Sumsets and Projective Curves

Abstract

AbstractThe 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 = { a 1 , … , a n } a monomial projective curve $$C_A\subset \mathbb P^{n-1}_{{\mathbf {k}}}$$ C A ⊂ P k n - 1 such that the Hilbert function of $$C_A$$ C A and the cardinalities of $$sA:=\{a_{i_1}+\cdots +a_{i_s}\mid 1\le i_1\le \cdots \le i_s\le n\}$$ s A : = { a i 1 + ⋯ + a i s ∣ 1 ≤ i 1 ≤ ⋯ ≤ i s ≤ n } agree. The singularities of $$C_A$$ C A determines the asymptotic behaviour of |sA|, equivalently the Hilbert polynomial of $$C_A$$ 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.