Torelli theorems for Quot schemes of vector bundles on curves
Abstract
For a vector bundle E on a smooth projective curve C, one defines the Quot scheme Qd(E,C) parametrizing subsheaves of E having length d torsion quotients. If Ei is a vector bundle of rank r≥ 2 on a smooth projective curve Ci for i=1,2, isomorphisms between ℙCi(Ei) preserving the projective bundle structures induce isomorphisms of Qd(Ei,Ci), called natural isomorphisms. We show that for d≥ 2 all isomorphisms between Qd(Ei,Ci) are natural, except for a non-natural involution of Q2(𝒪C⊕ r,C). This gives a complete description of the automorphism group of Qd(E,C), generalizing previous works of Biswas-Dhillon-Hurtubise and Gangopadhyay. This also shows that one can reconstruct the curve from the Quot scheme, a Torelli-type theorem. As a key step in our proof, we show that any isomorphism between symmetric powers of smooth projective curves preserving big diagonals is natural, which is interesting in its own right.
Related
- Torelli Theorem for moduli stacks of vector bundles on tame Deligne Mumford curves
- Torelli theorem for the moduli spaces of twisted vector bundles on smooth projective curves
- Tangent bundles of Quot schemes of vector bundles on curves
- Torelli theorem for the product of moduli spaces of vector bundles over a curve
- Ample vector bundles on non-proper schemes
BibTeX
@misc{bansal2026torelli,
title = {{Torelli theorems for Quot schemes of vector bundles on curves}},
author = {Ashima Bansal and Supravat Sarkar and Shivam Vats},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14283},
eprint = {2609.14283},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14283},
}
RIS
TY - UNPB
TI - Torelli theorems for Quot schemes of vector bundles on curves
AU - Ashima Bansal
AU - Supravat Sarkar
AU - Shivam Vats
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14283
UR - https://arxiv.org/abs/2609.14283
AB - For a vector bundle E on a smooth projective curve C, one defines the Quot scheme Q_d(E,C) parametrizing subsheaves of E having length d torsion quotients. If E_i is a vector bundle of rank r≥ 2 on a smooth projective curve C_i for i=1,2, isomorphisms between ℙ_C_i(E_i) preserving the projective bundle structures induce isomorphisms of Q_d(E_i,C_i), called natural isomorphisms. We show that for d≥ 2 all isomorphisms between Q_d(E_i,C_i) are natural, except for a non-natural involution of Q_2(𝒪_C^⊕ r,C). This gives a complete description of the automorphism group of Q_d(E,C), generalizing previous works of Biswas-Dhillon-Hurtubise and Gangopadhyay. This also shows that one can reconstruct the curve from the Quot scheme, a Torelli-type theorem. As a key step in our proof, we show that any isomorphism between symmetric powers of smooth projective curves preserving big diagonals is natural, which is interesting in its own right.
ER -