Seshadri constants and hyperelliptic curves on abelian varieties
Abstract
Given a principally polarized abelian variety, we give a sufficient condition for the van Geemen-van der Geer locus Γ00 to be positive dimensional. Along the way, we prove a sharp Castelnuovo-type inequality for hyperelliptic curves on abelian varieties, and characterize the cases in which we have equality. These results are consequences of a general relation between Seshadri constants and the surjectivity of high order Gauß-Wahl maps on curves inside abelian varieties, which we also prove. Applying such relation to Abel-Jacobi curves we obtain new results about the surjectivity of higher Gauß-Wahl maps on smooth curves, sharpening a previous result of Bertram, Ein and Lazarsfeld. In the process, we discuss further questions and conjectures.
Related
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BibTeX
@misc{alvarado2026seshadri,
title = {{Seshadri constants and hyperelliptic curves on abelian varieties}},
author = {Nelson Alvarado},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.09418},
eprint = {2606.09418},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.09418},
}
RIS
TY - UNPB
TI - Seshadri constants and hyperelliptic curves on abelian varieties
AU - Nelson Alvarado
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2606.09418
UR - https://arxiv.org/abs/2606.09418
AB - Given a principally polarized abelian variety, we give a sufficient condition for the van Geemen-van der Geer locus Γ_00 to be positive dimensional. Along the way, we prove a sharp Castelnuovo-type inequality for hyperelliptic curves on abelian varieties, and characterize the cases in which we have equality. These results are consequences of a general relation between Seshadri constants and the surjectivity of high order Gauß-Wahl maps on curves inside abelian varieties, which we also prove. Applying such relation to Abel-Jacobi curves we obtain new results about the surjectivity of higher Gauß-Wahl maps on smooth curves, sharpening a previous result of Bertram, Ein and Lazarsfeld. In the process, we discuss further questions and conjectures.
ER -