Construction of Regular Fat Linear Sets and Non-Representable q-Matroids
Abstract
Let q be a prime power. An $(r, i)$-regular fat 𝔽q-linear set has exactly r points of weight i≥2, called fat points. All of its remaining points have weight one, and at least one such point occurs. We solve two open problems posed by three of the authors of this paper. First, we derive divisibility constraints on the point weights using the scalar stabilizer of the defining subspace. Consequently, a regular fat linear set of rank 2i in PG(1,qn) has at most two or exactly q + 1 fat points. In the latter case, the fat points form an 𝔽q-subline. Second, we construct higher-dimensional regular fat linear sets whose fat points do not form an 𝔽q-subgeometry. In particular, for every k≥ 3 we obtain examples of rank 2k whose fat locus consists of an 𝔽q-subgeometry PG(k−2,q) in PG(k−1,q4) together with one external point. This construction is then generalized further. Finally, we apply our projective-line results to q-matroid representability and present an infinite family of non-representable rank-two q-matroids arising from partial spreads.
Related
BibTeX
@misc{qiu2026construction,
title = {{Construction of Regular Fat Linear Sets and Non-Representable q-Matroids}},
author = {Yufei Qiu and Valentino Smaldore and Zhenpeng Wang and Corrado Zanella and Ferdinando Zullo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17055},
eprint = {2610.17055},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17055},
}
RIS
TY - UNPB
TI - Construction of Regular Fat Linear Sets and Non-Representable q-Matroids
AU - Yufei Qiu
AU - Valentino Smaldore
AU - Zhenpeng Wang
AU - Corrado Zanella
AU - Ferdinando Zullo
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17055
UR - https://arxiv.org/abs/2610.17055
AB - Let q be a prime power. An $(r, i)$-regular fat 𝔽_q-linear set has exactly r points of weight i≥2, called fat points. All of its remaining points have weight one, and at least one such point occurs. We solve two open problems posed by three of the authors of this paper. First, we derive divisibility constraints on the point weights using the scalar stabilizer of the defining subspace. Consequently, a regular fat linear set of rank 2i in PG(1,q^n) has at most two or exactly q + 1 fat points. In the latter case, the fat points form an 𝔽_q-subline. Second, we construct higher-dimensional regular fat linear sets whose fat points do not form an 𝔽_q-subgeometry. In particular, for every k≥ 3 we obtain examples of rank 2k whose fat locus consists of an 𝔽_q-subgeometry PG(k−2,q) in PG(k−1,q^4) together with one external point. This construction is then generalized further. Finally, we apply our projective-line results to q-matroid representability and present an infinite family of non-representable rank-two q-matroids arising from partial spreads.
ER -