Castelnuovo–Mumford Regularity of Projective Monomial Curves via Sumsets

Let A={a0,…,an-1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A=\{a_0,\ldots ,a_{n-1}\}$$\end{document} be a finite set of n≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 4$$\end{document} non-negative relatively prime integers, such that 0=a0<a1<⋯<an-1=d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0=a_0<a_1<\cdots <a_{n-1}=d$$\end{document}. The s-fold sumset of A is the set sA of integers that contains all the sums of s elements in A. On the other hand, given an infinite field k, one can associate with A the projective monomial curve CA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_A$$\end{document} parametrized by A, CA={(vd:ua1vd-a1:⋯:uan-2vd-an-2:ud)∣(u:v)∈Pk1}⊂Pkn-1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \quad \mathcal {C}_A=\{(v^d:u^{a_1}v^{d-a_1}:\cdots :u^{a_{n-2}}v^{d-a_{n-2}}:u^d) \mid (u:v)\in \mathbb {P}^{1}_k\}\subset \mathbb {P}^{n-1}_k. \end{aligned}$$\end{document}The exponents in the previous parametrization of CA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_A$$\end{document} define a homogeneous semigroup S⊂N2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}\subset \mathbb {N}^2$$\end{document}. We provide several results relating the Castelnuovo–Mumford regularity of CA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_A$$\end{document} to the behavior of the sumsets of A and to the combinatorics of the semigroup S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}$$\end{document} that reveal a new interplay between commutative algebra and additive number theory.


Introduction
Let A = {a 0 , a 1 , . . ., a n−1 } ⊂ N be a set of non-negative integers where we assume that a 0 < • • • < a n−1 and set d := a n−1 .For every s ∈ N, the s-fold sumset of A, sA, is defined by 0A := {0} and for s ≥ 1, Additive number theory studies the sumsets of A. As we will see later in (0.1), for our purpose we will need to count the number of elements in sA.As observed in [21, (1.1) p.2], in order to compute |sA|, one may assume without loss of generality that a 0 = 0 and gcd(a 1 , . . ., a n−1 ) = 1.When this occurs, A is said to be in normal form.
Consider now the points a 0 = (0, d), a 1 = (a 1 , d − a 1 ), . . ., a n−1 = (d, 0) in N 2 , the set A = {a 0 , a 1 , . . ., a n−1 }, and the subsemigroup S of N 2 generated by A. Given an arbitrary infinite field k, one can associate to A the projective monomial curve C A parametrized by A: If A is in normal form, it is an algebraic curve of degree d and its defining ideal I(C A ) is the kernel of the homomorphism of k-algebras ϕ : k[x 0 , . . ., The ideal I(C A ) is homogeneous, binomial and prime, i.e., it is a homogeneous toric ideal.This provides a bridge between additive number theory and the geometry of monomial projective curves that has been recently explored in [11] and later generalized to higher dimension varieties in [6].We will follow here the same philosophy: our aim is to study some homological invariants of the projective monomial curve C A that we will now define, through the sumsets of A, and vice versa.

