Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 493-530 |
| Seitenumfang | 38 |
| Fachzeitschrift | Beitrage zur Algebra und Geometrie |
| Jahrgang | 64 |
| Ausgabenummer | 2 |
| Frühes Online-Datum | 25 Apr. 2022 |
| Publikationsstatus | Veröffentlicht - Juni 2023 |
Abstract
We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if E≠ O X is simple, then the natural map Ext1(E,E)→Ext1(E[n],E[n]) is injective for every n. Along with previous results, this implies that E↦ E [ n ] defines an embedding of the moduli space of stable bundles of slope μ∉ [- 1 , n- 1] on the curve X into the moduli space of stable bundles on the symmetric product X ( n ). The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of X with the loci in the moduli space of stable bundles on X ( n ) where the dimension of the tangent space jumps. We also prove that E [ n ] is simple if E is simple.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Geometrie und Topologie
- Mathematik (insg.)
- Algebra und Zahlentheorie
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in: Beitrage zur Algebra und Geometrie, Jahrgang 64, Nr. 2, 06.2023, S. 493-530.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Extension groups of tautological bundles on symmetric products of curves
AU - Krug, Andreas
N1 - Publisher Copyright: © 2022, The Author(s).
PY - 2023/6
Y1 - 2023/6
N2 - We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if E≠ O X is simple, then the natural map Ext1(E,E)→Ext1(E[n],E[n]) is injective for every n. Along with previous results, this implies that E↦ E [ n ] defines an embedding of the moduli space of stable bundles of slope μ∉ [- 1 , n- 1] on the curve X into the moduli space of stable bundles on the symmetric product X ( n ). The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of X with the loci in the moduli space of stable bundles on X ( n ) where the dimension of the tangent space jumps. We also prove that E [ n ] is simple if E is simple.
AB - We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if E≠ O X is simple, then the natural map Ext1(E,E)→Ext1(E[n],E[n]) is injective for every n. Along with previous results, this implies that E↦ E [ n ] defines an embedding of the moduli space of stable bundles of slope μ∉ [- 1 , n- 1] on the curve X into the moduli space of stable bundles on the symmetric product X ( n ). The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of X with the loci in the moduli space of stable bundles on X ( n ) where the dimension of the tangent space jumps. We also prove that E [ n ] is simple if E is simple.
KW - 14J60
KW - 14H60
KW - 14C05
KW - Symmetric products of curves
KW - Moduli of vector bundles
KW - Tautological bundles
KW - Extension groups
UR - http://www.scopus.com/inward/record.url?scp=85128708583&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2105.13740
DO - 10.48550/arXiv.2105.13740
M3 - Article
VL - 64
SP - 493
EP - 530
JO - Beitrage zur Algebra und Geometrie
JF - Beitrage zur Algebra und Geometrie
SN - 0138-4821
IS - 2
ER -