Sumsets regularity versus Castelnuovo-Mumford regularity for monomial curves

Abstract

Let $A=\{0=a_0<a_1<\dots<a_{k-1}=d\}$ be integers with greatest common divisor $1$, let $C_A$ be the projective monomial curve they define, and let $\mathrm{reg}(A)$ be the Castelnuovo-Mumford regularity of its coordinate ring. Gimenez and González-Sánchez proved $\mathrm{reg}(A)\leσ(A)+1$ for the sumsets regularity $σ(A)$; the reverse inequality $σ(A)\le\mathrm{reg}(A)$ was proposed as an open conjecture in a 2024 talk. We show that it fails and measure by how much. Let $c_1,c_2$ be the conductors of the two numerical semigroups attached to $A$ and $N=\lceil(c_1+c_2)/d\rceil$. For curves in $\mathbb{P}^3$ that are not arithmetically Cohen-Macaulay, $\mathrm{reg}(A)\ge(c_1+c_2)/d+2$, with equality exactly for the arithmetically Buchsbaum curves, and $σ(A)=\mathrm{reg}(A)$. For arithmetically Cohen-Macaulay curves in $\mathbb{P}^3$ we prove $9N\le10\,\mathrm{reg}(A)-2$, with $9/10$ best possible, and give an explicit family with $σ(A)-\mathrm{reg}(A)$ unbounded. In $\mathbb{P}^4$ a family with a band-shaped Apéry set has $\mathrm{reg}(A)/σ(A)\to(2+\sqrt2)/4$ along the solutions of a Pell equation, and gluing carries this bound to every $\mathbb{P}^r$ with $r\ge5$.