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

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.


Introduction
Through this paper, K denotes an algebraically closed field of characteristic zero, R = K[x 0 , . . . , x n ] and GL(n+1, K) denotes the group of invertible matrices of size (n+1)×(n+1) with coefficients in K.
In [20], Mezzetti, Miró-Roig and Ottaviani related the existence of homogeneous artinian ideals I ⊂ R generated by homogeneous forms F 1 , . . . , F r of degree d failing the weak Lefschtez property in degree d − 1 to the existence of rational projective varieties of P ( n+d n ) −r−1 We identify ω X d with the ideal relint(I d ) = (x a 0 0 · · · x an n ∈ R Γ | 0 = a 0 · · · a n ) of R Γ generated by the relative interior of the semigroup ring K[m 1 , . . . , m µ d ] (Proposition 4.1). We prove that relint(I d ) is generated by monomials of degree d and 2d (Theorem 4.2). This connection allows us to compute the Castelnuovo-Mumford regularity of R Γ .
Finally, we introduce a new family of smooth rational monomial projections of the Veronese variety ν d (P n ) ⊂ P ( n+d n )−1 naturally associated to level GT −varieties. A GT −variety X d is called level if the ideal relint(I d ) is generated only by monomials of degree d and, hence, R Γ is a level ring. An RL−variety X d associated to a level GT −variety X d is a monomial projection of the Veronese variety ν d (P n ) ⊂ P ( n+d n )−1 parameterized by the set of monomials M n,d \ {m = x a 0 0 · · · x an n ∈ relint(I d ) | deg(m) = d}. The name RL−variety is conceived to emphasize the relation with the relative interior and the levelness. We give examples of RL−varieties in any dimension. Inspired by the recent work of Alzati and Re ( [3]), we contribute to the classical open problem of computing the cohomology of the normal bundle of smooth rational varieties (Theorem 5.9). Most results and examples of this topic focus on smooth rational curves and surfaces, see for instance [2], [10] and [25]. We determine the cohomology table of the normal bundle of any RL−variety, shading new light on higher dimensions.
Let us see how this work is organized. In Section 2, we gather the basic definitions and results needed in the body of this paper. Section 3 is entirely devoted to find a set of homogeneous binomial generators of the homogeneous ideal I(X d ) of any GT −variety X d . We establish that the homogeneous coordinate ring of X d is isomorphic to R Γ and that I(X d ) is a homogeneous prime binomial ideal. Our main result (Theorem 3.11) proves that I(X d ) is generated by binomials of degree at most 3. We give families of examples of GT −varieties whose homogeneous ideals are minimally generated by binomials of degree 2 and 3. In Section 4, we investigate the algebraic structure of the canonical module ω X d of the homogeneous coordinate ring of X d . We identify ω X d with the ideal relint(I d ) of R Γ , which gives us a combinatorial description of ω X d . In Theorem 4.2 we show that relint(I d ) is generated by monomials of degree d and 2d. We further study GT −varieties X d where R Γ is a level ring, i.e. the canonical module relint(I d ) of R Γ is generated in only one degree. In particular, we study varieties whose homogeneous coordinate ring R Γ is level and relint(I d ) is minimally generated in degree d. Afterwards in Theorem 4.10, we characterize the Castelnuovo-Mumford regularity of R Γ .
Finally in Section 5, we introduce the notions of a level GT −variety and its associated RL−variety and we give examples of any dimension. Using the new methods of [3], we compute the cohomology table of the normal bundle of any RL−variety (see Theorem 5.9).
Acknowledgements. The authors are grateful to the anonymous referee for providing detailed comments which have improved the exposition of this paper.

Preliminaries
In this section, we introduce the main objects and results we use in the body of this paper. First, we define semigroups and normal semigroups, we relate them to invariant theory of finite groups and we see a geometrical interpretation of these objects. For more details the reader can look at [14], [4] and [27]. Finally, we define the weak Lefschetz property, we recall the notions of GT −systems and GT −varieties and we collect some basic results on this topic.
