Castelnuovo–Mumford Regularity of Projective Monomial Curves via Sumsets

Abstract

AbstractLet $$A=\{a_0,\ldots ,a_{n-1}\}$$ A = { a 0 , … , a n - 1 } be a finite set of $$n\ge 4$$ n ≥ 4 non-negative relatively prime integers, such that $$0=a_0<a_1<\cdots <a_{n-1}=d$$ 0 = a 0 < a 1 < ⋯ < a n - 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 $$\mathcal {C}_A$$ C A parametrized by A, $$\begin{aligned} \quad \mathcal {C}_A=\{(v^d:u^{a_1}v^{d-a_1}:\cdots :u^{a_{n-2}}v^{d-a_{n-2}}:u^d) \mid (u:v)\in \mathbb {P}^{1}_k\}\subset \mathbb {P}^{n-1}_k. \end{aligned}$$ C A = { ( v d : u a 1 v d - a 1 : ⋯ : u a n - 2 v d - a n - 2 : u d ) ∣ ( u : v ) ∈ P k 1 } ⊂ P k n - 1 . The exponents in the previous parametrization of $$\mathcal {C}_A$$ C A define a homogeneous semigroup $$\mathcal {S}\subset \mathbb {N}^2$$ S ⊂ N 2 . We provide several results relating the Castelnuovo–Mumford regularity of $$\mathcal {C}_A$$ C A to the behavior of the sumsets of A and to the combinatorics of the semigroup $$\mathcal {S}$$ S that reveal a new interplay between commutative algebra and additive number theory.