Sumsets and Projective Curves

Abstract

The aim of this note is to exploit a new relationship between additive combinatorics and the geometry of monomial projective curves. We associate to a finite set of non-negative integers $A=\{a_1,\ldots , a_n\}$ a monomial projective curve $C_A\subset \mathbb P^{n-1}_{{\mathbf {k}}}$ such that the Hilbert function of $C_A$ and the cardinalities of $sA:=\{a_{i_1}+\cdots +a_{i_s}\mid 1\le i_1\le \cdots \le i_s\le n\}$ agree. The singularities of $C_A$ determines the asymptotic behaviour of |sA|, equivalently the Hilbert polynomial of $C_A$, and the asymptotic structure of sA. We show that some additive inverse problems can be translate to the rigidity of Hilbert polynomials and we improve an upper bound of the Castelnuovo-Mumford regularity of monomial projective curves by using results of additive combinatorics.

1 Introduction

Let $A=\{a_1,\ldots , a_n\}$, $n\ge 2$, be a set of different non-negative integers; we assume that $a_1<\cdots <a_n$. Given a non-negative integer $s\ge 1$ the s-fold iterated sumset of A is

$$\begin{aligned} sA=\{a_{i_1}+\cdots +a_{i_s}\mid 1\le i_1\le \cdots \le i_s\le n\}, \end{aligned}$$

we set $0A=\{0\}$; notice that $1 A=A$.

Following Nathanson, a direct problem in additive combinatorics is a problem in which we try to determine the structure and properties of |sA|, $s\ge 0$, when the set A is known. On the other hand, an inverse problem in additive combinatorics is a problem in which we attempt to deduce properties of A from properties of sA, $s\ge 0$, [22].

The aim of this paper is to establish and to study a bridge between additive combinatorics and the geometry of monomial projective curves. We argue back and forth: we use results of monomial projective curves to recover or to improve results of additive combinatorics and vice versa, see Theorems 4.3 and 4.7. In particular, we show that some inverse problems can be translate in terms of the rigidity of Hilbert polynomials, see Sects. 4 and [9].

In this paper, we have selected some significative results of the geometry of monomial projective curves and additive combinatorics; there are a huge number of results and properties of both areas to link that we will consider elsewhere, see [5].

The contents of the paper is the following. In the second section, following [8], we attach to the set A a monomial projective curve $C_A\subset \mathbb P_{{\mathbf {k}}}^{n-1}$. The Hilbert function of $C_A$ and the cardinalities of sA, $s\ge 0$, agree. Some previous results can be found in Ref. [16].

In the section, three we use the data provided by the singularities of $C_A$ to determine the asymptotic behaviour of |sA|, equivalently the Hilbert polynomial of $C_A$, Proposition 3.1. As a consequence we can describe the asymptotic decomposition of sA of the so-called fundamental result of additive combinatorics, Propositions 3.3 and 3.4.

The Sect. 4 is devoted to recover, by considering generic hyperplane sections of $C_A$, some additive inverse results and to link them with rigid polynomials and rigid properties, Proposition 4.2, Theorem 4.3. We finish the paper improving an upper bound on the Castelnuovo-Mumford regularity of $C_A$ established in [1] using a result of Lev on the growth of |sA|, Theorem 4.7.

For the basic results on algebra, algebraic geometry or additive number theory we will use: [2, 13, 22]. The computations of this paper are performed by using Singular, [6].

Notations

In this paper ${\mathbf {k}}$ is an arbitrary infinite field. Let $R=\sum _{i\ge 0} R_i$ be an standard ${\mathbf {k}}=R_0$ algebra, i.e., $R={\mathbf {k}}[R_1]$. We denote by ${{\text {H\!F}}}_R$ the Hilbert function of R, i.e., ${{\text {H\!F}}}_R(i)={\text {dim}}_{{\mathbf {k}}} R_i$ for all $i\ge 0$. It is known that there exists a rational coefficient polynomial ${\text {H\!P}}_R$, Hilbert polynomial of R, such that ${\text {H\!P}}(i)={{\text {H\!F}}}(i)$ for $i\gg 0$.

