}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{m_{1},\ldots , m_{r}\}$$\end{document} of degree d which are invariants of Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda$$\end{document}. Id\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_d$$\end{document} is a monomial Togliatti system, provided r≤d+n-1n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \le \left( {\begin{array}{c}d+n-1\\ n-1\end{array}}\right)$$\end{document}, and in this case the projective toric variety Xd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_d$$\end{document} parameterized by (m1,…,mr)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m_{1},\ldots , m_{r})$$\end{document} is called a GT-variety with group Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda$$\end{document}. 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=2$$\end{document}, we compute explicitly the Hilbert function, polynomial and series of Xd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_d$$\end{document}. 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.


Introduction
In 1946 [28], Eugenio Togliatti classified the rational surfaces of ℙ N , N ≥ 5 , parameterized by cubics and representing a Laplace equation of order 2, i.e., whose osculating spaces have all dimension strictly less than the expected 5. Only for one of the surfaces found by Togliatti the apolar ideal to the ideal generated by the polynomials giving the parameterization is artinian, and it is the ideal J = (x 3 , y 3 , z 3 , xyz) ⊂ K [x, y, z] . In 2007 [2], Brenner and Kaid proved that, over an algebraically closed field of characteristic 0, J is the only ideal of the form (x 3 , y 3 , z 3 , f (x, y, z)) , with f ∈ k[x, y, z] homogeneous of degree 3, failing the weak Lefschetz property (see Sects. 2, 2.3, for the definition). In 2013, the connection between these two examples has been clarified and extended. In the article [19], it is proved that, given an artinian ideal I ⊂ k[x 0 , … ,  [1, 4-7, 17, 18, 20] and [24]. In [17] and [24] there are descriptions of the minimal monomial Togliatti systems with " low" number of generators, where minimal means that it does not contain any smaller Togliatti system. There is an interesting family of examples generalizing one aspect of the ideal J found by Togliatti. More precisely, we consider the following situation. We fix integers 2 ≤ n < d , 0 ≤ 0 ≤ ⋯ ≤ n < d such that GCD( 0 , … , n , d) = 1 and we fix e, a dth primitive root of 1. Let Λ ⊂ GL(n + 1, k) be the cyclic subgroup of order d generated by the diagonal matrix M d; 0 ,…, n ∶= diag(e 0 , … , e n ) . We denote by I d the artinian ideal generated by all monomials {m 1 , … , m r } of degree d which are invariants of Λ and by X d the image of the morphism I d ∶ ℙ n → ℙ r−1 defined by (m 1 , … , m r ) . With this notation, J is the ideal corresponding to Λ = ⟨M 3;0,1,2 ⟩ ⊂ GL (3, k) . The study of the ideals I d ⊂ k[x 0 , x 1 , x 2 ] started in [18], where it is also determined the geometry of the surface S d corresponding to Λ = ⟨M d;0,1,2 ⟩ ⊂ GL (3, k) . The minimal free resolution of S d is described, as well as it is proved that S d is an arithmetically Cohen-Macaulay surface generated by quadrics and cubics. Afterward in [6], some results are generalized for the three-fold F d corresponding to Λ = ⟨M d;0,1,2,3 ⟩ . The minimality of the ideals I d for any group Λ = ⟨M d; 0 ,…, n ⟩ is established in [4] and [7], and the argument relies on a careful study of the permanent of certain circulant matrices.
