Castelnuovo-Mumford regularity of projective monomial curves via sumsets

Abstract

Let $A=\{a_0,\dots ,a_{n-1}\}$ be a finite set of $n\ge 4$ non-negative relatively prime integers such that $0=a_0<a_1<\cdots <a_{n-1}=d$. 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 to $A$ the projective monomial curve ${\mathcal{C}}_A$ parametrized by $A$, \[ \mathcal{C}_A=\{(v^d:u^{a_1}v^{d-a_1}:\cdots :u^{a_{n-2}}v^{d-a_{n-2}}:u^d) \mid \ (u:v)\in\mathbb{P}^{1}_k\}\subset\mathbb{P}^{n-1}_k\,. \] The exponents in the previous parametrization of ${\mathcal{C}}_A$ define a homogeneous semigroup $\mathcal{S}\subset {\mathbb{N}}^2$. We provide several results relating the Castelnuovo-Mumford regularity of ${\mathcal{C}}_A$ to the behaviour of the sumsets of $A$ and to the combinatorics of the semigroup $\mathcal{S}$ that reveal a new interplay between commutative algebra and additive number theory.

19 pages, 2 figures, 1 table. Theorem 3.4 is new in this version. References [17], [20] and [23] are new in the bibliography. To appear in Mediterr. J. Math

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