PreprintCC BY 4.0arXiv
Irrational Seshadri constants from dihedral orbits
Abstract
We construct an irrational one-point Seshadri constant on the blow-up X9 of ℙ2 at nine very general points. The divisor L=9H−3(E1+⋯+E5)−2(E6+⋯+E9) is ample and satisfies ε(L;x)=2√5 at a very general point x∈ X9. To prove this, we establish that the Seshadri constant of 𝒪ℙ1×ℙ1(1,1) at ten very general points is 1/√5, completing the reflection approach proposed by Dionne and Roth for the ten-point case. We also prove that the same equality holds at a very general free orbit of a fixed dihedral group of order ten. This yields an irrational one-point Seshadri constant on the singular quotient surface. A plane model of its minimal resolution and a deformation of the blow-up centers then give the result on X9.
Related
- Dihedral reflections and an infinite series of irrational Seshadri constants
- 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{laface2026irrational,
title = {{Irrational Seshadri constants from dihedral orbits}},
author = {Antonio Laface and Luca Ugaglia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26521},
eprint = {2609.26521},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26521},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Irrational Seshadri constants from dihedral orbits
AU - Antonio Laface
AU - Luca Ugaglia
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.26521
UR - https://arxiv.org/abs/2609.26521
AB - We construct an irrational one-point Seshadri constant on the blow-up X_9 of ℙ^2 at nine very general points. The divisor L=9H−3(E_1+⋯+E_5)−2(E_6+⋯+E_9) is ample and satisfies ε(L;x)=2√5 at a very general point x∈ X_9. To prove this, we establish that the Seshadri constant of 𝒪_ℙ^1×ℙ^1(1,1) at ten very general points is 1/√5, completing the reflection approach proposed by Dionne and Roth for the ten-point case. We also prove that the same equality holds at a very general free orbit of a fixed dihedral group of order ten. This yields an irrational one-point Seshadri constant on the singular quotient surface. A plane model of its minimal resolution and a deformation of the blow-up centers then give the result on X_9.
ER -