In the present paper, we focus our attention on the arithmetic Cohen-Macaulay property (shortly aCM) of any variety X d , as well as surfaces parameterized by Togliatti systems I ⊂ k[x 0 , x 1 , x 2 ] . All these varieties are monomial projections of Veronese varieties. Any result in this direction should therefore be considered as a contribution to the longstanding problem of deciding whether projections of Veronese varieties are aCM, posed by Gröbner in [12]. Our first result is Theorem 3.1, stating the non-trivial fact that any monomial invariant of Λ of degree a multiple of d can be expressed as a product of monomial invariants of Λ of degree d. It relies on a result of Erdös, Ginzburg and Ziv ( [8]). By a GT-system we shall mean a Togliatti system I ⊂ k[x 0 , … , x n ] whose associated morphism I ∶ ℙ n → ℙ r−1 is a Galois covering with group ℤ∕dℤ . It follows that I d is a GT-system with group Λ , provided r ≤ d + n − 1 n − 1 , and in this case we call X d a GT-variety with group Λ.
Our main result proves that any variety X d is aCM, and so GT-varieties with group Λ are aCM (Theorem 3.3). We deduce it from Theorem 3.1, proving that the coordinate ring of X d is the ring of invariants R Λ , where Λ is the diagonal linear group of order d 2 generated by M d; 0 ,…, n and M d;1,…,1 = diag(e, … , e) . Afterward, we turn our attention to the Hilbert function of X d and we give a combinatorial description of it. In the case n = 2 , we are able to obtain Theorem 4.12 containing an explicit expression for the Hilbert polynomial and series, as well as a minimal free resolution of any GT-surface (Theorem 4.14). From this we provide a complete description of the homogeneous ideal of any GT-surface.
Finally, we address the general problem of the arithmetic Cohen-Macaulayness of surfaces parameterized by monomial Togliatti systems whose coordinate rings are not rings of invariants of finite linear groups. We give a counterexample showing that this property is not true in general. However, we provide a new class of Togliatti systems, whose varieties are aCM. These are not GT-systems, but are obtained as a different generalization of the ideal J. The proof relies on the study of the associated numerical semigroup, using a criterion due to Goto and Watanabe in [10] and Trung in [29].
Let us outline how this work is organized. Section 2 contains the basic definitions and results needed in the rest of this paper. We introduce semigroup rings and the rings of invariants by finite groups. Next, we present the basic facts on Galois coverings and quotient varieties by finite groups of automorphisms. Finally, we recall the notion of Togliatti systems and GT-systems introduced in [4,18] and [19].
The main results of this paper are collected in Sects. 3 and 4. In Sect. 3 we prove that any variety X d is aCM. In Sect. 4, we focus on the geometric properties of GT-surfaces. We explicitly determine their Hilbert function, polynomial and series. Fixed an integer 2 (see Theorem 4.12). We find a minimal free resolution of any GT-surface (Theorem 4.14), which allows us to conclude that its homogeneous ideal is a binomial prime ideal minimally generated by quadrics and cubics. We give the exact number of both types of generators (see Corollary 4.16).
Section 5 concerns the arithmetic Cohen-Macaulayness of surfaces parameterized by monomial Togliatti systems whose coordinate rings are not rings of invariants of finite linear groups.
Notation Throughout this paper, k denotes an algebraically closed field of characteristic zero, R = k[x 0 , … , x n ] and GL(n + 1, k) the multiplicative group of invertible (n + 1) × (n + 1) matrices with coefficients in k. If z, z ′ are positive integers, we denote by (z, z � ) the greatest common divisor of z and z ′ .