Denoting by k[C
Given a minimal graded free resolution (m.g.f.r.) of the graded k[x 0 , . . ., x n−1 ]-module k[C A ], where the F i 's are free modules, one has that for all i = 0, . . ., p, F i is generated by β i,j elements of degree j. is more chaotic.In this paper, the sumsets of A and the combinatorics of the semigroup S will be related to the Castelnuovo-Mumford regularity and the regularity of the Hilbert function of k[C A ], revealing a nice interplay between additive number theory and commutative algebra.Note that if n = 2, A = {0, 1} and if n = 3, C A is a hypersurface, so we will assume here that n ≥ 4.
The paper is structured as follows.In section 1, we recall some results in additive number theory, in particular the fundamental Structure Theorem and its relation to monomial curves.We define the sumsets regularity σ(A) of a finite set of integers in normal form A as the least integer such that, for all larger integers, the decomposition in the Structure Theorem holds.Several upper bounds for σ(A) that appear in the literature are recalled, in particular the Granville-Walker bound recently obtained in [14].In section 2, we analyze the structure of the semigroup S and see that the sumsets regularity of A defined in the previous section could also be called the conductor of the semigroup S. We focus on two important finite subsets of the semigroup S that will play in fundamental role later: its Apery set and its exceptional set.Both subsets can be used to characterize the Cohen-Macaulay property for k[C A ] as shown in Proposition 2.6.Section 3 contains our main results.We start by completing the characterization of the elements in the Structure Theorem given in [11,Prop. 3.4] and express the sumsets regularity of A in terms of some invariants of the monomial curve C A in Theorem 3.1.As a direct consequence, we give a new upper bound for the sumsets regularity in Theorem 3.4.We also give a combinatorial way for computing the Castelnuovo-Mumford regularity of k[C A ] in terms of the Apery and the exceptional sets of S (Theorem 3.7) and provide both upper and lower bounds for the Castelnuovo-Mumford regularity of k[C A ] in terms of the conductor of S (Theorem 3.16).In section 4, we prove in Theorem 4.3 a general result that allows to read on the Betti diagram the value of the difference between the Castelnuovo-Mumford regularity and the regularity of the Hilbert function of k[C A ]. Applied to the monomial curve C A , we deduce in Theorem 4.6 a way to characterize when the regularity is attained at the last step of a m.g.f.r.Finally, in section 5 we use our results to relate a recent result in additive number theory, the Granville-Walker bound for the sumsets regularity, to a classical result in algebraic geometry, the Gruson-Lazarsfeld-Peskine bound for the Castelnuovo-Mumford regularity in the particular case of monomial curves.More precisely, we show how to obtain the first bound from the second and vice versa.
The computations in the examples given in this paper are performed by using Singular [7] and, in particular, the library mregular.lib[2].We also used the package NumericalSgps [8] of GAP.

Notations. In this paper,
⌊x⌋ is the greatest integer less than or equal to x (floor function), while ⌈x⌉ is the least integer greater than or equal to x (ceil function).If d ∈ N and A ⊂ N, we denote d − A := {d − a : a ∈ A}.Furthermore, we will assume that all the semigroups have an identity, i.e., we don't distinguish between semigroup and monoid.
If R = ⊕ s∈N R s is a standard graded k-algebra, we denote by HF R and HP R its Hilbert function and Hilbert polynomial respectively.The least integer r such that, for all integer s ≥ r, HF R (s) = HP R (s) is called the regularity of the Hilbert function of R and we will denote it by r(R).The Castelnuovo-Mumford regularity of R will be denoted by reg(R) and we will use the abbreviation m.g.f.r. for minimal graded free resolution.
Finally, when we draw part of a semigroup S ⊂ N 2 as in Figures 2.1 and 2.2, filled circles represent points in S while empty squares represent points outside S, i.e., gaps of S.

The Structure Theorem
In this section we give an overview of some results in additive number theory and their connection to monomial curves.Let's first recall the so-called Structure Theorem, one of the main results in additive number theory.
is a finite set in normal form, then there exist integers c 1 , c 2 ∈ N and finite subsets for all s ≥ max{1, s 0 } where The elements in the Structure Theorem have recently been characterized in [11,Prop. 3.4] in terms of the curve C A and some of its invariants.If A is a finite set in normal form, it is well known that C A has two possible singular points, P 1 = (1 : 0 :  11,Prop. 3.4]).Following notations in Theorem 1.1, for i = 1, 2 the following claims hold: (1) c i is the conductor of S i . ( Definition 1.3.The least integer σ such that the decomposition (1.1) in Theorem 1.1 holds for all s ≥ σ will be called the sumsets regularity of A and we will denote it by σ(A).
Theorem 1.1 provides an upper bound for σ(A) that is generally far from its real value: σ(A) ≤ (n − 2)(d − 1)d.After Nathanson's proof, other proofs of Theorem 1.1 have been published, [25,13,14].In these articles, the authors give the following better upper bounds for σ(A): Note that in [25,13,14], the union in equation (1.1) is not shown to be disjoint but this is shown in [19] for the Granville-Walker bound and, as Besides giving a great upper bound for σ(A), Granville and Walker also characterize the sets A for which this bound is attained.
and only if, either A or d − A belongs to one of the following two families: for some a such that 2 ≤ a ≤ d − 2.
Note for any A belonging to one of the two families in Theorem 1.4, the monomial curve C A is smooth.

