Sumsets and monomial projective curves

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 $A=\{a_1,\cdots ,a_n\}$ a monomial projective curve $C_A\subset {\mathbb{P}}_k^{n-1}$ 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\mathrm{.}Thesingularitiesof$C_A$ determines the asymptotic behaviour of $\mathrm{\mid }sA\mathrm{\mid }$, 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.