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={a_1,ldots , a_n} a monomial projective curve C_Asubset mathbb P^{n-1}_{{mathbf {k}}} such that the Hilbert function of C_A and the cardinalities of sA:={a_{i_1}+cdots +a_{i_s}mid 1le i_1le cdots le i_sle n} agree. The singularities of C_A determines the asymptotic behaviour of |sA|, 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.