The structure of the homogeneous semigroup and its relation to the sumsets
As already observed, associated to a set of integers A = {a 0 = 0 < a 1 < • • • < a n−1 = d} in normal form, one has the set where a i = (a i , d − a i ) for all i = 0, . . ., n − 1, that we will call its homogenization.A semigroup S in N 2 generated by a set A of this form will be said to be homogeneous of degree d.
It is trivial to verify that the sumsets of A are completely determined by those of A since, for each s ∈ N, sA = {(α, sd − α) : α ∈ sA} .In particular, for any s ∈ N, |sA| = |sA|.Furthermore, the semigroup S generated by A satisfies that S = ⊔ ∞ s=0 sA.Note that each sA lies on the "line" L s := {(x, y) ∈ N 2 : x + y = sd}.
We can apply the Structure Theorem to improve our knowledge on the sumsets of A and the semigroup S. By Theorem 1.1 and Proposition 1.2, we have that for all s ≥ σ(A), sA consists on a central interval and, outside that interval, a copy of the non-trivial part of the semigroups S 1 and S 2 , i.e., for all s ≥ σ(A), Furthermore, σ(A) is the least integer such that this decomposition is satisfied for all s ≥ σ(A).More precisely, for s ≥ σ(A), when we go from sA to (s + 1)A, gaps coming from S 1 move up while gaps coming from S 2 move to the right, and there are no other gaps in (s + 1)A than the ones coming from sA, as shown in Figure 2.1.And σ(A) is the least integer such that this occurs.For this reason, the regularity of the sumsets of A, σ(A), could also be called the conductor of the homogeneous semigroup S and denoted by σ(S).If no confusion arises, from now on we will simply denote this number by σ, i.e., σ = σ(S) = σ(A).

Central interval
Structure of the sumsets of A. For s ≥ σ, we distinguish three disjoint areas: the central interval and the copies of the non-trivial parts of S 1 and S 2 .
We can relate the conductor of the semigroup S to the Hilbert function regularity of k[C A ] on the one hand, and to the conductors of the semigroups S 1 and S 2 on the other.This relation will become more precise later in Theorem 3.1.
be a finite set in normal form and σ be its sumsets regularity.
Let's now focus on the three semigroups S 1 , S 2 and S. For i = 1, 2, we define the Apery set of S i with respect to d as Ap i := {a ∈ S i : a − d / ∈ S i }.We know that Ap i is a complete set of residues modulo d, and hence Ap Definition 2.3.The Apery set AP S of S and the exceptional set E S of S are defined as follows: Remark 2.4.As a consequence of Theorem 1.1, one gets that, if σ is the conductor of S, then The Cohen-Macaulayness of k[C A ] is characterized in terms of AP S and E S as we will show in Proposition 2.6 .Let's previously prove the following easy lemma.Lemma 2.5.For all i = 1, . . ., d − 1, the following claims hold: ( ∈ AP S and there exist natural numbers x > r i and y > t d−i such that (x, t d−i ) ∈ AP S and (r i , y) ∈ AP S .
Proof.(1) is trivial.In order to prove (2), take i ∈ {1, 2, . . ., d − 1}.Since r i ∈ S 1 , there exists a natural number y > t d−i such that (r i , y) ∈ S and if we choose the least y ∈ N satisfying this property, then (r i , y) ∈ AP S .The proof of the existence of x is analogous.
( We focus now on the distribution of points (x, y) in AP S and E S on the levels given by the sumsets of A.
Proof.Let's count the number of elements in AP s for all s ∈ N. Note that and since E 0 = E 1 = ∅ and sA = ∅ if s < 0, one gets that the formula holds if s ≤ 1.Consider now s ≥ 2. Since for each element s ∈ (s − 1)A, neither s + a 0 nor s + a n−1 belong to AP s , every element in (s − 1)A provides two elements in sA that do not belong to AP s and any other element in sA belongs to AP s .But we are counting some of those elements twice, precisely the s ∈ sA such that s − a 0 ∈ (s − 1)A and s − a n−1 ∈ (s − 1)A.Now for such an element s, either s − a 0 − a n−1 / ∈ (s − 2)A and hence s ∈ E s , or (x, y) − a 0 − a n−1 ∈ (s − 2)A.This provides the following formula, and the result follows.
Remark 2.10.As a consequence of the previous theorem and Remark 2.4, we obtain that Remark 2.12.The result in Corollary 2.11 holds in a more general setting.For a graded (or local) k-algebra R of Krull dimension two, the differences between two consecutive elements in the sequence (HF R (s)− HF R (s − 1)) ∞ s=0 are the coefficients of its h-polynomial that are known to be non-negative when R is Cohen-Macaulay [24].Thus, the sequence (HF R (s) − HF R (s − 1)) ∞ s=0 is increasing.
Note that if one removes the Cohen-Macaulay hypothesis, then the result in Corollary 2.11 may be wrong as the first example below shows.But this property does not characterize arithmetically Cohen-Macaulay curves as the second example shows.

