Sumsets and monomial projective curves

TL;DR

This paper reveals a new connection between additive combinatorics and the geometry of monomial projective curves, linking sumset cardinalities to the Hilbert function and singularities of these curves.

Contribution

It introduces a novel relationship between sumsets and the geometry of monomial projective curves, enabling translation of additive inverse problems into geometric rigidity questions.

Findings

The Hilbert function of the curve matches the sumset cardinalities.

Singularities of the curve influence the sumset growth and Hilbert polynomial.

An improved upper bound on the Castelnuovo-Mumford regularity is achieved.

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 $A=\{a_1,\cdots, a_n\}$ a monomial projective curve $C_A\subset \mathbb P^{n-1}_{k}$ 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. 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.