Theta divisor

introduction

It is a well known fact that the Theta divisor on the Jacobian of a non-singular curve is a determinantal variety, i.e. is defined by the zero set of a determinant.
It is a classical result that the evaluation at the n-torsion points, \(n\geq 4\) of Riemann's theta function completely determines the abelian variety embedded in \(\mathbb{P}^{n^g-1}\). (See Mumford's Tata lectures 3)

Grushevsky, Samuel, and Klaus Hulek. “Geometry of Theta Divisors --- a Survey.” arXiv:1204.2734 [math], April 12, 2012. http://arxiv.org/abs/1204.2734.

Emily Clader, Samuel Grushevsky, Felix Janda, Dmitry Zakharov, Powers of the theta divisor and relations in the tautological ring, arXiv:1605.05425 [math.AG], May 18 2016, http://arxiv.org/abs/1605.05425
Jungkai Chen, Zhi Jiang, Zhiyu Tian, Irregular varieites with geometric genus one, theta divisors, and fake tori, arXiv:1604.07503 [math.AG], April 26 2016, http://arxiv.org/abs/1604.07503
Humberto A. Diaz, The motive of a smooth Theta divisor, http://arxiv.org/abs/1603.04345v1
Auffarth, Robert, Gian Pietro Pirola, and Riccardo Salvati Manni. “Torsion Points on Theta Divisors.” arXiv:1512.09296 [math], December 31, 2015. http://arxiv.org/abs/1512.09296.
Izadi, Elham, and Jie Wang. “The Irreducibility of the Primal Cohomology of the Theta Divisor of an Abelian Fivefold.” arXiv:1510.00046 [math], September 30, 2015. http://arxiv.org/abs/1510.00046.
Kass, Jesse Leo, and Nicola Pagani. “Extensions of the Universal Theta Divisor.” arXiv:1507.03564 [math], July 13, 2015. http://arxiv.org/abs/1507.03564.
Krämer, Thomas. “Cubic Threefolds, Fano Surfaces and the Monodromy of the Gauss Map.” arXiv:1501.00226 [math], December 31, 2014. http://arxiv.org/abs/1501.00226.
Rahmati, Mohammad Reza. “Motive of Theta Divisor I.” arXiv:1411.3375 [math], October 30, 2014. http://arxiv.org/abs/1411.3375.