Dihedral reflections and an infinite series of irrational Seshadri constants
Abstract
Laface and Ugaglia recently constructed an irrational one-point Seshadri constant on the blow-up of ℙ2 at nine very general points by combining a dihedral orbit on ℙ1×ℙ1, a sequence of de Jonquières transformations, and a reflection argument along a (−4)-curve with balanced normal bundle. We show that the same mechanism extends uniformly to every odd integer n≥ 5. For n=2k+1 we prove ε(𝒪ℙ1×ℙ1(n−4,1);p1,…,p2n)=√((n−4)/n) at 2n very general points, and already at a very general free orbit of a fixed dihedral group of order 2n. For every odd n≥ 7 this produces an explicit ample line bundle on the blow-up of ℙ2 at k+7=(n+13)/2 very general points whose one-point Seshadri constant at a very general point equals 2√(n(n−4)). More precisely, after one quadratic transformation we obtain the ample divisor Ln=(3n−4)H−nE1−(n−2)(E2+⋯+E5)−4(E6+⋯+Ek+5)−2(Ek+6+Ek+7), with Ln2=4n(n−4) and ε(Ln;x)=√(Ln2). We also isolate an abstract balanced-reflection principle underlying the construction: a nef class on a special fiber can be reflected across a rational curve of square −2a whenever the curve has normal bundle 𝒪ℙ1(−a)⊕ 2 in the total space, and the reflected class is nef on very general fibers.
Related
- Irrational Seshadri constants from dihedral orbits
- Weighted Seshadri constants and ellipsoid embeddings
- Seshadri constants and hyperelliptic curves on abelian varieties
- Lifting negative curves from finite fields and Seshadri constants at ten points
- Lefschetz Pencils on surfaces and an application to Seshadri constants
BibTeX
@misc{malara2026dihedral,
title = {{Dihedral reflections and an infinite series of irrational Seshadri constants}},
author = {Grzegorz Malara and Łukasz Merta and Justyna Szpond and Marcin Zieliński},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01783},
eprint = {2610.01783},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01783},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dihedral reflections and an infinite series of irrational Seshadri constants
AU - Grzegorz Malara
AU - Łukasz Merta
AU - Justyna Szpond
AU - Marcin Zieliński
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01783
UR - https://arxiv.org/abs/2610.01783
AB - Laface and Ugaglia recently constructed an irrational one-point Seshadri constant on the blow-up of ℙ^2 at nine very general points by combining a dihedral orbit on ℙ^1×ℙ^1, a sequence of de Jonquières transformations, and a reflection argument along a (−4)-curve with balanced normal bundle. We show that the same mechanism extends uniformly to every odd integer n≥ 5. For n=2k+1 we prove ε(𝒪_ℙ^1×ℙ^1(n−4,1);p_1,…,p_2n)=√((n−4)/n) at 2n very general points, and already at a very general free orbit of a fixed dihedral group of order 2n. For every odd n≥ 7 this produces an explicit ample line bundle on the blow-up of ℙ^2 at k+7=(n+13)/2 very general points whose one-point Seshadri constant at a very general point equals 2√(n(n−4)). More precisely, after one quadratic transformation we obtain the ample divisor L_n=(3n−4)H−nE_1−(n−2)(E_2+⋯+E_5)−4(E_6+⋯+E_k+5)−2(E_k+6+E_k+7), with L_n^2=4n(n−4) and ε(L_n;x)=√(L_n^2). We also isolate an abstract balanced-reflection principle underlying the construction: a nef class on a special fiber can be reflected across a rational curve of square −2a whenever the curve has normal bundle 𝒪_ℙ^1(−a)^⊕ 2 in the total space, and the reflected class is nef on very general fibers.
ER -