Given a set B of non-negative integers $b_1,\dots ,b_n$ we denote by $\langle b_1,\ldots ,b_n\rangle$ the sub-semigroup of $\mathbb N$ generated by B. Given a multi-index $\alpha =(\alpha _1,\ldots ,\alpha _n) \in \mathbb N^n$ we define its total order by $|\alpha |=\sum _{i=1}^n \alpha _i$ and the total order with respect to A by $|\alpha |_A=\sum _{i=1}^n a_i \alpha _i$.

2 The Bridge Between Additive Number Theory and Projective Curves

We first show that we can consider several straight simplifications on the set A and an easy property on the growth of |sA|, see [22],

Lemma 2.1

Given a set of non-negative integers $A=\{a_1,\ldots , a_n\}$, $n\ge 2$, with $a_1<\cdots <a_n$, it holds:

• (1) To compute |sA| we may assume that $a_1=0$ and $\gcd (a_2,\ldots , a_n)=1$,

• (2) under the above conditions, $|(s+1)A|\ge |sA|+n-1$ for all $s\ge 0$.

Proof

• (1) Let us consider $A'=\{0, (a_2-a_1)/d,\ldots , (a_n-a_1)/d\}$ where $d=\gcd (a_2-a_1,\ldots , a_n-a_1)$. It is easy to see that $|sA| =|sA'|$ for all $s\ge 0$.

• (2) Assume that A satisfies the conditions of (1). Since the maximum of sA is $sa_n$ we deduce that $s a_n+a_2,\ldots , sa_n+a_n\in (s+1)A\setminus sA$, so we get the claim: $|(s+1)A|\ge |sA|+n-1$ for all $s\ge 0$.

Given a general set of non-negative integers A, the associated set $A'$ of the proof of the previous Lemma, is called the normal form of A, see [22]. From now on we assume that a set A satisfies Lemma 2.1 (1).

Next, we recall the key construction of [8].

Definition 2.2

We denote by R(A) the ${\mathbf {k}}$-subalgebra of ${\mathbf {k}}[t,w]$ generated by $t^{a_i} w$, $i=1,\dots , n$. We consider ${\mathbf {k}}[t,w]$ endowed with the grading defined by ${\text {deg}}(t)=0$, ${\text {deg}}(w)=1$.

Let $\phi ={\mathbf {k}}[X_1,\ldots , X_n]\longrightarrow {\mathbf {k}}[t,w]$ the degree zero ${\mathbf {k}}$-algebra morphism defined by $\phi (X_i)=t^{a_i} w$. We have ${\text {Im}}(\phi )=R(A)$ and the homogeneous piece of degree s of R(A), i.e. $R(A)_s$, admits the ${\mathbf {k}}$-basis

$$\begin{aligned} t^{\alpha }w^s, \quad \alpha \in sA. \end{aligned}$$

(1)

From this fact we get:

Proposition 2.3

[8, Sect. 2] For all $s\ge 0$ it holds ${{\text {H\!F}}}_{R(A)}(s)=|sA|$.

In the following result a system of generators of ${\text {Ker}}(\phi )$ is computed:

Proposition 2.4

[4, Proposizione 2.2], [8, Proposition 6.4] The kernel of $\phi$ is generated by the binomials $X^\alpha -X^\beta$, $\alpha , \beta \in \mathbb N^n$, such that $|\alpha |=|\beta |$ and $|\alpha |_A=|\beta |_A$.

Next, we link R(A) with a suitable monomial projective curve.