Preliminaries
In this section, we introduce the main objects and results we shall use. First, we define semigroups and normal semigroups, and we present three results on the Cohen-Macaulayness of semigroup rings needed in the sequel (see [3,10,15] and [31]). Second, we prove that quotient varieties by the action of finite groups of automorphisms are Galois coverings and we translate this result from the point of view of Invariant Theory. For a further exposition in Invariant Theory of finite groups, see for instance [3] and [26]. Finally, we introduce the weak Lefschetz property and the notions of Togliatti systems and GT-systems.

Semigroup rings and rings of invariants
By a semigroup, we mean a finitely generated subsemigroup H = ⟨h 1 , … , h t ⟩ of ℤ n+1 . We denote by L(H) the additive subgroup of ℤ n+1 generated by H and by r the rank of L(H) in ℤ n+1 . We also denote by k[H] ⊆ R the semigroup ring associated to H, i.e., the graded k-algebra whose basis elements correspond to the monomials X h j , j = 1, … , t , where X h j denotes the monomial x a 0 0 ⋯ x a n n with h j = (a 0 , … , a n ) . By a basis of k[H] we mean a set of Concerning normal semigroups, Hochster proves the following result.
◻ A large family of normal semigroups comes from Invariant Theory, precisely those associated to finite abelian groups acting linearly on R. We take Λ = ℤ∕ℤd 1 ⊕ ⋯ ⊕ ℤ∕ℤd r and we choose d i -th primitive roots of unity e i , i = 1, … , r . Λ can be linearly represented in GL(n + 1, k) by means of r diagonal matrices diag(e We consider the ring of invariants R Λ ∶= {p ∈ R | (p) = p for all ∈ Λ} . A polynomial p ∈ R Λ if and only if all its monomials belong to R Λ . By Noether's degree bound (see [26, 1.2 Theorem.]), R Λ has a finite basis consisting of monomials of degree at most the order of Λ . Let X h 1 , … , X h t be a monomial basis of R Λ and H = ⟨h 1 , … , h t ⟩ . Then R Λ ≅ k [H] . Furthermore, a monomial x a 0 0 ⋯ x a n n ∈ R Λ if and only if (a 0 , … , a n ) satisfies the system of congruences: Now, if w ∈ L(H) is such that zw ∈ H for some z ∈ ℤ ≥0 , then w ∈ H . So H is normal and k[H] is a CM ring.
By [16,Proposition 13], the ring of invariants of any finite group acting linearly on R is CM. This is a particular case of [16, Proposition 12] that we present next. Let A be a subring of R: a Reynolds operator is a A-linear map ∶ R → A such that |A = id A . We have:

Theorem 2.3 Let A be a subring of R such that there exists a Reynolds operator and R is integral over A. Then A is a Cohen-Macaulay ring.
Proof See [16,Proposition 12].
◻ Let G ⊂ GL(n + 1, k) be a finite group acting on R. We denote by R G the ring of invariants of G. One can easily check that the map ∶ R → R G , defined by (p) = �G� −1 ∑ g∈G g(p) , is a Reynolds operator. Furthermore, any element p ∈ R is a solution of equation (1) a 0 u 0,i + ⋯ + a n u n,i ≡ 0 (mod d i ), i = 1, … , r.
which is a polynomial in Y with coefficients in R G . So R is integral over R G and, by Theorem 2.3, R G is CM.
Partially motivated by the results of Proposition 2.2 and Theorem 2.3, Goto, Suzuki and Watanabe, and Trung proved: Theorem 2.4 Let H be a semigroup and assume that there exist ℚ-linearly independent elements f 1 , … , f m ∈ H such that z ⋅ H ⊂ ⟨f 1 , … , f m ⟩ , for some positive integer z. The following conditions are equivalent.
In particular, set

Galois coverings and quotient varieties
We recall that a covering of a variety X consists of a variety Y and a finite morphism f ∶ Y → X . The group of deck transformations G ∶= Aut(f ) is defined to be the group of automorphisms of Y commuting with f. We say that f ∶ Y → X is a covering with group Aut(f).

Definition 2.6
When a group G acts on a variety X, there is a natural way of constructing Galois coverings.

Definition 2.7
Let G be a group acting on a variety X. The quotient of X by G is defined to be a variety Y with a surjective morphism p ∶ X → Y such that any morphism ∶ X → Z to a variety Z factors through p if and only if (x) = (g(x)) , for all x ∈ X and g ∈ G.
Remark 2.8 If it exists, the quotient variety is unique up to isomorphism and is denoted by X/G. In particular, the morphism p ∶ X → X∕G verifies that if x, y ∈ X , then p(x) = p(y) if and only if g(x) = y , for some g ∈ G.
The irreducibility of X allows us to conclude that f = g i , for some g i ∈ G . Therefore, Aut( ) = G and it is clear that given (x) ∈ X∕G , the fiber −1 ( (x)) = G x , so Aut( ) = G acts transitively on −1 ( (x)) . ◻ A finite group of automorphisms of the affine space n+1 can be regarded as a finite group G ⊂ GL(n + 1, k) acting on R. Let {f 1 , … , f t } be a basis of R G , also called a set of fundamental invariants of G, and let k[w 1 , … , w t ] be the polynomial ring in the new variables w 1 , … , w t . We denote by syz(f 1 , … , f t ) the kernel of the morphism from A n+1 to A t defined by w i → f i , i = 1, … , t . We have: Proposition 2.12 Let G ⊂ GL(n + 1, k) be a finite group acting on n+1 , let {f 1 , … , f t } be a set of fundamental invariants of G and let ∶ n+1 → t be the morphism defined by (f 1 , … , f t ) . Then, is a Galois covering of ( n+1 ) with group G.