Semigroup rings and rings of invariants. By a semigroup we mean a finitely generated additive subsemigroup H = h 1 , . . . , h t ⊂ Z n+1 ≥0 . L(H) is the additive subgroup of Z n+1 generated by H. We denote by K[H] ⊆ R the semigroup ring associated to H, i.e. the graded K−algebra generated by the monomials X h j = x a j 0 0 · · · x a j n n ∈ R associated to the points h j = (a j 0 , . . . , a j n ) ∈ H, j = 1, . . . , t. Definition 2.1. A semigroup H ⊂ Z n+1 is called normal if it satisfies the following condition: if zh ∈ H for some h ∈ L(H) and 0 = z ∈ Z ≥0 , then h ∈ H.
A large family of normal semigroups comes from invariant theory, precisely those associated to finite abelian groups acting on R. To be more precise, let Λ ≃ Z/d 1 Z ⊕ · · · ⊕ Z/d r Z and choose d i -th primitive roots e i of 1 ∈ K , i = 1, . . . , r. Therefore Λ can be linearly represented in GL(n + 1, K) by means of r diagonal matrices diag(e u 0,i i , . . . , e u n,i i ), where u j,i ∈ Z ≥0 , 0 ≤ j ≤ n, 1 ≤ i ≤ r. Let R Λ := {p ∈ R | λ(p) = p for all λ ∈ Λ} be the ring of invariants of Λ acting on R. Since Λ acts diagonally, each monomial x a 0 0 · · · x an n ∈ R is mapped into a multiple of itself by every λ ∈ Λ, and a polynomial p ∈ R Λ if and only if all its monomials are invariants of Λ. Thus, by the Noether's degree bound (see [27,Theorem 2.1.4]), R Λ has a finite basis consisting of monomials of degree at most the order of Λ. By a basis of R Λ we mean a set of elements {θ 1 , . . . , θ l } ⊂ R Λ which minimally generates R Λ as a K−algebra, i.e. R Λ = K[θ 1 , . . . , θ l ]. Let X h 1 , . . . , X ht 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 an n ∈ R Λ if and only if (a 0 , . . . , a n ) satisfies the system of congruences: (1) a 0 u 0,i + · · · + a n u n,i ≡ 0 (mod d i ), i = 1, . . . , r.
Now, if w ∈ L(H) is such that zw ∈ H for some z ∈ Z ≥0 , then w ∈ H, so H is normal. By [14,Theorem 1] or [15,Proposition 13], K[H] is Cohen Macaulay. More generally, let G ⊂ GL(n + 1, K) be a finite group. Geometrically, the ring R G of invariants can be regarded as the coordinate ring of the quotient of A n+1 by G. To be more precise, set {f 1 , . . . , f t } a basis of R G , often 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 . Then the quotient of A n+1 by G is given by the morphism π : A n+1 → π(A n+1 ) ⊂ A t , such that π(a 0 , . . . , a n ) = (f 1 (a 0 , . . . , a n ), . . . , f t (a 0 , . . . , a n )). Even further, π is a Galois covering of π(A n+1 ) with group G. For further details on quotients varieties we refer the reader to [24]. The ideal I(π(A n+1 )) of the quotient variety is called the ideal of syzygies among the invariants f 1 , . . . , f t ; it is the kernel of the homomorphism defined by w i → f i , i = 1, . . . , t. We denote it by syz(f 1 , . . . , f t ). We summarize all these facts in the following proposition.
The cardinality of a general orbit G(a), 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 π : P n → P t−1 is a morphism, then the projective version of Proposition 2.2 is true.
GT-systems and GT-varieties. Let I ⊂ R be a homogeneous artinian ideal. We say that I has the weak Lefschetz property (WLP) if there is a linear form L ∈ R 1 such that, for all integers j, the multiplication map ×L : (R/I) j−1 → (R/I) j has maximal rank. In [20], Mezzetti, Miró-Roig and Ottaviani proved that the failure of the WLP is related to the existence of varieties satisfying at least one Laplace equation of order greater than 2. More precisely, they proved: Theorem 2.3. Let I ⊂ R 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: (i) I fails the WLP in degree d − 1; (ii) F 1 , . . . , F r become K−linearly dependent on a general hyperplane H of P n ; (iii) the n-dimensional variety Y = ϕ(P n ), where ϕ : P n P ( n+d n )−r−1 is the rational morphism associated to (I −1 ) d , satisfies at least one Laplace equation of order d − 1.