Regularity, sumsets and semigroups
We start this section by giving a characterization of σ, the sumsets regularity of A, which is also the conductor of the semigroup S, in terms of the curve C A and its invariants.This result already appears in [12] and it concludes the characterization of the elements in Structure Theorem given in [11,Prop. 3.4].( d ⌉, the sumset sA decomposes as the union of three disjoint subsets As a direct consequence of Proposition 3.2 we recover the well-known fact that for any n ≥ 4, the rational normal curve, i.e., the curve C A given by A = [0, n − 1], is the only projective monomial curve in P n−1 k which is both smooth and arithmetically Cohen-Macaulay.
By combining the Erdös-Graham bound for the condutor of a numerical semigroup and the bound for the Castelnuovo-Mumford regularity of a projective monomial curve given by L'vovsky, we obtain the following new bound for the sumsets regularity.This bound is different from the already known bounds recalled in Section 1. Indeed, in Example 3.
Then, the least integer σ such that the decomposition (1.1) in Theorem 1.1 holds for all s ≥ σ, i.e., the sumsets regularity of A, satisfies  (1) where the symbol ∼ = means that there exists an isomorphism of Z-graded modules.
When C A is arithmetically Cohen-Macaulay, S ′ = S by Proposition 2.6 ( 6) so H 1 m (k[S]) = 0 as we already know.For i = 1, 2, let Then, by the equivalent definition for the the Castelnuovo-Mumford regularity given in [10], one has that . The proof of Theorem 3.7 will then be a consequence of the following two lemmas that relate the local cohomology modules ) and H 2 m (k[S]) to the numbers m (E S ) and m (AP S ).Note that the relation m (E S ) = end H 1 m (k[S]) + 1 stated in Lemma 3.9 also holds when C A is arithmetically Cohen-Macaulay since both numbers are −∞ in this case.Lemma 3.9.If S ′ = S, i.e., if C A is not arithmetically Cohen-Macaulay, then max{s : Therefore, E ′ S ⊂ S ′ \ S and we get that max{s : E ′ s = ∅} ≤ max{s : (S ′ \ S) ∩ L s = ∅}.Conversely, let (x, y) ∈ (S ′ \ S)∩L s be an element such that s is maximum.Then, (x, y)+a 0 ∈ S and (x, y) + a n−1 ∈ S, and hence (x, y) ∈ E ′ s .So max{s : E ′ s = ∅} ≥ max{s : (S ′ \ S) ∩ L s = ∅} and the equality max{s : E s+2 = ∅} = max{s : (S ′ \ S) ∩ L s = ∅} follows.By Lemma 3.8 (1), it implies that m (E S ) = end H 1 m (k[S]) + 1. Observe that in the previous proof, we show that E ′ S ⊂ S ′ \ S. Equality, which would be a result stronger than the one stated in Lemma 3.9, is wrong in general.Using the example given in [16,Example 3.2], we show that those two sets may be different.
Therefore, in this case one has that m (AP S ) = end H 2 m (k[S]) + 2. Proof.Let (x, y) ∈ AP s+2 be an element such that s is maximal and consider the element ), one can assume without loss of generality that x − d / ∈ S 1 .Then, there exists ) + 1 by Lemma 3.9, and end Note that there exist curves such that the maximum in Theorem 3.7 is equal to m (E S ) and not equal to m (AP S ), and vice versa.For instance, if C A is arithmetically Cohen-Macaulay, then m (AP S ) > m (E S ) = −∞.But there also exist non-arithmetically Cohen-Macaulay curves such that m (AP S ) > m (E S ).In order to illustrate these facts, we use the same curves as in Example 2.13.Example 3.13.