Definition 2.5

Given a set $A=\{a_1=0,a_2,\ldots , a_n\}$ such that $a_1<\cdots < a_n$ and $\gcd (a_2,\ldots , a_n)=1$ we consider the monomial curve $C_A$ of $\mathbb P_{{\mathbf {k}}}^{n-1}$ defined by the Kernel of

$$\begin{aligned} \begin{array}{cccc} \psi &{} {\mathbf {k}}[X_1,\ldots , X_n] &{} \longrightarrow &{} {\mathbf {k}}[u,v] \\ &{} X_i &{} \mapsto &{} u^{a_n-a_i}v^{a_i} \end{array} \end{aligned}$$

If we consider the standard grading of ${\mathbf {k}}[u,v]$ we get that ${\text {Ker}}(\psi )=I_{A}$ is a homogeneous ideal of ${\mathbf {k}}[X_1,\ldots , X_n]$. We denote by ${\mathbf {k}}[C_A]:={\mathbf {k}}[X_1,\ldots ,X_n]/{\text {Ker}}(\phi )$ the homogeneous coordinate ring of $C_A$. We write ${{\text {H\!F}}}_{C_A}={{\text {H\!F}}}_A$ and ${\text {H\!P}}_{C_A}={\text {H\!P}}_A$.

Proposition 2.6

For all set $A=\{a_1=0,a_2,\ldots , a_n\}$ we have that ${\text {Ker}}(\phi )=I_{A}$. Hence $R(A)\cong {\mathbf {k}}[C_A]$ as graded k-algebras.

Proof

We first prove that ${\text {Ker}}(\phi )\subset I_A$. Let’s consider a binomial $X^\alpha -X^\beta$, $\alpha , \beta \in \mathbb N^n$, with $|\alpha |=|\beta |$ and $|\alpha |_A=|\beta |_A$. Then

$$\begin{aligned} \psi (X^\alpha -X^\beta )=u^{a_n |\alpha |-|\alpha |_A}v^{|\alpha |_A}-u^{a_n |\beta |-|\beta |_A}v^{|\beta |_A}=0, \end{aligned}$$

by Proposition 2.4 we get that ${\text {Ker}}(\phi )\subset {\text {Ker}}(\psi )=I_A$.

Next, we prove that $I_A\subset {\text {Ker}}(\phi )$. Let $F\in I_A$ be a polynomial, so

$$\begin{aligned} F(u^{a_n}, u^{a_n-a_2}v^{a_2},\ldots , u^{a_n-a_{n-1}}v^{a_{n-1}}, v^{a_n})=0. \end{aligned}$$

If $X^\alpha$, $\alpha \in \mathbb N^n$, is a monomial of F then

$$\begin{aligned} X^\alpha (u^{a_n}, u^{a_n-a_2}v^{a_2},\ldots , u^{a_n-a_{n-1}}v^{a_{n-1}}, v^{a_n})= u^{a_n|\alpha |-|\alpha |_A}v^{|\alpha |_A} \end{aligned}$$

Hence we may assume that F is a homogeneous polynomial

$$\begin{aligned} F=\sum _{i=1}^d \lambda _i X^{\alpha _i} \end{aligned}$$

such that $|\alpha _i|_A=c$, $a_n|\alpha _i|=c+d$ and $\lambda _i\in {\mathbf {k}}\setminus \{0\}$.

Since $F(u^{a_n}, u^{a_n-a_2}v^{a_2},\ldots , u^{a_n-a_{n-1}}v^{a_{n-1}}, v^{a_n})=0$ we deduce that $\sum _{i=1}^d \lambda _i =0,$ so

$$\begin{aligned} F=\sum _{i=1}^{d-1} \lambda _i ( X^{\alpha _i}- X^{\alpha _d}) \in {\text {Ker}}(\phi ). \end{aligned}$$

Remark 2.7

We write $\mathcal B_A=\frac{{\mathbf {k}}[C_A]}{X_1 {\mathbf {k}}[C_A]}$, notice that $\mathcal B_A$ is a graded algebra of dimension one since the coset of $X_1$ is a non-zero divisor of ${\mathbf {k}}[C_A]$; $\mathcal B_A$ is the homogeneous coordinate ring of the hyperplane section of $C_A$ defined by $X_1=0$. Both algebras ${\mathbf {k}}[C_A]$ and $\mathcal B_A$ are standard algebras, i.e. generated by their homogeneous pieces of degree one, i.e. ${\mathbf {k}}[C_A]_1$ and $(\mathcal B_A)_1$, respectively. In general $\mathcal B_A$ is non Cohen–Macaulay as the classic example of Macaulay shows, see Example 4.8.