Proof. See [20,Theorem 3.2]. Motivated by the above results, Mezzetti, Miró-Roig and Ottaviani introduced the following definitions (see [20] and [18]): Definition 2.4. Let I ⊂ R be an artinian ideal generated by r ≤ n+d−1 n−1 forms of degree d. We say that: (i) I is a Togliatti system if it fails the WLP in degree d − 1.
(ii) I is a monomial Togliatti system if, in addition, I can be generated by monomials.
In particular, a Togliatti system is called smooth if the variety Y in Theorem 2.3(iii) is smooth. The name is in honour of Togliatti who proved that for n = 2 the only smooth Togliatti system of cubics is [28] and [29]). The systematic study of Togliatti systems was initiated in [20] and it has been continuing in [19], [18], [1], [22] and [21]. Precisely in [18], it was introduced the notion of GT-system with group a finite cyclic group. Recently in [8], this notion has been generalized as follows.
Definition 2.5. A GT-system with a finite group G is an artinian ideal I d ⊂ R generated by r forms F 1 , . . . , F r of degree d such that: (i) I d is a Togliatti system. (ii) The morphism ϕ I d : P n → P r−1 defined by (F 1 , . . . , F r ) is a Galois covering with group G.
If conditions (i) and (ii) holds, we say that ϕ I d (P n ) is a GT −variety with group G.
GT −systems with group a finite cyclic group have been extensively studied in [19], [7] and [6], while in [8], the authors investigate GT −systems with the dihedral group acting on K[x 0 , x 1 , x 2 ]. In the last two references, invariant theory techniques have been applied to tackle both objects. Fix G ⊂ GL(n + 1, K) a finite group of order d. Assume that the ring R G has a basis B formed by homogeneous invariants of G of degree d and set I d the ideal generated by B. Keeping this notation, we have the following. Proposition 2.6. If |B| ≤ d+n−1 n−1 , then I d is a GT −system with group G. Proof. Since I d contains a homogeneous system of parameters of R G , it is an artinian ideal. By Proposition 2.2, the associated morphism ϕ I d is a Galois covering with group G. By Theorem 2.3, it is enough to prove that I d fails the WLP in degree d − 1, i.e. for any linear form L ∈ R 1 , the multiplication map ×L : (R/I d ) d−1 → (R/I d ) d is not injective. Let L ∈ R 1 and consider F := Id =g∈G g(L) ∈ R d−1 . We have that L · F = g∈G g(L) ∈ R G and, hence, F ∈ ker(×L).
Let us see an illustrative example.
In [6], [8], and previously in [9], the authors focus on the geometry of GT −varieties with group a finite cyclic group or a dihedral group. In [6, Theorem 3.2], it is proved that all GT −varieties with group a finite cyclic group are aCM varieties and in [8,Proposition 4.3], it is shown its analogous for GT −surfaces with a dihedral group. Furthermore, in [6, Theorem 4.14] and [8,Theorem 4.6], it is determined a minimal free resolution of GT −surfaces with group a finite cyclic group and a dihedral group, respectively. Reference [9] is devoted to find a minimal set of homogeneous binomial generators of the homogeneous ideal of certain GT −threefolds with group a finite cyclic group. Our goal is to extend the results obtained so far for GT −surfaces and GT −threefolds to arbitrary n-dimensional GT −varieties with group a finite cyclic group.

On the homogeneous ideal of GT-varieties
In this section, we look for a system of generators of the homogeneous ideal of GT −varieties with group a finite cyclic group. Using combinatorial techniques, we prove that all these ideals are generated by homogeneous binomials of degree 2 and 3. We begin introducing some notations.
Along this section, we fix a finite cyclic group Γ := M d;α 0 ,...,αn ⊂ GL(n + 1, K) of order d and we consider the ring R Γ of invariants of Γ endowed with the natural grading The cyclic extension Γ ⊂ GL(n + 1, K) of Γ is the finite abelian group of order d 2 generated by M d;α 0 ,...,αn and M d;1,...,1 . We consider the ring R Γ of invariants of Γ with the grading set of all monomial invariants of Γ of degree d, ordered lexicographically. We denote I d ⊂ R the monomial artinian ideal generated by M d . By ϕ I d : P n → P µ d −1 , we denote the associated morphism and we set X d := ϕ I d (P n ) ⊂ P µ d −1 its image. In [6], it is established the following.
