Hyperelliptic curves on (1,d) abelian surfaces for d≤ 15

@misc{borowka2026hyperelliptic,
  title = {{Hyperelliptic curves on $(1,d)$ abelian surfaces for $d\leq 15$}},
  author = {Paweł Borówka and Paolo Grossi and Anatoli Shatsila},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40989},
  eprint = {2610.40989},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40989},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Hyperelliptic curves on (1,d) abelian surfaces for d≤ 15
AU  - Paweł Borówka
AU  - Paolo Grossi
AU  - Anatoli Shatsila
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40989
UR  - https://arxiv.org/abs/2610.40989
AB  - Bryan, Oberdieck, Pandharipande and Yin predicted the number of hyperelliptic curves on general abelian surfaces under a transversality condition. Some of the numbers have already been confirmed. In this paper we construct singular hyperelliptic curves on (1,d) polarised abelian surfaces for d≤ 15 confirming a few predicted numbers, namely of (geometric) genus g=d and of geometric genus k+3 on (1,2k) and (1,2k+1) polarised surfaces. Moreover, we have found that all genus 7 hyperelliptic curves on general (1,8) polarised surfaces are double coverings of curves locally embedded into (1,4) polarised surfaces. More interestingly, there is a unique genus 9 hyperelliptic curve (up to translation) locally embedded into a general (1,12) polarised surface and it is a Klein covering of a curve locally embedded into a (1,3) polarised surface.
ER  -