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Geometry of a semistable degeneration of moduli of vector bundles on curves
Abstract
We study the Gieseker moduli space ℳX0 of stable rank-$2$, degree-$1$ vector bundles on an irreducible curve X0 with one node. Its normalization is a two-step blow-up of the moduli space of generalized parabolic bundles on the normalization X̃0, and ℳX0 is recovered by gluing along a natural involution on the boundary. From this we compute Pic(ℳX0) and, via a Harder--Narasimhan stratification, the virtual Poincaré polynomial of ℳX0. Since the normalization morphism is small, the intersection cohomology of ℳX0 equals the ordinary cohomology of the smooth normalization, which gives a closed formula in terms of the arithmetic genus.
Related
- 𝔸1-connectedness of moduli of semistable bundles and symplectic bundles on a curve
- Torelli theorem for the product of moduli spaces of vector bundles over a curve
- Fourier-Mukai partners and autoequivalences of moduli spaces of vector bundles
- A Note On the moduli space of rank two vector bundles on Hirzebruch surfaces
- Torelli Theorem for moduli stacks of vector bundles on tame Deligne Mumford curves
BibTeX
@misc{das2026geometry,
title = {{Geometry of a semistable degeneration of moduli of vector bundles on curves}},
author = {Sourav Das},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08589},
eprint = {2610.08589},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08589},
}
RIS
TY - UNPB
TI - Geometry of a semistable degeneration of moduli of vector bundles on curves
AU - Sourav Das
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08589
UR - https://arxiv.org/abs/2610.08589
AB - We study the Gieseker moduli space ℳ_X_0 of stable rank-$2$, degree-$1$ vector bundles on an irreducible curve X_0 with one node. Its normalization is a two-step blow-up of the moduli space of generalized parabolic bundles on the normalization X̃_0, and ℳ_X_0 is recovered by gluing along a natural involution on the boundary. From this we compute Pic(ℳ_X_0) and, via a Harder--Narasimhan stratification, the virtual Poincaré polynomial of ℳ_X_0. Since the normalization morphism is small, the intersection cohomology of ℳ_X_0 equals the ordinary cohomology of the smooth normalization, which gives a closed formula in terms of the arithmetic genus.
ER -