◻
The cardinality of a general orbit G(a), a ∈ n+1 , is called the degree of the covering. Moreover, if we can find a homogeneous set of fundamental invariants {f 1 , … , f t } of G such that ∶ ℙ n → ℙ t−1 is a morphism, then the projective version of Proposition 2.12 is true.

Lefschetz properties and Togliatti systems
Let I ⊂ R be a homogeneous artinian ideal. The weak Lefschetz property (WLP for short) is an important property of these ideals, which has attracted much interest in the last years, see for instance [2,13,19,[21][22][23]. We recall the definition. We say that I has the WLP if there is a linear form L ∈ R 1 such that, for all integers j, the multiplication map has maximal rank. We say that I fails the WLP in degree j 0 if for any linear form L ∈ R 1 , the multiplication map ×L ∶ (R∕I) j 0 → (R∕I) j 0 +1 has not maximal rank. In 2013 [19], Mezzetti, Miró-Roig and Ottaviani established a close connection between algebraic and geometric language showing that the failure of the WLP for ideals generated by forms of the same degree is related to the existence of varieties whose all osculating spaces of a certain order have dimension less than expected. To state the precise statement, we shortly recall the definition of the Macaulay's inverse system I −1 of I and the language of osculating spaces and Laplace equations.
In addition to R, we consider a second polynomial ring R = k[X 0 , … , X n ] . We have the apolarity action of R on R by partial differentiation, i.e., if F ∈ R and h ∈ R , then On the geometric side, we recall that, if X is a rational projective variety with a birational parameterization ℙ n ⤏ X ⊂ ℙ r−1 given by r n + s s − 1} , but it could be lower. If strict inequality holds for all smooth points of X, and dim (s) x, then X is said to satisfy Laplace equations of order s. Indeed, in this case the partials of order s of F 1 , … , F r are linearly dependent, which gives differential equations of order s satisfied by F 1 , … , F r . In [19] the following theorem is proved.
be an artinian ideal generated by r forms F 1 , … , F r of degree d and let I −1 be its Macaulay inverse system. If r ≤ n + d − 1 n − 1 , then the following conditions are equivalent.
is the rational map associated to ( smooth, and monomial if I can be generated by monomials. The name is in honour of Eugenio Togliatti, who proved that for n = 2 the only smooth Togliatti system of cubics is the monomial ideal (see [2,18,27,28]). The corresponding variety Y, parameterized by (I −1 ) 3 , is a smooth surface in ℙ 5 , known as Togliatti surface; its 2-osculating spaces have all dimension ≤ 4 instead of the expected dimension 5. The systematic study of Togliatti systems I was initiated in [19], where one can find in particular a classification of monomial Togliatti systems with " low" number of generators; for further results the reader can see [1,17,18,20,24]. In [18] the authors introduced the notion of Galois-Togliatti system (shortly GT-system), which we recall now.

Definition 2.14
is a Galois covering with cyclic group ℤ∕dℤ.
In the sequel, the image of the morphism I d will be denoted by X d . The varieties X d and Y, introduced in Theorem 2.13 are called apolar. The first example of GT-system is the ideal (2). The corresponding pair of apolar varieties is formed by the Togliatti surface Y ⊂ ℙ 5 and the cubic surface X 3 ⊂ ℙ 3 .