reg(k[C
In particular, we obtain that reg(k The Castelnuovo-Mumford regularity of the semigroup ring k[S] can also be bounded from above and from below in terms of σ, the conductor of S.These bounds will be given in Theorem 3.16 where we distinguish two cases depending on the value of σ in Theorem 3.1.Let's first prove a lemma that will be needed in the proof.Recall that for i = 1, 2, the Fröbenius number of S i , denoted by F (S i ), is the largest gap of S i , i.e., F Proof.One has that F (S 1 ) + d ∈ Ap 1 and consider y ∈ Ap 2 such that F (S 1 ) + d + y ≡ 0 (mod d).Note that y = 0.By Lemma 2.5, there are two options: either (F (S 1 ) + d, y) ∈ AP S , or there exist y ′ ≥ y such that (F (S 1 ) + d, y ′ ) ∈ AP S .In both cases, there exists y ≥ 1 such that (F (S 1 ) + d, y) ∈ AP S and, analogously, there exists x ≥ 1 such that (x, F (S 2 ) + d) ∈ AP S .By Theorem 3.7, We have the following bounds for the Castelnuovo-Mumford regularity of k[C A ]: (1 In both cases, the upper bound is a consequence of Theorem 3.  1.
Table 1.Examples where the bounds in Theorem 3.16 are attainted.

A r(k[C
The following result is more precise than the one stated in Theorem 3.16 in a particular case.It gives, in this case, the precise relationship between the three regularities, in the sense of Castelnuovo-Mumford, of the Hilbert function and of the sumsets. On the other hand, Using the previous results, we can give a new proof for the bound obtained by J. Elias in [11] for arithmetically Cohen-Macaulay curves.Proposition 3.20 ([11,Thm. 4.7]).Proof.The case λ = 1 is proved just before the proposition so assume that λ ≥ 2. Since for all µ = 1, 2, . . ., λ − 1, i (−1) i β i,reg +n−µ = 0, by equation (4.2) one gets that HP A (t) = HF A (t) for all t ≥ reg −λ + 1, i.Note that the previous result is general.In its proof, we do not use that the ring is the coordinate ring of a monomial projective curve.This proves the following result that improves [9, Thm.(1) If we focus on the secondary diagonals of the Betti diagram starting from the bottom right of the table, the number λ in the previous theorem is the label of the first diagonal such that the alternating sum of the Betti numbers on this diagonal is not 0.
(2) If p denotes the projective dimension of the module M , the previous result implies that β p,reg(M )+p = 0, i.e., the regularity is attained at the last step of a m.g.f.r. of M , if, and only if, λ = n − p, i.e., D = n − 1 − p.This occurs, in particular, whenever M is a Cohen-Macaulay module so, in this case, reg(M ) − r(M ) = n − 1 − p which is a well-known fact; see, e.g., [9,Cor. 4.8].

The relation between known bounds for σ and reg(k[C A ])
In this final section, we show how the bound for σ recently obtained by Granville and Walker in [14] and the classical bound for reg(k[C A ]) given by the Gruson-Lazarsfeld-Peskine Theorem [15] are related.As a consequence of some of our results, we obtain that each of these bounds can be deduced from the other.
Let's first recall these two bounds.Consider A = {a 0 = 0 < a 1 < • • • < a n−1 = d} ⊂ N a set in normal form.In section 1, we presented several upper bounds for σ, the sumsets regularity of A, which we also called the conductor of the homogeneous semigroup S. The best bound is the one given in [14, Thm.1] by Granville and Walker: Let's first show that the Granville-Walker bound (5.1) can be deduced from (5.2) using Theorem 3.1.We start with the following result that bounds one of the terms in Theorem 3.1.Conversely, in order to show that (5.2) can be deduced from (5.1), we will use the additional result of Granville and Walker recalled in Theorem 1.4 where all the sets A in normal form such that the bound in (5.1) is attained are characterized.We distinguish three cases: A ] := k[x 0 , . . ., x n−1 ]/I(C A ) the homogeneous coordinate ring of C A , one has that k[C A ] is isomorphic to Imϕ = k[S], the semigroup ring of S. If HF A denotes the Hilbert function of k[C A ], by [11, Prop.2.3] one has that (0.1) |sA| = HF A (s), for all s ∈ N.
The non-zero integers β i,j are invariants of the module k[C A ] called its graded Betti numbers, and we can arrange them in the Betti diagram of k[C A ], a table whose entry in column i and row j isβ i,i+j .The size of the Betti diagram of k[C A ] is measured by two important invariants of k[C A ]: the label of the last column is the projective dimension of k[C A ], pd(k[C A ]) = p,the index of the last free module in any m.g.f.r. of k[C A ], while the label of the last row is its Castelnuovo-Mumford regularity,(0.2) reg(k[C A ]) := max{j − i : β i,j = 0, 0 ≤ i ≤ p, j ≥ 0} .The projective dimension is controlled by the Auslander-Buchsbaum formula: pd(k[C A ]) = n − depth(k[C A ]).As the Krull dimension of k[C A ]is 2 and the ideal I(C A ) is prime, the depth of k[C A ] can only be 1 or 2. Thus, the projective dimension is either n − 2 if k[C A ] is Cohen-Macaulay, or n − 1 otherwise.The behaviour of the Castelnuovo-Mumford regularity of k[C A ]