Example 2.8

Let us consider the set $A=\{0,2,4,5,7\}$. The associated monomial curve $C_A$ is defined by the parameterization $(u,v)\mapsto (u^7,u^5v^2,u^3v^4,u^2v^5,v^7)$. Then the defining ideal of $C_A$ is minimally generated by $x_2^2-x_1 x_3, x_2 x_4-x_1 x_5, x_3 x_4-x_2 x_5, x_2 x_3^2-x_1 x_4^2, x_3^3-x_1 x_4 x_5, x_4^3-x_3^2 x_5$, [6]. The Hilbert function of $C_A$ is ${{\text {H\!F}}}_A=\{1,5,12,19, 26, 33, \ldots \}$ and the Hilbert polynomial ${\text {H\!P}}_A(s)=7s-2$.

3 Sumsets and Monomial Projective Curves

We first recall some well known results on curves applied to the projective curve $C_A$, [13]. The monomial projective curve $C_A$ is rational with two eventually singular points $P_1=(1,0,\ldots ,0), P_2=(0,\ldots ,0,1)\in \mathbb P^{n-1}_{{\mathbf {k}}}$. In the affine open neighborhood $X_1=1$ of $P_1$ the curve $C_A$ is defined by the parameterization $v\mapsto (v^{a_2},\ldots , v^{a_n})$; and in the open affine neighborhood $X_n=1$ of $P_2$ the curve is defined by the parameterization $u\mapsto (u^{a_n}, u^{a_n-a_2}, \ldots ,u^{a_n-a_{n-1}})$. The point $P_1$ is non-singular iff $a_2=1$ and $P_2$ is non-singular iff $a_n-a_{n-1}=1$.

We denote by $p_a(C_A)$ the arithmetic genus of $C_A$, i.e.,

$$\begin{aligned} {\text {H\!P}}_A(0)=1-p_a(C_A). \end{aligned}$$

Since $C_A$ is rational its geometric genus is zero and

$$\begin{aligned} p_a(C_A)=\sum _{P\in \mathrm{Sing}(C_A)} \delta (C_A,P), \end{aligned}$$

where $\delta (C_A,P)$ is the singularity order of $P\in Sing(C_A)$, i.e.

$$\begin{aligned} \delta (C_A,P)={\text {dim}}_{{\mathbf {k}}}\frac{\overline{\mathcal O_{C_A,P}}}{\mathcal O_{C_A,P}} \end{aligned}$$

where the over-line stands for the integral closure of ${\mathcal O_{C_A,P}}$ in its field of fractions. Summarizing, we get

$$\begin{aligned} {\text {H\!P}}_A(0)=1-\delta (C_A,P_1)-\delta (C_A,P_2). \end{aligned}$$

Since $C_A$ is a monomial curve in an affine neighbourhood of $P_1$ (resp. $P_2$) we have

$$\begin{aligned} \delta (C_A,P_1)=\mathrm{Card}(\mathbb N\setminus \langle a_2,\ldots , a_n \rangle ) \end{aligned}$$

and

$$\begin{aligned} \delta (C_A,P_2)=Card(\mathbb N\setminus \langle a_n-a_{n-1},\ldots , a_n-a_2,a_n \rangle ). \end{aligned}$$

We know that the Hilbert polynomial ${\text {H\!P}}_A(s)$ and the Hilbert function ${{\text {H\!F}}}_A(s)$ agree for $s\gg 0$. The first integer $s_0$ such that ${{\text {H\!F}}}_A(s)={\text {H\!P}}_A(s)$ for all $s\ge s_0$ is called the regularity of the Hilbert function and it is denoted by $r(C_A)$.

The Castelnuovo–Mumford regularity $\mathrm{reg}(C_A)$ of $C_A$, see [7], for monomial projective curves is upper bounded in terms of the set A. From [19, Proposition 5.5], see also [14],

$$\begin{aligned} \mathrm{reg}(C_A)\le \rho (A):=1+{\text {Max}}\{ (a_i-a_{i-1})+(a_j-a_{j-1}) ;2\le i<j\le n\} \end{aligned}$$

since $r(C_A)\le \mathrm{reg}(C_A)$ we get that ${{\text {H\!F}}}_A(s)={\text {H\!P}}_A(s)$ for all $s\ge \rho (A)$.

