Multi-graded generic initial ideals, regularity, and the optimal colorful fractional Helly theorem for d-Leray complexes
Abstract
A celebrated result of Bayer and Stillman from 1987 states that for a homogeneous ideal I of a polynomial ring S, the regularities of S/I and S/GIN(I) are the same under the reverse lexicographic monomial ordering, where GIN(I) is the generic initial ideal. If R is a polynomial ring whose variables are subdivided into disjoint blocks of variables X1,…,Xc, there is a natural multi-grading on R, and one can analogously define a multi-graded version of the generic initial ideal for any multi-homogeneous ideal I of R. However, the full strength of the Bayer--Stillman Theorem fails in the multi-graded setting; there are multi-homogeneous ideals I such that the regularities are not preserved after passing to the multi-graded generic initial ideal no matter the choice of monomial ordering. We prove lower bounds on the regularity of R/I in terms of almost regular sequences of the multi-graded generic initial ideal of I restricted to each block of variables. Again, we use the reverse lexicographic monomial ordering, but interestingly, the lower bound result requires a particular choice of ordering on the variables. As an application, we prove the optimal fractional Helly theorem for d-Leray simplicial complexes, a problem stemming from the work of Kim in 2017.
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BibTeX
@misc{mcginnis2026multigraded,
title = {{Multi-graded generic initial ideals, regularity, and the optimal colorful fractional Helly theorem for $d$-Leray complexes}},
author = {Daniel McGinnis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.25891},
eprint = {2608.25891},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25891},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multi-graded generic initial ideals, regularity, and the optimal colorful fractional Helly theorem for d-Leray complexes
AU - Daniel McGinnis
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2608.25891
UR - https://arxiv.org/abs/2608.25891
AB - A celebrated result of Bayer and Stillman from 1987 states that for a homogeneous ideal I of a polynomial ring S, the regularities of S/I and S/GIN(I) are the same under the reverse lexicographic monomial ordering, where GIN(I) is the generic initial ideal. If R is a polynomial ring whose variables are subdivided into disjoint blocks of variables X_1,…,X_c, there is a natural multi-grading on R, and one can analogously define a multi-graded version of the generic initial ideal for any multi-homogeneous ideal I of R. However, the full strength of the Bayer--Stillman Theorem fails in the multi-graded setting; there are multi-homogeneous ideals I such that the regularities are not preserved after passing to the multi-graded generic initial ideal no matter the choice of monomial ordering. We prove lower bounds on the regularity of R/I in terms of almost regular sequences of the multi-graded generic initial ideal of I restricted to each block of variables. Again, we use the reverse lexicographic monomial ordering, but interestingly, the lower bound result requires a particular choice of ordering on the variables. As an application, we prove the optimal fractional Helly theorem for d-Leray simplicial complexes, a problem stemming from the work of Kim in 2017.
ER -