k and P 2 =
(0 : • • • : 0 : 1) ∈ P n−1 k .Moreover, if δ(C A , P ) denotes the singularity order of P , then δ(C A , P 1 ) = |N \ S 1 | and δ(C A , P 2 ) = |N \ S 2 |, where S 1 and S 2 denote the numerical semigroups generated by A and d − A, respectively.Using that C A has degree d, one gets ([11, Prop.3.1]) that for all s ) If sd − c 2 < c 1 , the central interval in the previous decomposition of sA does not exist, and hence σ ≥ ⌈ c 1 +c 2 d ⌉.Proof.Both results are consequences of the discussion before Figure 2.1.(

Figure 2 .
2 shows how points in E S and AP S look like when one draws the semigroup S.

Theorem 3 . 1 .
The least integer σ such that the decomposition (1.1) in Theorem 1.1 holds for all s ≥ σ is σ = max r(k[C A ]), c 1 + c 2 d where r(k[C A ]) is the regularity of the Hilbert function of k[C A ] and c i is the conductor of the numerical semigroup S i for i = 1, 2. Proof.By Lemma 2.1, σ ≥ max r(k[C A ]), ⌈ c 1 +c 2 d ⌉ .Conversely, for s ≥ max r(k[C A ]), ⌈ c 1 +c 2 d ⌉ , one has that (2.1) is satisfied by applying (1.2).Moreover, since sd − c 2 ≥ c 1 , one has that Since both sets sA and C 1 ⊔ [c 1 , sd − c 2 ] ⊔ (sd − C 2 ) are finite and have the same cardinality, they are equal, so max r(k[C A ]), ⌈ c 1 +c 2 d ⌉ ≥ σ and the resut follows.Given a subset A ⊂ N in normal form, it is not easy to know in advance whether σ = r(k[C A ]) or σ = ⌈ c 1 +c 2 d ⌉.But in some cases it is, as Proposition 3.2 and Corollary 3.21 show.Proposition 3.2.

