Moduli spaces of rank 2 sheaves on ℙ3 via Quot schemes
Abstract
We study the Gieseker--Maruyama moduli space Mc3 of stable rank 2 torsion-free sheaves on ℙ3 with Chern classes c1=−1, c2=1, and odd c3 ≤ 1. Every such sheaf E has reflexive hull E∗∗ parametrised by M1, and the quotient E∗∗/E is a 0-dimensional sheaf of length n = (1−c3)/2. We prove that Mc3 is isomorphic to the relative Quot scheme Quotℙ3 × M1/M1(ℛ,n), where ℛ is the universal reflexive sheaf over M1. Our technique combines (i) Bridgeland-type wall-crossing in Donaldson--Thomas theory, (ii) the theory of good moduli spaces, (iii) deformation theory in dgla and Ext-quiver language. We deduce that Mc3 is connected, and we analyse its \emph{smoothable component} Mc3main, proving it is rational and generically smooth of dimension 4n+3. We further prove that dim Sing Mc3 ≥ 4n−2 for all n ≥ 2. Finally, we provide a geometric characterisation for the cases n≤ 2. We show that M−1 is a smooth rational projective variety of dimension 7 and compute its motive in the Grothendieck ring of varieties, while M−3 is shown to be an integral, normal variety of dimension 11 that is regular in codimension 4.
Related
- Birational maps of moduli spaces of stable sheaves on a K3 surface induced by Fourier-Mukai transforms
- Stability of representation cohomology sheaves on moduli spaces of bundles
- Limits of Hyperkähler Manifolds and Moduli Spaces of Sheaves on K3 Surfaces
- On leading Gopakumar--Vafa invariants of local ℙ2 and leading Betti numbers of moduli spaces of one-dimensional sheaves on ℙ2
- Tangent bundles of Quot schemes of vector bundles on curves
BibTeX
@misc{almeida2026moduli,
title = {{Moduli spaces of rank 2 sheaves on $\mathbb P^3$ via Quot schemes}},
author = {Charles Almeida and Nicolò Bignami and Marcos Jardim and Andrea T. Ricolfi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01961},
eprint = {2611.01961},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01961},
}
RIS
TY - UNPB
TI - Moduli spaces of rank 2 sheaves on ℙ^3 via Quot schemes
AU - Charles Almeida
AU - Nicolò Bignami
AU - Marcos Jardim
AU - Andrea T. Ricolfi
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01961
UR - https://arxiv.org/abs/2611.01961
AB - We study the Gieseker--Maruyama moduli space M_c_3 of stable rank 2 torsion-free sheaves on ℙ^3 with Chern classes c_1=−1, c_2=1, and odd c_3 ≤ 1. Every such sheaf E has reflexive hull E^∗∗ parametrised by M_1, and the quotient E^∗∗/E is a 0-dimensional sheaf of length n = (1−c_3)/2. We prove that M_c_3 is isomorphic to the relative Quot scheme Quot_ℙ^3 × M_1/M_1(ℛ,n), where ℛ is the universal reflexive sheaf over M_1. Our technique combines (i) Bridgeland-type wall-crossing in Donaldson--Thomas theory, (ii) the theory of good moduli spaces, (iii) deformation theory in dgla and Ext-quiver language. We deduce that M_c_3 is connected, and we analyse its \emph{smoothable component} M_c_3^main, proving it is rational and generically smooth of dimension 4n+3. We further prove that dim Sing M_c_3 ≥ 4n−2 for all n ≥ 2. Finally, we provide a geometric characterisation for the cases n≤ 2. We show that M_−1 is a smooth rational projective variety of dimension 7 and compute its motive in the Grothendieck ring of varieties, while M_−3 is shown to be an integral, normal variety of dimension 11 that is regular in codimension 4.
ER -