Notice that $\rho _A\le a_n-n+3$. This inequality can be deduced from the upper bound of the Castelnuovo–Mumford regularity conjectured by Eisenbud and Goto and proved by Gruson–Lazarsfeld–Peskine in the case of smooth curves, [12]. If $C_A$ is non-singular then we have a better upper bound of the Castelnuovo–Mumford regularity, [14, Theorem 2.7],

$$\begin{aligned} \mathrm{reg}(C_A)\le 1+{\text {Max}}\{ (a_i-a_{i-1}) ;2\le i<j\le n\}. \end{aligned}$$

The following result describes the asymptotic behaviour of |sA|, see [11, 16, 23].

Proposition 3.1

Given a set $A=\{a_1=0, a_2,\ldots , a_n\}$ of integers such that $a_0< a_1< \cdots <a_n$ with $\gcd (a_2,\ldots , a_n)=1$ it holds

$$\begin{aligned} |sA|={{\text {H\!F}}}_A(s)=s a_n +1- \delta (C_A,P_1)-\delta (C_A,P_2) \end{aligned}$$

for all $s\ge \rho (A)$.

Proof

We know that $C_A$ is a degree $a_n$ projective curve, so

$$\begin{aligned} {\text {H\!P}}_A(s)=s a_n+ {\text {H\!P}}_A(0)= s a_n +1- \delta (C_A,P_1)-\delta (C_A,P_2). \end{aligned}$$

Since ${{\text {H\!F}}}_A(s)={\text {H\!P}}_A(s)$ for all $s\ge \rho (A)$ and we know that $|sA|={{\text {H\!F}}}_A(s)$ for all $s\ge 0$, we get the claim.

Corollary 3.2

$\mathcal B_A$ is a one-dimensional standard graded algebra of multiplicity $a_n$.

Proof

Since $X_1$ is a non-zero divisor of ${\mathbf {k}}[C_A]$, Remark 2.7, we get the claim from the last proposition.

The often called fundamental result of additive combinatorics claims:

Proposition 3.3

[22, Theorem 1.1] Given a set $A=\{a_1=0, a_2,\ldots , a_n\}$ of integers such that $a_0< a_1< \cdots <a_n$ with $\gcd (a_2,\ldots , a_n)=1$, there exists a positive integer $\sigma$, non-negative integers $c_1, c_2$ and finite sets $C_1\subset [0,c_1-2]$ and $C_2\subset [0,c_2-2]$ such that

$$\begin{aligned} sA=C_1\sqcup [c_1,s a_n-c_2]\sqcup (\{s a_n\}-C_2) \end{aligned}$$

for all $s\ge \sigma$.

Notice that from the above identity of sets we deduce

$$\begin{aligned} |sA|=a_n s + 1-(c_1-|C_1|+c_2-|C_2|) \end{aligned}$$

for $s\ge \sigma$. From Proposition 3.1 we get that

$$\begin{aligned} \delta (C_A,P_1)+\delta (C_A,P_2)=c_1-|C_1|+c_2-|C_2|. \end{aligned}$$

Let $\Gamma _1$ be the semigroup generated by $a_1,\ldots , a_n$ and let $\Gamma _2$ be the semigroup generated by $a_n-a_{n-1},\ldots , a_n-a_2, a_n$. Notice that $\Gamma _i$ is the semigroup of the curve singularity germ $(C_A, P_i)$, $i=1,2$.

Next, we determine the set $C_i$ and the integer $c_i$, $i=1,2$, in terms of the eventual singular points of the projective curve $C_A$. Notice that $C_i=\emptyset$ iff $P_i$ is a non-singular point of $C_A$, $i=1,2$.

Proposition 3.4

Following the notations of Proposition 3.3, we have that, $i=1,2$,

$$\begin{aligned} \delta (C_A,P_i)=c_i-|C_i| \end{aligned}$$

$c_i$ is the conductor of $\Gamma _i$ and $C_i=\Gamma _i\cap [0,c_i-2]$.

