Sumsets regularity versus Castelnuovo-Mumford regularity for monomial curves
Abstract
Let A={0=a0<a1<…<ak−1=d} be integers with greatest common divisor $1$, let CA be the projective monomial curve they define, and let reg(A) be the Castelnuovo-Mumford regularity of its coordinate ring. Gimenez and González-Sánchez proved reg(A)≤σ(A)+1 for the sumsets regularity σ(A); the reverse inequality σ(A)≤reg(A) was proposed as an open conjecture in a 2024 talk. We show that it fails and measure by how much. Let c1,c2 be the conductors of the two numerical semigroups attached to A and N=⌈(c1+c2)/d⌉. For curves in ℙ3 that are not arithmetically Cohen-Macaulay, reg(A)≥(c1+c2)/d+2, with equality exactly for the arithmetically Buchsbaum curves, and σ(A)=reg(A). For arithmetically Cohen-Macaulay curves in ℙ3 we prove 9N≤10 reg(A)−2, with 9/10 best possible, and give an explicit family with σ(A)−reg(A) unbounded. In ℙ4 a family with a band-shaped Apéry set has reg(A)/σ(A)→(2+√2)/4 along the solutions of a Pell equation, and gluing carries this bound to every ℙr with r≥5.
Related
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BibTeX
@misc{chaudhary2026sumsets,
title = {{Sumsets regularity versus Castelnuovo-Mumford regularity for monomial curves}},
author = {Prashant Chaudhary and Kapil Khandelwal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24121},
eprint = {2610.24121},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24121},
}
RIS
TY - UNPB
TI - Sumsets regularity versus Castelnuovo-Mumford regularity for monomial curves
AU - Prashant Chaudhary
AU - Kapil Khandelwal
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24121
UR - https://arxiv.org/abs/2610.24121
AB - Let A={0=a_0<a_1<…<a_k−1=d} be integers with greatest common divisor $1$, let C_A be the projective monomial curve they define, and let reg(A) be the Castelnuovo-Mumford regularity of its coordinate ring. Gimenez and González-Sánchez proved reg(A)≤σ(A)+1 for the sumsets regularity σ(A); the reverse inequality σ(A)≤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=⌈(c_1+c_2)/d⌉. For curves in ℙ^3 that are not arithmetically Cohen-Macaulay, reg(A)≥(c_1+c_2)/d+2, with equality exactly for the arithmetically Buchsbaum curves, and σ(A)=reg(A). For arithmetically Cohen-Macaulay curves in ℙ^3 we prove 9N≤10 reg(A)−2, with 9/10 best possible, and give an explicit family with σ(A)−reg(A) unbounded. In ℙ^4 a family with a band-shaped Apéry set has reg(A)/σ(A)→(2+√2)/4 along the solutions of a Pell equation, and gluing carries this bound to every ℙ^r with r≥5.
ER -