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 $0=a_0$ < $\cdots$ < $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 $C_A$ parametrized by $A$: \[C_A=\{(v^d:u^{a_1}v^{d-a_1}:\cdots:u^{a_{n-2}}v^{d-a_{n-2}}:u^d)\}\] where $(u:v)$ covers the whole projective line over $k$. In this talk, we will focus on the relation between the Castelnuovo-Mumford regularity of $C_A$ and the behaviour of the sumsets $sA$ and show how this provides a nice interplay between Commutative Algebra and Additive Number Theory. This talk is based on a joint work with Philippe Gimenez.