Proof

We only have to prove the result for $i=1$. Notice that if $s\ge {\text {Max}}\{\sigma , (c_1+c_2)/a_n\}$ then

$$\begin{aligned}{}[c_1,c_1+a_2]\subset sA. \end{aligned}$$

Moreover, since $sA\subset (s+1)A$, $s\ge 1$, we have for all $s\gg 0$ that

$$\begin{aligned}{}[c_1,c_1+a_2]\subset sA \cap [0,c_1+a_2]=\Gamma _1 \cap [0,c_1+a_2]. \end{aligned}$$

From this we get that $c_1$ is the conductor of $\Gamma _1$ and that

$$\begin{aligned} C_1=\Gamma _1\cap [0,c_1-2]. \end{aligned}$$

Example 3.5

We consider the set $A=\{0,2,4,5,7\}$ of Example 2.8. The decomposition of 5A is

$$\begin{aligned} 5A=\{0,2\}\sqcup [4,33]\sqcup \{35\} \end{aligned}$$

so $c_1=4$, $C_1=\{0,2\}$, $c_2=2$ and $C_2=\{0\}$. In this case we have $\Gamma _1=\{0,2,4,5,\ldots \}$, $\Gamma _2=\{0,2,3,\ldots \}$ and $\delta _1=2$, $\delta _1=1$.

4 Rigid Hilbert Polynomials and Additive Inverse Problems

In this section, we link the inverse problems with the rigidity of Hilbert polynomials and functions, [9, 10]. In particular, we will recover several upper and lower bounds of the function |sA| from some properties of the Hilbert function of $C_A$.

Definition 4.1

Let $H:\mathbb N\longrightarrow \mathbb N$ be a numerical function asymptotically polynomial, i.e. there exists a polynomial $p(T)\in \mathbb Z[T]$ such that $H(s)=p(s)$ for $s\gg 0$. Let $\mathcal C$ be a class of graded ${\mathbf {k}}$-algebras. We say that p(T) is a rigid polynomial for the class $\mathcal C$ if for all graded ${\mathbf {k}}$ algebra D of $\mathcal C$ if ${\text {H\!P}}_D=p$ then ${{\text {H\!F}}}_D=H$, see [9].

From Lemma 2.1 (2) we get:

Proposition 4.2

[22, Theorems 1.3] Given a set $A=\{a_1=0, a_2,\ldots , a_n\}$ of integers such that $a_0< a_1< \cdots <a_n$ with $\gcd (a_2,\ldots , a_n)=1$, for all $s\ge 0$ it holds

$$\begin{aligned} s(n-1)+1 \le |sA| \le \left( {\begin{array}{c}s+n-1\\ s\end{array}}\right) . \end{aligned}$$

Proof

From Lemma 2.1 (2) we deduce the left hand inequality. The right hand inequality follows from Proposition 2.6.

In the next result we get [22, Theorems 1.2, 1.6 and 1.8]; in particular we prove that $p(T)=(n-1)T+1$ is a rigid polynomial for the class of ${\mathbf {k}}[C_A]$ algebras and that the condition $|sA|= s(n-1)+1$, for some $s\ge 2$, is a rigid property, i.e., determines the whole Hilbert function, see [10].

Theorem 4.3

[22, Theorems 1.2, 1.6, 1.8] Given a set $A=\{a_1=0, a_2,\ldots , a_n\}$ of integers such that $a_0< a_1< \cdots <a_n$ with $\gcd (a_2,\ldots , a_n)=1$, the following conditions are equivalent:

• (1) $|sA|=s(n-1)+1+ o(s)$ for infinitely many s, where o(s) is an arithmetic function such that $\lim _{s\rightarrow \infty } o(s)=0$,

• (2) $|sA|= s(n-1)+1$ for all $s\gg 0$,

• (3) $|sA|= s(n-1)+1$ for some $s\ge 2$,

• (4) $A=\{0,1,\ldots , n-1\}$,

• (5) $|sA|= s(n-1)+1$ for all $s\ge 0$.

