Castelnuovo–Mumford regularity of projective monomial curves via sumsets

Abstract

Let A={a0,…,an−1} be a finite set of n≥4 non-negative relatively prime integers, such that 0=a0<a1<⋯<an−1=d. The s-fold sumset of A is the set sA of integers that contains all the sums of s elements in A. On the other hand, given an infinite field k, one can associate with A the projective monomial curve CA

parametrized by A, CA={(vd:ua1vd−a1:⋯:uan−2vd−an−2:ud)∣(u:v)∈P1k}⊂Pn−1k.

The exponents in the previous parametrization of CA define a homogeneous semigroup S⊂N2. We provide several results relating the Castelnuovo–Mumford regularity of CA to the behavior of the sumsets of A and to the combinatorics of the semigroup S that reveal a new interplay between commutative algebra and additive number theory.