Proof. See  is satisfied, we refer to I d as a GT −system and to X d as a GT −variety.
Let w 1 , . . . , w µ d be new variables and set S := K[w 1 , . . . , w µ d ]. We denote by I(X d ) ⊂ S the homogeneous ideal of X d . By Proposition 2.2, I(X d ) is the kernel of the morphism ρ : S → R given by ρ(w i ) = m i . It holds that I(X d ) is the homogeneous binomial prime ideal generated by For any k ≥ 2, we denote by W d,k the set of all binomials of W d of degree exactly k. Our goal is to prove that I(X d ) is generated by binomials of degree 2 and 3, i.e. the ideal 3 . Through families of examples in 3.8 we observe that this bound is sharp. We start with some definitions.
Definition 3.5. Fixed k ≥ 2, we define a suitable k−binomial to be a non-zero binomial Definition 3.6. Given a suitable k−binomial w α = w α + − w α − ∈ W d,k , we denote by supp + (w α ) (respectively supp − (w α )) the support of the monomial w α + (respectively support of w α − ). We say that w α is non trivial if supp + (w α ) ∩ supp − (w α ) = ∅. Otherwise, we say that w α is trivial.
By an I(X d ) k −sequence from w α + to w α − we mean a finite sequence (w 1 , . . . , w t ) of monomials of S satisfying the following two conditions: Example 3.8. Let Γ = M 6;0,1,2,3 ⊂ GL(4, K) be a finite group of order 6. There are µ 6 = 16 monomial invariants of Γ, we have: 2 , x 6 3 }, By Theorem 3.2(iii), the ideal I 6 generated by them is a GT −system and its associated variety X 6 is a GT −variety. The homogeneous binomials w 1 w 15 − w 2 4 and w 3 w 12 w 15 − w 6 w 9 w 14 are non trivial suitable binomials of degree 2 and 3, respectively. Indeed, ). Finally, {w 3 w 12 w 15 , w 5 w 9 w 15 , w 6 w 9 w 14 } is an I(X 6 ) 3 −sequence from w 3 w 12 w 15 to w 6 w 9 w 14 .
The following result characterizes the inclusion of ideals W d,k ⊂ W d,k−1 , k ≥ 3. It is key to prove our main results, which we state later.
Proof. We apply the same arguments as in [9,Proposition 5.4].
Theorem 3.11. Let Γ = M d;α 0 ,...,αn ⊂ GL(n + 1, K) be a cyclic group of order d and X d ⊂ P µ d −1 the variety parameterized by the ideal I d = (m 1 , . . . , m µ d ) generated by all monomial invariants of Γ of degree d. Then, the homogeneous ideal I(X d ) of X d is generated by quadrics and cubics. Precisely, Proof. First, we prove that for all k ≥ 4, any non trivial suitable k−binomial admits an We consider the monomials m i 1 , m j 1 and for each 0 ≤ s ≤ n we define: This gives rise to a non-zero monomial m = x c 0 0 · · · x cn n ∈ R of degree strictly smaller than d. Clearly, m divides m i 2 · · · m i k (see (2)). Thus we consider are monomials of degree d and in particular: Applying the same argument as before, we factorize Therefore (w i 1 · · · w in , w 2 , w 3 , w j 1 · · · w jn ) is an I(X d ) k −sequence, from which it follows that W d,k ⊂ W d,k−1 . The argument we have developed only requires that (k − 2)d ≥ 2d, which it is satisfied for all k ≥ 4. Thus we have proved that for all k ≥ 4, which completes the proof.
Despite W d,2 always belongs to a minimal set of homogeneous binomial generators of I(X d ), it is not the case for W d, 3 . Even further, this fact depends on the action of the group Γ = M d;α 0 ,...,αn , as we illustrate in the following examples.
. Therefore, if a minimal set of homogeneous binomial generators of I(V d ) contains a binomial of degree 3, the same holds for I(X d ).

