PreprintCC BY-NC-SA 4.0arXiv
Hyperelliptic curves on K3 surfaces
Abstract
We study hyperelliptic curves on K3 surfaces. We show that any number of general points on a K3 surface of Picard rank one can be connected by a hyperelliptic curve. As a by-product we recover the Beauville--Voisin decomposition of the small diagonal, together with some new bounds on its torsion order in the integral Chow group. On the arithmetic side, given a K3 surface X of geometric Picard rank one defined over a number field k, we show that there exists a finite extension k⊂ k' such that quadratic points are dense in Xk', extending a theorem of Corvaja--Zucconi to K3 surfaces of arbitrary degree.
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BibTeX
@misc{moretti2026hyperelliptic,
title = {{Hyperelliptic curves on $K3$ surfaces}},
author = {Federico Moretti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17536},
eprint = {2610.17536},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17536},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Hyperelliptic curves on K3 surfaces
AU - Federico Moretti
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17536
UR - https://arxiv.org/abs/2610.17536
AB - We study hyperelliptic curves on K3 surfaces. We show that any number of general points on a K3 surface of Picard rank one can be connected by a hyperelliptic curve. As a by-product we recover the Beauville--Voisin decomposition of the small diagonal, together with some new bounds on its torsion order in the integral Chow group. On the arithmetic side, given a K3 surface X of geometric Picard rank one defined over a number field k, we show that there exists a finite extension k⊂ k' such that quadratic points are dense in X_k', extending a theorem of Corvaja--Zucconi to K3 surfaces of arbitrary degree.
ER -