= 6 .
3 (2), both numbers s EG 0 and s L 0 introduced in Theorem 3.4 are strictly lower than the Granville-Walker bound s GW 0 recalled in Section 1: s EG 0 = 4, s L 0 = 5, and s GW 0 This new bound deserves to be studied in the future.Theorem 3.4.
Combining these two bounds, one gets that ⌈ c 1 + c 2 d ⌉ ≤ s EG 0 .On the other hand, using the known fact that r(k[C A ]) ≤ reg(k[C A ]) ([9, Thm 4.2]) and that reg(k[C A ]) ≤ s L 0 by [20, Prop.5.5], the upper bound follows from Theorem 1.1.In order to express reg(k[C A ]) in terms of AP S and E S , let's introduce the following notations.Definition 3.5.For any set A ⊂ N in normal form, consider the homogeneous semigroup S ⊂ N 2 associated.We define • m (E S ) := max ({s ∈ N : E s+1 = ∅}) (and m (E S ) := −∞ if E S = ∅), and • m (AP S ) := max ({s ∈ N : AP s = ∅}).Remark 3.6.(1) Note that the maxima in Definition 3.5 are attained because AP S and E S are finite subsets of N 2 by Remark 2.4.In fact, m (E S ) ≤ σ and m (AP S ) ≤ σ + 1. (2) Both m (E S ) and m (AP S ) can be expressed in terms of the sumsets of A as follows: • m (E S ) = max ({s ∈ N : ∃α ∈ sA such that α − d ∈ sA \ (s − 1)A}), and • m (AP S ) = max ({s ∈ N : ∃α ∈ sA such that α / ∈ (s − 1)A and α − d / ∈ (s − 1)A}).The following result gives a combinatorial way for computing the Castelnuovo-Mumford regularity of k[C A ]. Theorem 3.7.The Castelnuovo-Mumford regularity of the projective monomial curve C A is reg(k[C A ]) = max{m (E S ) , m (AP S )} .In order to prove this result, let's recall some known facts on the local cohomology modules of the coordinate ring of C A , k[C A ] ∼ = k[S].For k[C A ], there are at most two non-trivial local cohomology modules, H 1 m (k[S]) and H 2 m (k[S]), where m denotes the irredundant ideal.Furthermore, these two modules are completely characterized in terms of the semigroup S.
) [1, Ex. 4.3].For A = {0, 5, 9, 11, 20}, m (E S ) = 4 and m (AP S ) = 5, so C A is not arithmetically Cohen-Macaulay, and reg(k[C A ]) = 5 = m (AP S ) > m (E S ).Remark 3.14.Let m be the maximal homogeneous ideal of k[S] ∼ = k[C A ] and q := u d , v d .We know that q is a minimal reduction of m.Denote by red(k[C A ]) the reduction number of m with respect to q, i.e., red(k[C A ]) = min{s ∈ N : m s+1 = qm s }, which can be computed as red(k[C A ]) = min{s ∈ N : (s + 1)A = sA + {a 0 , a n−1 }} .By the discussion at the beginning of section 2, it is clear that red(k[C A ]) = m (AP S ), and we can characterize when reg(k[C A ]) = red(k[C A ]) in a combinatorial way: by Theorem 3.7, 7 and Remark 3.6 (1).If σ = r(k[C A ]) ≥ ⌈ c 1 +c 2 d ⌉, then we apply the known fact r(k[C A ]) ≤ reg(k[C A ]), see [9, Thm 4.2], and in the other case, the lower bound is the one given in Lemma 3.15.Example 3.17.In order to illustrate that all the upper and lower bounds in Theorem 3.16 are sharp, the values of r(k[C A ]), ⌈ c 1 +c 2 d ⌉, σ and reg(k[C A ]) in four different examples are displayed in Table

Proof.
Set s 0 := ⌈ d−1 n−2 ⌉.By Corollary 2.11, the sequence (|sA| − |(s − 1)A|) s∈N is increasing and its limit is d.Indeed, as observed in the proof of this corollary, |sA| − |(s − 1)A| = s j=0 | AP j | for all s ∈ N. On the other hand, | AP S | = d by Proposition 2.6 (4) and, by [18, Thm.1], |sA| − |(s − 1)A| ≥ d if s ≥ s 0 .Therefore, | AP s | = 0 for all s > s 0 , and hence reg(k[C A ]) ≤ s 0 by Theorem 3.7.Finally, as a consequence of Theorem 3.16, one gets a sufficient condition for σ to be equal to ⌈ c 1 +c 2 d ⌉ in Theorem 3.1.The condition is expressed in terms of the difference between the Castelnuovo-Mumford regularity and the regularity of the Hilbert function of k[C A ]. We will see in the next section how this condition can be characterized in terms of the Betti numbers of k[C A ]. Corollary 3.21.

