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 $A=\left\{a_1, \ldots, a_n\right\}$ a monomial projective curve $C_A \subset \mathbb{P}_{\mathbf{k}}^{n-1}$ such that the Hilbert function of $C_A$ and the cardinalities of $s A:=\left\{a_{i_1}+\cdots+a_{i_s} \mid 1 \leq i_1 \leq \cdots \leq i_s \leq n\right\}$ agree. The singularities of $C_A$ determines the asymptotic behaviour of $|s A|$, equivalently the Hilbert polynomial of $C_A$, and the asymptotic structure of $S A$. 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.