Example 2.15
Fix integers n = 2 , d = 5 , fix e a 5th primitive root of 1 and let Λ = ⟨diag(1, e, e 3 )⟩ ⊂ GL(3, k) be a cyclic group of order 5. The homogeneous component of degree 5 of R Λ is generated by the invariant monomials In total we have r = 5 monomials so the inequality r ≤ n + d − 1 n − 1 is satisfied. One proves that the ideal I 5 ⊂ R generated by these monomials fails the WLP in degree 4 and the morphism I 5 ∶ ℙ 2 → ℙ 4 is a Galois covering of degree 5 with cyclic group ℤ∕5ℤ (see Corollary 3.4). Actually I 5 (ℙ 2 ) is the quotient surface by the action of the finite group of automorphisms of ℙ 2 generated by diag(1, e, e 3 ).
In the following, we will study GT-systems I d generated by forms of degree d which are invariants of a finite diagonal cyclic subgroup of GL(n + 1, K) of order d. Note that Definition 2.14 does not assume that the ideal is monomial. For examples of non-monomial Togliatti systems, the reader can look at [5]. However, the Togliatti systems we will study in Sects. 3, 4 and 5 are all monomial.

The arithmetic Cohen-Macaulayness of GT-varieties
In this section, we study the ideals generated by all monomials {m 1 , … , m d } of degree d which are invariants of a finite diagonal cyclic group Λ ⊂ GL(n + 1, k) of order d. They are We study the varieties associated to them, which we call GT-varieties with group Λ ; in particular we prove that they are aCM.
To this end, we fix integers 2 ≤ n < d and diag(e 0 , … , e n ) , where e is a dth primitive root of 1. We consider the cyclic group Λ = ⟨M d; 0 ,…, n ⟩ ⊂ GL(n + 1, k) of order d, and the abelian group Λ ⊂ GL(n + 1, k) of order d 2 generated by M d; 0 ,…, n and M d;1,…,1 = diag(e, … , e) . As usual R Λ (respectively R Λ ) represents the ring of invariants of Λ (respectively Λ ). Let {m 1 , … , m d } be the set of all monomials of degree d which are invariants of Λ and denote by I d the monomial artinian ideal generated by them. Let I d ∶ ℙ n → ℙ d −1 be the morphism associated to I d and define Our first result shows that This will allow us to prove that any variety X d is aCM and that I d is a monomial a n …, n } a sequence of integers where 0 is repeated a 0 times, 1 is repeated a 1 times, and so on. Since t ≥ 2 , S contains more than 2d − 1 elements. Hence by [8,Theorem] and [9], there exists a subsequence S ′ ⊂ S of d elements summing to a multiple rd of d. ∈ R Λ . So we have that a 1 − a 0 + a 2 − a 0 and a 1 − a 0 + 2(a 2 − a 0 ) are multiples of 3, which implies that a 1 − a 0 and a 2 − a 0 are multiples of 3. Now we assume a 0 a 1 a 2 = 0 . We may suppose that a 0 = 0 and a 1 a 2 ≠ 0 . We have that a 1 + a 2 and a 1 + 2a 2 are multiples of 3, which gives that a 1 and a 2 are multiples of 3.

Theorem 3.3 X d is a toric aCM variety.
Proof By definition, X d is parameterized by monomials and hence it is toric. By Theorem 3.1, we have that {m 1 , … , m d } is a set of fundamental invariants of Λ . Therefore, the is an invariant of Λ , so ×L(p) = 0 and ×L is not injective. ◻

Definition 3.5 An ideal I d as in Corollary 3.4 is called a GT-system with group Λ.
We present examples of families of monomial GT-systems, which also motivates our next definition. k) . In [18] the authors prove that d ≤ d + 1 . Hence, by Corollary 3.4, I d is a monomial GT-system. (ii) Fix integers 3 = n < d and let Λ = ⟨M d;0,1,2,3 ⟩ ⊂ GL(4, k) . In [6] it is proved that So by Corollary 3.4, I d is a monomial GT-system.

Definition 3.7
We call GT-variety with group Λ any projective variety I d (ℙ n ) associated to a a GT-system I d with group Λ = ⟨M d; 0 ,…, n ⟩ ⊂ GL(n + 1, k).