4 .
1 by Theorem 3.16, so D ≤ 1.The shape of the Betti diagram In this section, we relate both regularities reg(k[C A ]) and r(k[C A ]) in terms of the Betti diagram of k[C A ] that can be used to characterize the difference D:= reg(k[C A ]) − r(k[C A ]). Recall that the projective dimension of k[C A ] is either n − 2 if the ring k[C A ] is Cohen-Macaulay, or n − 1 otherwise.Proposition 4.2.If λ ≥ 1 be the least positive integer such that i (−1) i β i,reg +n−λ = 0, then r(k[C A ]) = reg(k[C A ]) − λ + 1, i.e., D = λ − 1, and HP A (reg −λ) − HF A (reg −λ) = i (−1) n+i−1 β i,reg +n−λ .
4.2]: Theorem 4.3.Let M be a finitely generated graded module over k[x 0 , . . ., x n−1 ], and denote by D the difference between the Castelnuovo-Mumford regularity and the regularity of the Hilbert function of M , i.e., D := reg(M ) − r(M ).Then, D + 1 is the least non-negative integer λ ≥ 0 such that i (−1) i β i,reg(M )+n−λ = 0, where the β ij are the graded Betti numbers of M .Remark 4.4.
e., D = 1) in Proposition 4.2 includes all arithmetically Cohen-Macaulay curves since β n−1,reg +n−1 = β n−1,reg +n−2 = 0 and β n−2,reg +n−2 = 0 in this case.But there are also non-arithmetically Cohen-Macaulay curves C A such that D = reg(k[C A ]) − r(k[C A ]) = 1 as we will see in the next example.Example 4.5.Different values of D = reg(k[C A ]) − r(k[C A ]) and different shapes for the Betti diagram of k[C A ] are obtained in the following four examples of monomial curves in P 4 k .

Proof.
The equivalence (1)⇔(3) is a direct consequence of Proposition 4.2 as observed in Remark 4.4(2).Therefore, we only have to prove (1)⇔(2).It is well known that the maximal degree of the minimal (n− depth(R/I))-syzygies of k[C A ] is equal to end H depth(R/I) m (k[S]) + n.This is, e.g., a consequence of [5, Cor.2.2].If k[C A ]is not Cohen-Macaulay, then by Theorem 3.7, its proof, and Lemma 3.9, one has that reg(k[C A ]) = m (E S ) if and only if end H 1 m (k[S]) + 1 = m (E S ) ≥ end H 2 m (k[S]) + 2, i.e., if and only if the Castelnuovo-Mumford regularity is attained at the last step of a m.g.f.r. of k[C A ] by (3.1) and the previous observation.This proves the equivalence between (1) and (2).

Finally
, let's focus on monomial curves in P 3 k .Since these curves have codimension 2, they have some additional properties.Proposition 4.7.Let A ⊂ N be a set in normal form with |A| = 4 and consider the associated monomial curve C A ⊂ P 3 k .(1) The Castelnuovo-Mumford regularity is attained at the last step of a m.g.f.r. of k[C A ]. (2) Setting D := reg(k[C A ]) − r(k[C A ]), one has that 0 ≤ D ≤ 1.More precisely,D = 0 ⇐⇒ k[C A ] is not Cohen-Macaulay ⇐⇒ reg(k[C A ]) = m (E S ) ≥ m (AP S ) , D = 1 ⇐⇒ k[C A ] is Cohen-Macaulay ⇐⇒ reg(k[C A ]) = m (AP S ) > m (E S ) .Proof.(1)is a particular case of [3, Cor.2.13].By Proposition 4.2 and Remark 4.4 (2), this implies that either D = 0 if C A is not arithmetically Cohen-Macaulay, or D = 1 if C A is arithmetically Cohen Macaulay.(2) then follows from Theorem 4.6.

(5. 1 ) σ ≤ d − n + 2 .
On the other hand, a classical and important result in algebraic geometry provides an upper bound for the Castelnuovo-Mumford regularity of any reduced and irreducible projective curve in terms of its degree and codimension.It is the Gruson-Lazarsfeld-Peskine Theorem; see, e.g.,[9, Thm.5.1].Applied to the monomial projective curve C A , the Gruson-Lazarsfeld-Peskine Theorem claims that (5.2) reg(k[C A ]) ≤ d − n + 2 .
[17]f k[S] is not Cohen-Macaulay, the ring k[S ′ ] is called the Cohen-Macaulayfication of k[S].This is because S = S ′ by Proposition 2.6 (6) and k[S ′ ] is the least Cohen-Macaulay intermediate between k[S] and its field of fractions; see [4, Remark 4.7].(2)For a general affine semigroup ring S, the Cohen-Macaulay property of the semigroup ring k[S] may depend on the characteristic of the field k, as shown in[17].However, by Proposition 2.6, it is clear that this is not the case for a homogeneous semigroup S ⊂ N 2 .