The canonical module of GT-varieties
The algebraic structure of the canonical module ω X of the homogeneous coordinate ring of an aCM projective variety X plays a central role in its geometry (see [4,5,16]). For example, it can lead us to derive information on the Hilbert function and series, as well as on the Castelnuovo-Mumford regularity, of the homogeneous coordinate ring of X. Let Γ = M d;α 0 ,...,αn ⊂ GL(n + 1) be a cyclic group of order d (see Notation 3.1). We denote by I d = (m 1 , . . . , m µ d ) ⊂ R the ideal generated by all monomial invariants of Γ of degree d, ϕ I d : P n → P µ d −1 the morphism induced by I d and X d = ϕ I d (P n ) ⊂ P µ d −1 its image. X d is an aCM projective variety and its homogeneous coordinate ring S/ I(X d ) is isomorphic to R Γ (see Theorem 3.2). In this section, we deal with the canonical module ω X d of S/ I(X d ). We identify ω X d with an ideal of R Γ and we prove that it is generated by monomials of degree d and 2d. We focus on varieties X d the homogeneous coordinate rings S/ I(X d ) of which are level ring, i.e. their canonical modules ω X d are generated in only one degree. Afterwards, we characterize the Castelnuovo-Mumford regularity of R Γ .
Therefore H d is a normal semigroup (see Section 2) with H + d := {(a 0 , . . . , a n ) ∈ H d | a i > 0, i = 0, . . . , n} = ∅, for instance (d, . . . , d) ∈ H + d . It holds that H + d coincides with the relative interior of H d . We set relint(I d ) := (x a 0 0 · · · x an n ∈ R Γ | 0 = a 0 · · · a n ) the ideal of R Γ generated by all monomials associated to H + d . We have the following. For a complete exposition of the canonical module of normal semigroup rings we refer the reader to [4].
We denote by C d,k ⊂ relint(I d ) the set of all monomials of degree kd. We have: Proof. It is enough to show that for any monomial m ∈ C d,k , k ≥ 3, there exists a monomial m ′ ∈ C d,k−1 which divides m. This proves that for k ≥ 3, C d,k ⊂ C d,k−1 ⊂ · · · ⊂ C d,2 .
We fix an integer k ≥ 3, a monomial m = x a 0 0 · · ·x an n ∈ C d,k and we set m 1 = m/(x 0 · · · x n ) = x a 0 −1 0 · · · x an−1 n . Since d ≥ n + 1 and k ≥ 3, m ′ is a monomial of degree kd − (n + 1) ≥ 2d. We define the sequence of integers L = (α 0 , a 0 −1 . . . , α 0 , . . . , α n , an−1 . . . , α n ), by [11,Theorem] there exists a subsequence L ′ = (α 0 , b 0 . . ., α 0 , . . . , α n , bn . . ., α n ) ⊂ L of d integers the sum of which is a multiple of d. Therefore, L ′ gives rise a monomial m 2 := x b 0 0 · · · x bn n ∈ R Γ of degree d which divides m 1 . Hence, we can factorize m = m 2 m ′ and by construction m ′ ∈ C d,k−1 is the required monomial. Example 4.3. Let Γ = M 6;0,1,2,3 ⊂ GL(4, K) be a cyclic group of order 6. We have that Only the following four monomials x 2 0 x 4 1 x 4 2 x 2 3 , x 2 0 x 3 1 x 6 2 x 3 , x 0 x 6 1 x 3 2 x 2 3 , x 0 x 5 1 x 5 2 x 3 ∈ C 6,2 do not belong to the ideal C 6,1 ⊂ R Γ . From this observation and Theorem 4.2, we obtain that the canonical module of R Γ is the ideal: We recall that R Γ is a level ring if its canonical module relint(I d ) is generated in only one degree and R Γ is a Gorenstein ring if it is a level ring and relint(I d ) is principal. As a consequence of Theorem 4.2, we have that R Γ is a level ring if and only if relint(I d ) = C d,1 or C d,1 = ∅. In [6, Corollary 4.13(ii)], it is shown that the homogeneous coordinate ring of any GT −surface is level. However, the same assertion is not true for GT −varieties of higher dimensions, see for instance [9]. We investigate further this property, which will play an important role in the last section of this work. Let us first present an interesting family of examples of Gorenstein GT −varieties.
