Schottky versus super Schottky in genus 4
Abstract
The study of super Riemann surfaces and their moduli led Witten and Felder, Kazhdan and Polishchuk to ask what is the smallest d such that the d-th power of the Schottky ideal is contained in the super Schottky ideal. Felder, Kazhdan and Polishchuk proved that d=g in odd genus g≥5 and that d∈{g−1,g} in even genus g≥4; Y.~Shen subsequently proved that d=g in even genus g≥6. Thus the only remaining open case was g=4. Here we close this gap by showing that in genus $4$ too, the smallest power of the Schottky ideal contained in the super Schottky ideal is d=g=4. The question is equivalent to one concerning the codifferential of the super period map. The quadratic odd contribution sends a conormal vector to a bivector, represented by a skew-symmetric (2g−2)×(2g−2) matrix. The question is to find such a vector for which, at a generic point, this matrix has maximal rank 2g−2, or $6$ in our case. Our computation uses the standard exact sequences on C× C attached to the sheaves OO(a,b,c), together with a degeneration to a vanishing theta-null and some facts about the Szegő kernel. At a curve with a vanishing theta-null, a regularized version of the codifferential can be described by an explicit multiplication map, allowing an easy computation of the rank. This rank turns out to be $4$. A first-order calculation on an explicit deformation of a cyclic trigonal vanishing theta-null differentiates the global Gaussian-map identity for the regularized conormal bivector. Including the variation of the Gaussian map gives a nonzero first normal symbol. The resulting first-order form is nonzero on the null space of the limiting skew-symmetric matrix, so the rank jumps to $6$ on nearby curves, showing that d=4.
Related
- On the Super-Schottky Ideal and Powers of the Classical Schottky Ideal for g ≥ 5
- Nonnegative conorms, regular matroids, and the tropical Schottky problem
- Higher dimensional Kleinian groups of small Hausdorff dimension are classical Schottky groups II
- The Coxeter-Todd abelian sixfold, Gamma00, and big Schottky
- Fast computation and convergence analysis of the infinite-product representation of the Schottky--Klein prime function
BibTeX
@misc{donagi2026schottky,
title = {{Schottky versus super Schottky in genus 4}},
author = {Ron Donagi and Simone Noja},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16131},
eprint = {2609.16131},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16131},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Schottky versus super Schottky in genus 4
AU - Ron Donagi
AU - Simone Noja
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16131
UR - https://arxiv.org/abs/2609.16131
AB - The study of super Riemann surfaces and their moduli led Witten and Felder, Kazhdan and Polishchuk to ask what is the smallest d such that the d-th power of the Schottky ideal is contained in the super Schottky ideal. Felder, Kazhdan and Polishchuk proved that d=g in odd genus g≥5 and that d∈{g−1,g} in even genus g≥4; Y.~Shen subsequently proved that d=g in even genus g≥6. Thus the only remaining open case was g=4. Here we close this gap by showing that in genus $4$ too, the smallest power of the Schottky ideal contained in the super Schottky ideal is d=g=4. The question is equivalent to one concerning the codifferential of the super period map. The quadratic odd contribution sends a conormal vector to a bivector, represented by a skew-symmetric (2g−2)×(2g−2) matrix. The question is to find such a vector for which, at a generic point, this matrix has maximal rank 2g−2, or $6$ in our case. Our computation uses the standard exact sequences on C× C attached to the sheaves OO(a,b,c), together with a degeneration to a vanishing theta-null and some facts about the Szegő kernel. At a curve with a vanishing theta-null, a regularized version of the codifferential can be described by an explicit multiplication map, allowing an easy computation of the rank. This rank turns out to be $4$. A first-order calculation on an explicit deformation of a cyclic trigonal vanishing theta-null differentiates the global Gaussian-map identity for the regularized conormal bivector. Including the variation of the Gaussian map gives a nonzero first normal symbol. The resulting first-order form is nonzero on the null space of the limiting skew-symmetric matrix, so the rank jumps to $6$ on nearby curves, showing that d=4.
ER -