Abstract:
Given $A=\{a_0,\ldots,a_{n-1}\}$ a finite set of $n\geq 4$ non-negative integers that we will assume to be in normal form, i.e., such that $0=a_0< a_1 <\cdots < a_{n-1}=d$ and relatively prime, the $s$-fold sumset of $A$ is the set $sA$ of integers obtained by collecting 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 $A$, that is, the Zariski closure of \[ \{(v^d:u^{a_1}v^{d-a_1}:\cdots:u^{a_{n-2}}v^{d-a_{n-2}}:u^d) \mid (u:v) \in \mathbb{P}_k^{ 1}\} \subset \mathbb{P}_k^{\, n} \, .\] This allows us to establish a bridge between Additive Number Theory and Commutative Algebra and obtain some results connecting the Castelnuovo-Mumford regularity of $\mathcal{C}_A$ and the behaviour of the sumsets $sA$. This talk is based on a joint work with Philippe Gimenez