PreprintCC BY 4.0arXiv
Theta-duality and Prym-Torelli for cyclic covers of hyperelliptic curves
Abstract
Let f:C̃→ C be an étale cyclic cover of odd prime degree d of a hyperelliptic curve of genus g≥ 2. We revisit the theta-duality reconstruction argument for the generic injectivity of the associated Prym map by Naranjo-Ortega-Pirola-Spelta and identify an additional residual locus arising from the fixed divisor of the line bundles occurring in that argument. For d≥5, this locus does not affect reconstruction, giving injectivity under the numerical assumption (d−1)(g−1)≥ 7. For d=3, the geometry is governed instead by trigonal pencils on the quotient of C̃ by a lift of the hyperelliptic involution; this yields generic degree $2$ in genus $5$ and injectivity for g≥6.
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BibTeX
@misc{shatsila2026thetaduality,
title = {{Theta-duality and Prym-Torelli for cyclic covers of hyperelliptic curves}},
author = {Anatoli Shatsila},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14135},
eprint = {2609.14135},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14135},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Theta-duality and Prym-Torelli for cyclic covers of hyperelliptic curves
AU - Anatoli Shatsila
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.14135
UR - https://arxiv.org/abs/2609.14135
AB - Let f:C̃→ C be an étale cyclic cover of odd prime degree d of a hyperelliptic curve of genus g≥ 2. We revisit the theta-duality reconstruction argument for the generic injectivity of the associated Prym map by Naranjo-Ortega-Pirola-Spelta and identify an additional residual locus arising from the fixed divisor of the line bundles occurring in that argument. For d≥5, this locus does not affect reconstruction, giving injectivity under the numerical assumption (d−1)(g−1)≥ 7. For d=3, the geometry is governed instead by trigonal pencils on the quotient of C̃ by a lift of the hyperelliptic involution; this yields generic degree $2$ in genus $5$ and injectivity for g≥6.
ER -