Hilbert function of GT-surfaces
In this section, we give a combinatorial description of the Hilbert function of any GT-variety X d with group Λ = ⟨M d; 0 ,…, n ⟩ ⊂ GL(n + 1, k) in terms of the invariants of Λ . For the particular case of GT-surfaces, we explicitly compute their Hilbert function, polynomial and series. We also determine a minimal free resolution of their homogeneous ideals. As a corollary, we obtain that the homogeneous ideal of any GT-surface is minimally generated by quadrics and cubics.
The following well-known result is needed.  For each r ∈ {0, … , n t} , we define |( * )| t,r to be the number of solutions of ( * ) t,r in ℤ n+1 ≥0 . We can rewrite Proposition 4.3 as follows.
In Theorem 3.3, we proved that S∕I(X d ) is CM; moreover, since X d is toric, we have that its ideal is generated by binomials: . We now consider a minimal graded free S-resolution N • of S∕I(X d ).
where N l ≅ ⨁ f l j≥l S(−j − l) b l,j and b l,f l > 0 , 1 ≤ l ≤ d − n − 1. As usual, the Cohen-Macaulay type of S∕I(X d ) is the dimension of the free S-module N d −n−1 . We recall that S∕I(X d ) is level if N d −n−1 is generated in only one degree and that S∕I(X d ) is Gorenstein if it is level and dim(N d −n−1 ) = 1 . We denote by reg(X d ) ∶= f d −n−1 + 1 the Castelnuovo-Mumford regularity of S∕I(X d ) . The ideal I(X d ) is minimally generated by b 1,j binomials of degree j + 1 , j = 1, … , f 1 . We set i = min{1 ≤ j ≤ f 1 | b 1,j ≠ 0} . We highlight two combinatorial ways of computing b 1,i which follow from Proposition 4.3. For completeness we include a simple proof. Let {m t 1 , … , m t N } ⊂ R Λ be the set of all monomials of degree td. Each m t j is a product of t monomials of degree d in R Λ (see Theorem 3.1). We denote by |m t j | the number of different ways of expressing m t j as product of t monomials of degree d.

Proposition 4.6 With the above notation, we have:
Proof Computing the Hilbert function of X d in degree i + 1 from N • , we obtain that which implies the first equality. By Proposition 4.6, b 1,i = is the number of all possible combinations of i + 1 monomials of degree d in R Λ . Thus From now on we focus on GT-surfaces. We fix an integer d ≥ 3 and a cyclic group Λ = ⟨M d;0,a,b ⟩ ⊂ GL(3, k) of order d with 0 < a < b . From Example 3.6(i) it follows that the ideal I d generated by all monomials {m 1 , … , m d } ⊂ R Λ of degree d is a monomial GT-system with group Λ , so the associated variety X d is a GT-surface with group Λ . In the rest of this section we will use the following notation.

Notation 4.8 We put
We denote by and the uniquely determined integers such that 0 < ≤ d ′ and b = a � + d � .

Remark 4.11
(ii) Proof (i) By Lemma 4.9, we only have to count the number of solutions (y 0 , y 1 , y 2 ) ∈ ℤ 3
A consequence of Theorem 4.14 is the following.

Remark 4.17
(ii) Fix d = 6 and let X 6 be a GT-surface with group Λ = ⟨M 6;0,a,b ⟩GL(3, k) . We have: . A minimal graded free S-resolution of S∕I(X 6 ) with (a, b, 6) = 4 has the shape: A minimal graded free S-resolution of S∕I(X 6 ) with (a, b, 6) = 5 has the shape: In this case, X 6 is an arithmetically Gorenstein surface of ℙ 6 . (iii) Fix d = 8 and let X 8 be a GT-surface with group Λ = ⟨M 8,0,a,b ⟩ . We have: As in the previous case, we obtain the following resolutions:

A new family of aCM surfaces parameterized by monomial Togliatti systems
Let n, d be positive integers and fix e, a dth primitive root of 1. We denote by Γ ⊂ GL(n + 1, k) the finite diagonal group of order d generated by M d;1,…,1 ∶= diag(e, … , e) . The Veronese is the projective variety whose homogeneous coordinate ring is the ring of invariants R Γ . The set M n,d ⊂ R of all monomials of degree d is a k-algebra basis of R Γ . By a monomial projection of V n,d , we mean a projective variety given parameterically by a subset of M n,d . In [12], Gröbner posed the problem of determining which monomial projections of Veronese varieties are aCM. Since then, there have been many efforts to solve this still open problem, see for instance [14,29] and [30]. In Sect. 3, we proved that all GT-varieties with finite linear diagonal cyclic group are aCM. However, not all surfaces parameterized by monomial Togliatti systems are aCM. For instance, the Togliatti system I = {x 5 0 , x 5 1 , x 5 2 , gives rise to a non   1, 1) . Inductively for t ≥ 2 , we define H 3t ∶= ⟨(3t, 0, 0), (0, 3t, 0), Let us illustrate the above definition with the following three examples. We denote by J 3t ⊂ R the monomial artinian ideal associated to H 3t . All ideals J 3t have 3t = 3t + 1 generators. It is easy to check by induction that they are Togliatti systems. Indeed, the first ideal J 3 is of course the monomial GT-system (2) with group ⟨M 3;0,1,2 ⟩ ⊂ GL (3, k) . On the other hand, for any t,  (3, k) , and let ∶ R → R G be the Reynolds operator. We have that for all t > 1 , (3, 3(t − 1), 0) ∉ H 3t (see Lemma 5.7), or equivalently and we get a contradiction. Our goal is to prove that all k[H 3t ] are CM rings. To this end, we want to apply Theorem 2.4. But first we need some preparation. We fix t > 1 and we put f 1 = (3t, 0, 0), f 2 = (0, 3t, 0), f 3 = (0, 0, 3t).
(ii) By construction H 3t ⊂ H 3 , so H 3t ⊂ H 3 . This means that for all u = (a 1 , a 2 , a 3 ) ∈ H 3t there exist f ≥ 1 and r ∈ {0, … , 2tf } such that u is a solution of the system: The converse is not true: (3, 3(t − 1), 0) ∉ H 3t but it belongs to H 3 . (iii) All generators of H 3t different from f 1 , f 2 , f 3 have all three components different from 0.

Remark 5.5 By construction, we can describe
where A i ∈ ℤ ≥0 for i = 1, … , 3t + 1 and h j is a generator of H 3(t−1) , for We give a couple of examples. Any u ∈ H 3t represents a monomial of degree a multiple of 3t, namely (3t)f. For

Remark 5.9
If w = (a 1 , a 2 , a 3 ) ∈ H 3t only has one nonzero component, namely a i , then We are now ready to prove the main theorem of this section.
We claim that this inclusion is a consequence of the following condition: Condition ( * ) : if w = (a 1 , a 2 , a 3 ) ∈ H 3 is such that a 1 a 2 a 3 ≠ 0 and w + f i ∈ H 3t for some i ∈ {1, 2, 3} , then either w ∈ H 3t or w + f j , w + f k ∉ H 3t for {i, j, k} = {1, 2, 3}.
Proof of the claim. We have already shown the same statement for elements w with a 1 a 2 a 3 = 0 in Corollary 5.8 and Remark 5.9. Since H 1 ⊂ H 3t ⊂ H 3 , an element w ∈ H 1 satisfying w + f j , w + f k ∈ H 3t , for some j, k ∈ {1, 2, 3} such that j ≠ k , belongs to H 3t . This proves the claim.
We may assume that A 1 = 0 , otherwise the result is trivial. We observe the following. Let u = m + h j = s j m + (3(t − s j ), 0, 0) and v = m + h i = s i m + (3(t − s i ), 0, 0) , with s j , s i > 0 , be two generators of H 3t . Therefore we can write u , respectively. So after doing suitable transformations on the summands of w + f 1 , we reduce it to one of the following forms.
Proof It follows from the same proof as Theorem 5.10 replacing m by km.