Proof. We denote I d ⊂ R (respectively I kd ⊂ R) the ideal generated by all monomials of degree d (respectively kd) which are invariants of Γ (respectively Γ k ). We write relint(I d ) = m . We want to prove that any monomial m ′ ∈ C d,2kd is divisible by a monomialm ∈ C d,kd . We fix m ′ = x a 0 0 · · · x an n ∈ C d,2kd . Notice that m ′ is also an invariant of Γ, so m ′ ∈ C d,k and by hypothesis m divides m ′ . We define m 1 = m ′ m ; since m 1 ∈ R Γ is a monomial of degree (2k − 1)d, by Theorem 3.2(i), there are monomials m 2 , . . . , m 2k ∈ R Γ of degree d such that m 1 = m 2 · · · m 2k , and hence m ′ = mm 2 · · · m 2k . For each monomial m i , 1 ≤ i ≤ 2k, there is a unique integer r i ≥ 0 such that the lattice point l m i is a solution of the system ( * ) 1,r i induced by Γ. By Lemma 3.10, there is a subsequence (r i 1 , . . . , r i k ) ⊂ (r 1 , . . . , r 2k ) of k integers the sum of which is a multiple of k. Therefore, we obtain that m i 1 · · · m i k ∈ R Γ k and m = m ′ /(m i 1 · · · m i k ) ∈ relint(I kd ) is the required monomial.
Proof. It follows directly from Propositions 4.4 and 4.5.
The rest of this section concerns the Castelnuovo-Mumford regularity reg(R Γ ) of R Γ with Γ = M d;α 0 ,...,αn ⊂ GL(n+1, K) a cyclic group of order d as in Notation 3.1. We characterize reg(R Γ ) in terms of relint(I d ). First we need some preparation. For sake of completeness we prove the following. Proof. We consider the graded quotient algebra . , x d n td . Therefore, for t ≥ n + 1 we have that A t = 0 and for 1 ≤ t ≤ n, a K−basis of A t is formed by the set of all monomials m = x a 0 0 · · · x an n ∈ R Γ of degree td such that a 0 < d, . . . , a n < d. We write θ 1 , . . . , θ D the set of all such monomials and θ 0 = 1. Then, it is clear that R Γ = θ 0 , θ 1 , . . . , θ D as a K[x d 0 , . . . , x d n ]-module.
The equality reg(R Γ ) = n + 1 holds if and only if C d,1 = ∅.
Proof. The right inequality follows immediately from (3). We set m = x d 0 · · · x d n and m ′ = x d−1 0 · · · x d−1 n . Lemma 3.10 assures the existence of a monomial of degree (n − 1)d in R Γ dividing m ′ , and hence it assures the existence of secondary invariants of degrees smaller or equal to (n − 1)d, so e n−1 > 0 which gives us the left inequality. Now, reg(R Γ ) = n + 1 if and only if e n > 0. If e n > 0, then there exists a secondary invariant θ of degree nd and we obtain m/θ ∈ C d,1 . Conversely, let p = x a 0 0 · · · x an n ∈ C 1,d . Notice that necessarily a i < d, i = 0, . . . , n, thus m/p is a secondary invariant of degree nd.
To end this section, we present some examples illustrating the last results. They also bring to light how the Hilbert series and regularity of R Γ can be deduced by just looking at the set of invariants of Γ of degree smaller or equal to nd, and vice versa.

Cohomology of normal bundles of RL-varieties
In this section, we introduce a new family of smooth rational monomial projections of the Veronese variety ν d (P n ) ⊂ P ( n+d n ) −1 which naturally arises from level GT −varieties (see Definition 5.1) and we study their canonical modules. We called them RL−varieties to stress the link with the notions of the relative interior and levelness. We devote the rest of this work to determine the cohomology of the normal bundle of any RL−variety (see Theorem 5.9). Both, the coordinate ring and the canonical module of GT −varieties, play an important role on our computations and the proof of Theorem 5.9 is inspired by [3].
In Section 4, we have seen that the canonical module ω X d of a GT −variety X d with group a finite cyclic group Γ = M d;α 0 ,...,αn ⊂ GL(n + 1, K) of order d is identified with the ideal relint(I d ) = (x a 0 0 · · · x an n ∈ R Γ | 0 = a 0 · · · a n ) and we have proved that relint(I d ) is generated by monomials of degree d and 2d. We begin with the following definition.  (ii) Fix integers 4 ≤ n, 1 ≤ k with n even. For d := k(n + 1) and for the finite cyclic group Γ = M d;0,1,2,...,n ⊂ GL(n + 1, K) of order d, the associated GT −variety X d is level (see Corollary 4.6).