Proof

By Proposition 3.1 we get that (1) implies (2). On the other hand, (2) trivially implies (3).

Assume (3), i.e., $|sA|= s(n-1)+1$ for some $s\ge 2$. Notice that

$$\begin{aligned} (s-1)A \cup \{(s-1) a_n +a_2,\ldots , (s-1) a_n +a_n\}\subset sA \end{aligned}$$

and, since $(s-1) a_n$ is the maximum of $(s-1)A$, we have

$$\begin{aligned} (s-1)A \cap \{(s-1) a_n +a_2,\ldots , (s-1) a_n +a_n\}=\emptyset . \end{aligned}$$

By Proposition 4.2 we have $|(s-1)A|\ge (s-1)(n-1)+1$, so

$$\begin{aligned} (s-1)A \cup \{(s-1) a_n +a_2,\ldots , (s-1) a_n +a_n\}= sA. \end{aligned}$$

(2)

We know that ${\mathbf {k}}[C_A]_s$ has as ${\mathbf {k}}$-basis the monomials $t^{\alpha }w^s, \alpha \in sA$ and $X_1{\mathbf {k}}[C_A]_{s-1}$ is generated by $t^{\alpha +a_1}w^s, \alpha \in (s-1)A$. By (2) we have that $(s-1)A+a_1\subset (s-1)A$ so the ${\mathbf {k}}$-vector space

$$\begin{aligned} (\mathcal B_A)_s=\frac{{\mathbf {k}}[C_A]_s}{X_1{\mathbf {k}}[C_A]_{s-1}} \end{aligned}$$

is generated by the cosets of

$$\begin{aligned} t^{(s-1)a_n+a_i}w^s, \quad i=2,\dots ,n. \end{aligned}$$

This fact implies that

$$\begin{aligned} X_n^{s-1} (\mathcal B_A)_1 =(\mathcal B_A)_s \end{aligned}$$

Since the algebra $\mathcal B_A$ is standard we get, multiplying both sides by $(\mathcal B_A)_{(r-1)(s-1)}$, that

$$\begin{aligned} X_n^{(s-1)r} (\mathcal B_A)_1= (\mathcal B_A)_{r(s-1)+1} \end{aligned}$$

for all $r\ge 1$. Since ${\text {dim}}_{{\mathbf {k}}}((\mathcal B_A)_t)=n-1$, for $t\gg 0$ we obtain, Proposition 3.2,

$$\begin{aligned} n-1\ge {\text {dim}}_{{\mathbf {k}}}(\mathcal B_A)_{r(s-1)+1}= a_n \end{aligned}$$

for $r\gg 0$. Hence $a_n\le n-1$ and we get (4).

The remaining implications are easy computations.

Remark 4.4

The curve $C_A$ for $A=\{0,\ldots , n-1\}$ is the rational normal curve of $\mathbb P^{n-1}_{{\mathbf {k}}}$, i.e., the curve defined by $(u,v)\mapsto (u^{n-1},u^{n-2}v,\ldots , uv^{n-2},v^{n-1})$.

Remark 4.5

From Lemma 2.1 (1) we get for a general set A that $|sA|= s(n-1)+1$ for all $s\ge 0$ if and only if A is a n-term arithmetic progression, i.e., $A=q_0+ q_1 [0,\ldots , n-1]$ for $q_0\in \mathbb N$ and $q_1\in \mathbb N\setminus \{0\}$.

Next we use a result on additive combinatorics in order to improve an upper bound of the Castelnuovo–Mumford regularity of rational projective curves. We first recall the following result of Lev:

Proposition 4.6

[18, Theorem 1] Given $A=\{a_1=0, a_2,\ldots , a_n\}$ with $\gcd (a_2,\ldots , a_n)=1$, it holds:

$$\begin{aligned} |sA|-|(s-1)A|\ge \min \{a_n, s(n-2)+1\} \end{aligned}$$

for all $s\ge 2$.

