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Number of equations defining the Kummer varieties of hyperelliptic curves
Abstract
The Jacobian of an hyperelliptic curve C of genus g over ℂ is equipped with a symmetric theta divisor Θ that defines a principal polarization. The line bundle 𝒪(2Θ) defines an embedding of the Kummer variety K inside the projective space ℙ2g−1, which factorizes inside a certain orthogonal Grassmannian. We describe a minimal free resolution of K inside this Grassmannian using geometric representation theory. As a consequence, we obtain a formula for the number of quartic equations defining K in the projective space ℙ2g−1.
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BibTeX
@misc{partouche2026number,
title = {{Number of equations defining the Kummer varieties of hyperelliptic curves}},
author = {Nathan Partouche},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35097},
eprint = {2610.35097},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35097},
}
RIS
TY - UNPB
TI - Number of equations defining the Kummer varieties of hyperelliptic curves
AU - Nathan Partouche
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35097
UR - https://arxiv.org/abs/2610.35097
AB - The Jacobian of an hyperelliptic curve C of genus g over ℂ is equipped with a symmetric theta divisor Θ that defines a principal polarization. The line bundle 𝒪(2Θ) defines an embedding of the Kummer variety K inside the projective space ℙ^2^g−1, which factorizes inside a certain orthogonal Grassmannian. We describe a minimal free resolution of K inside this Grassmannian using geometric representation theory. As a consequence, we obtain a formula for the number of quartic equations defining K in the projective space ℙ^2^g−1.
ER -