≥0
of one of the systems ( * * ) r are (d, 0, 0), (0, 0, d) and {(γ, d−2γ, γ) | γ = 0, . . . , d 2 }. Therefore we have Since x 0 · · · x n is an invariant of Γ 1 ⊂ GL(n + 1, K), as in the proof of Proposition 4.4 it follows that R Γ 1 is Gorenstein; by Proposition 4.5 we only have to check that µ d ≤ n−1+d n−1 . For n ≥ 3, it holds that From now onwards, we fix a level GT −variety X d with group a finite cyclic group Γ = M d;α 0 ,...,αn ⊂ GL(n + 1, K) of order d and we denote I d its associated GT −system. We denote η d = |C d,1 |, i.e. the number of monomials of degree d in relint(I d ) and we set N d := n+d d − η d − 1. By f d : P n → P N d we denote the morphism induced by the inverse system relint(I d ) −1 . We denote X d = f d (P n ) ⊂ P N and we call X d the RL−variety associated to X d . We have the following. x j for all i, j ∈ {0, . . . , n}, which is a sufficient condition for f d to be an embedding.
In [3], the authors develop a new method to compute the cohomology of the normal bundle of smooth rational projections of the Veronese variety ν n,d (P n ) ⊂ P ( n+d n ) −1 for which the parametrization is an embedding. Let X d be an RL−variety associated to a level GT −variety X d with group Γ. By Proposition 5.5, any RL−variety X d is of this kind. Furthermore, the relation between X d and relint(I d ) ⊂ R Γ allows us to apply this approach to any RL−variety X d . In this setting, we have the following presentation of the normal bundle N X d of the RL−variety X d ⊂ P N d (see [3, (3.3)]): Taking the long sequence of cohomology for (4), we determine the cohomology K−vector spaces H i (X d , N X d (−k)) in most cases, as the following result shows.
Proposition 5.6. Let X d ⊂ P N d be an RL−variety of dimension n ≥ 2, we have: (i) for all 0 < i < n − 1 and k ∈ Z, Proof. We fix k ∈ Z. We twist (4) by O P n (−k) and then we consider the long exact sequence of cohomology. For any i and k we obtain k)) → From the additivity of the cohomology, it follows the vanishing H i (X d , N X d (−k)) = 0 for all 0 < i < n − 1. In addition, we obtain the presentation The result follows from the Bott formulas for the cohomology of P n (see [23]).
Thus far, we have determined the dimension of H i (X d , N X d (−k)) for any i and k except: To compute them, we apply Proposition 5.6(i) to the long exact sequence of cohomology (5).
For any k we obtain the exact sequence As an immediate result, we get that for all k < d + n + 1: We focus on computing H n−1 (X d , N X d (−k)) for k ≥ d + n + 1. We need some preparation. By ∂ x 0 , . . . , ∂ xn we denote the linear operators acting on R as partial derivatives. Let m ∈ R l be a monomial and we write m = x a 0 0 · · · x an n . We denote ∂ m the composition of linear operators ∂ x 0 a 0 · · · ∂ x 0 · · · ∂ xn an · · · ∂ xn .
Lemma 5.7. Let m be a monomial of degree k − d − n − 1, let q and q ′ be monomials of degree k − n − 1 such that m divides both q and q ′ , and let 0 ≤ i = j ≤ n be integers. Then x i ∂ m q and x j ∂ m q ′ are linearly independent if and only if for any monomial m ′ = m of degree k − d − n − 1 which divides q and q ′ , x i ∂ m ′ q and x j ∂ m ′ q ′ are linearly independent.
Proof. We write m ′ = x b 0 0 · · · x bn n , m = x c 0 0 · · · x cn n , q = x a 0 0 · · · x an n , q ′ = x a ′ 0 0 · · · x a ′ n n . Assume that x i ∂ m q and x j ∂ m q ′ are linearly independent and there is m ′ = m, which divides q and q ′ , such that x i ∂ m ′ q and x j ∂ m ′ q ′ are linearly dependent. Therefore we have the equality which implies a l = a ′ l , 0 ≤ l = i, j ≤ n, a i = a ′ i − 1 and a j = a ′ j + 1. Then we obtain x an−cn n for some A, B ∈ K \ {0}, which is a contradiction.