In the following result we improve [1, Theorem 2.7], see also [17], where an upper bound of the Castelnuovo–Mumford regularity is given for a monomial projective curve $C_A$ under the hypothesis that A is an arithmetic sequence. We know that

$$\begin{aligned} {{\text {H\!F}}}_{\mathcal B_A}(s)={{\text {H\!F}}}_A(s)-{{\text {H\!F}}}_A(s-1)=|sA|-|(s-1)A| \end{aligned}$$

so last result shows that the Hilbert function of the one-dimensional graded algebra $\mathcal B_A$ grows rapidly. This is the key point in the proof of the following result where we assume that ${\mathbf {k}}[C_A]$ is Cohen–Macaulay. See [3, 15] for several criteria implying the Cohen–Macaulayness of ${\mathbf {k}}[C_A]$.

Theorem 4.7

Given $A=\{a_1=0, a_2,\ldots , a_n\}$ with $\gcd (a_2,\ldots , a_n)=1$. If the two-dimensional ring ${\mathbf {k}}[C_A]$ is Cohen–Macaulay then

$$\begin{aligned} \mathrm{reg}({\mathbf {k}}[C_A]) \le \lceil \frac{a_n-1}{n-2}\rceil \end{aligned}$$

Proof

We write $s_0=\lceil \frac{a_n-1}{n-2}\rceil$. Since ${\mathbf {k}}[C_A]$ is Cohen–Macaulay we have $r(C_A)+1=\mathrm{reg}({\mathbf {k}}[C_A])$ and that $\mathcal B_A$ is a one-dimensional Cohen–Macaulay ring. Hence we have

$$\begin{aligned} {{\text {H\!F}}}_{\mathcal B_A}(s)\le a_n \end{aligned}$$

for all $s\ge 1$, [21, Chapter XII]. From this inequality and Proposition 4.6, we get

$$\begin{aligned} s(n-2)+1\le {{\text {H\!F}}}_{\mathcal B_A}(s)\le \min \left\{ a_n, \left( {\begin{array}{c}s+n-2\\ s\end{array}}\right) \right\} \end{aligned}$$

for $s=1,\ldots , s_0-1$; and

$$\begin{aligned} {{\text {H\!F}}}_{\mathcal B_A}(s)=a_n \end{aligned}$$

for $s\ge s_0$, i.e. $r(\mathcal B_A)\le s_0$. Since $r(C_A)+1 =r(\mathcal B_A)$ we get the claim:

$$\begin{aligned} \mathrm{reg}({\mathbf {k}}[C_A])=r(C_A)+1 =r(\mathcal B_A)\le s_0. \end{aligned}$$

Example 4.8

(Macaulay’s example) In this example we consider the example of a non-singular, non-Cohen–Macaulay monomial projective curve given by Macaulay, [20]. In this case the set is $A=\{0,1,3,4\}$. The monomial curve $C_A$ associated to A is defined by the parameterization $(u,v)\mapsto (u^4,u^3v,uv^3,v^4)$. A computation with Singular [6] give us that ${{\text {H\!F}}}_A=\{1,4,9,13,17,21,\ldots \}$ and ${\text {H\!P}}_A(s)=4 s +1$. Since the points $P_1, P_2$ are non-singular points of $C_A$, we deduce last identity from Proposition 3.1 as well.

Example 4.9

We consider a especial case of [17, Case A]. Let us consider the set $A=\{0,7,8,9,10\}$. From [17, Theorem 2.1] we know that ${\mathbf {k}}[C_A]$ is Cohen–Macaulay and that $r(C_A)=5$ that agrees with the upper bound of the Theorem 4.7. The defining ideal of $C_A$ is minimally generated by: $x_3^2- x_2 x_4, x_3 x_4- x_2 x_5, x_4^2- x_3 x_5, x_2^4- x_1 x_3 x_5^2, x_2^3 x_3- x_1 x_4 x_5^2, x_2^3 x_4- x_1 x_5^3.$ A straight computation shows

$$\begin{aligned} {{\text {H\!F}}}_A=\{|sA|, s=0,1,\ldots \}=\{1,5, 12, 22, 32, 42, 52, 62, 72, \ldots \} \end{aligned}$$

and the Hilbert polynomial of $C_A$ is ${\text {H\!P}}_A=10s-8$.