Lefschetz properties for monomial complete intersections
Abstract
We give a complete characterization of the weak Lefschetz property (WLP) for monomial complete intersections over a field of positive characteristic. Richard Stanley observed that in characteristic zero WLP, and in fact the strong Lefschetz property (SLP), holds for all degree sequences, as a consequence of the hard Lefschetz theorem, and the same result was explained by Junzo Watanabe using the representation theory of sl2. In positive characteristic, many partial results are known, most notably the classification for constant degree sequences due to Brenner--Kaid and Kustin--Vraciu. Our approach is based on a cohomological and representation-theoretic interpretation of WLP. Combined with an analysis of cohomology characters, this leads to a complete numerical criterion for WLP, expressed by simple inequalities involving the p-adic digits of the exponents. We also give a new proof of the known classification of SLP using Renaud's algorithm for multiplication in the Green--Han--Monsky ring.
Related
- The Non-Lefschetz Locus of Monomial Complete Intersections in Positive Characteristic
- Lefschetz Properties of complete intersesctions and maximal variation of hyperplane sections
- Strong Lefschetz property of the Kirchhoff algebra of the complete tripartite graph K1,1,n
- On complete intersection projective closures of monomial curves
- K-stability and birational rigidity of Fano 3-fold weighted complete intersections
BibTeX
@misc{kyomuhangi2026lefschetz,
title = {{Lefschetz properties for monomial complete intersections}},
author = {Annet Kyomuhangi and Emanuela Marangone and Claudiu Raicu and Ethan Reed},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09534},
eprint = {2609.09534},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09534},
}
RIS
TY - UNPB
TI - Lefschetz properties for monomial complete intersections
AU - Annet Kyomuhangi
AU - Emanuela Marangone
AU - Claudiu Raicu
AU - Ethan Reed
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09534
UR - https://arxiv.org/abs/2609.09534
AB - We give a complete characterization of the weak Lefschetz property (WLP) for monomial complete intersections over a field of positive characteristic. Richard Stanley observed that in characteristic zero WLP, and in fact the strong Lefschetz property (SLP), holds for all degree sequences, as a consequence of the hard Lefschetz theorem, and the same result was explained by Junzo Watanabe using the representation theory of sl_2. In positive characteristic, many partial results are known, most notably the classification for constant degree sequences due to Brenner--Kaid and Kustin--Vraciu. Our approach is based on a cohomological and representation-theoretic interpretation of WLP. Combined with an analysis of cohomology characters, this leads to a complete numerical criterion for WLP, expressed by simple inequalities involving the p-adic digits of the exponents. We also give a new proof of the known classification of SLP using Renaud's algorithm for multiplication in the Green--Han--Monsky ring.
ER -