An RL−variety X d ⊂ P N d of dimension n ≥ 2 is a smooth rational variety embedded in P N d . In [3], the authors introduce a new method to compute the cohomology of the normal bundle of varieties of this kind. With the notation of [3], we write the embedding The RL−variety X d = f d (P(U)) is the projection in P N d of the Veronese variety ν d (P(U)) ⊂ P ( n+d n ) −1 from the projective space P(T ) of dimension n+d n − N d where T ∨ is identified with the K−vector subspace of R d generated by all the monomials of degree d in relint(I d ) = (x a 0 0 · · · x an n ∈ R Γ | 0 = a 0 · · · a n ). Let 0 ≤ i = j ≤ n, l ≥ 1 and t ≥ 1 be integers. We denote D i,j : S l U ⊗ S t U → S l−1 U ⊗ S t−1 U the linear map Proposition 5.8. Let X d ⊂ P N d be an RL−variety of dimension n ≥ 2 associated to a level GT −variety X d with group Γ = M d;α 0 ,...,αn ⊂ GL(n + 1, K). Then, Proof. By [3, Theorem 2], we obtain h n−1 (X d , N X d (−d − n − 1)) = dim(µ −1 (T )), where µ : U ⊗ S d−1 U → S d U is the multiplication map, and for all k ≥ d + n + 2: 0≤i,j,r,s≤n (ker(D i,j • D r,s )).
In particular, for k = d + n + 1, d + n + 2 we can conclude that h n−1 (X d , N X d (−d − n − 1)) = η d + n(d − 1) d n + d − 1 n and H n−1 (X d , N X d (−d − n − 2)) = U ⊗ T. Moreover, for k ≥ d + n + 3 we have H n−1 (X d , N X d (−k)) ∼ = {x 0 ⊗ q 0 + · · · + x n ⊗ q n ∈ R 1 ⊗ R k−n−2 | x 0 ∂ m (q 0 )+ · · · +x n ∂ m (q n ) ∈ relint(I d ) 1 for all monomial m ∈ R k−d−n−1 }, where relint(I d ) 1 denotes the K−vector subspace of R d generated by the set C d,1 ⊂ relint(I d ) of monomials of degree d. We want to prove that H n−1 (X d , N X d (−k)) = 0 for all k ≥ d+n+3. Assume that there exist q 0 , . . . , q n ∈ R k−n−2 and a monomial m ∈ R k−d−n−1 such that 0 = u m := x 0 ∂ m (q 0 ) + · · · + x n ∂ m (q n ) ∈ relint(I d ) 1 . Therefore, any monomial that appears in u m belongs to relint(I d ) ⊂ Γ. Let q ∈ R k−n−2 be a monomial such that 0 = x i ∂ m q is a monomial that occurs in u m . By Lemma 5.7, we have that if x 0 ⊗q 0 +· · ·+x n ⊗q n ∈ H n−1 (X d , N X d (−k)), then for any monomial m ′ ∈ R k−d−n−1 , x i ∂ m ′ q ∈ relint(I d )) ⊂ R Γ 1 = R Γ d . We will show that there always exists a monomial m ′ ∈ R k−d−n−1 dividing q such that x i ∂ m ′ q / ∈ R Γ . Thus, it concludes H n−1 (X d , N X d (−k)) = 0 for all k ≥ d + n + 3. Notice that, by Proposition 5.3, M d;α 0 ,...,αn has three indices α i , α j , α l two by two distinct. We consider monomials q = x a 0 0 · · · x an n ∈ R k−n−2 and m = x b 0 0 · · · x bn n ∈ R k−d−n−1 such that m divides q and x i ∂ m q ∈ relint(I d ). In particular, we have that b j < a j for all 0 ≤ j = i ≤ n and b i ≤ a i − 1.
x 3 ] and its associated GT −threefold X 4 is level with relint(I 4 ) = (x 0 x 1 x 2 x 3 ) (see Example 5.2(iii)). We present the cohomology table from degree −9 to 0 of the normal bundle N X 4 of the smooth rational variety X 4 parametrized by the inverse system relint(I 4 ) −1 .