Extension groups of tautological bundles on symmetric products of curves

Andreas Krug

doi:10.48550/arXiv.2105.13740

Details

Original language	English
Pages (from-to)	493-530
Number of pages	38
Journal	Beitrage zur Algebra und Geometrie
Volume	64
Issue number	2
Early online date	25 Apr 2022
Publication status	Published - Jun 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.

Keywords

14J60, 14H60, 14C05, Symmetric products of curves, Moduli of vector bundles, Tautological bundles, Extension groups

ASJC Scopus subject areas

Mathematics(all)
Geometry and Topology
Mathematics(all)
Algebra and Number Theory

Cite this

Extension groups of tautological bundles on symmetric products of curves. / Krug, Andreas.
In: Beitrage zur Algebra und Geometrie, Vol. 64, No. 2, 06.2023, p. 493-530.

Research output: Contribution to journal › Article › Research › peer review

Krug, A 2023, 'Extension groups of tautological bundles on symmetric products of curves', Beitrage zur Algebra und Geometrie, vol. 64, no. 2, pp. 493-530. https://doi.org/10.48550/arXiv.2105.13740, https://doi.org/10.1007/s13366-022-00644-0

Krug, A. (2023). Extension groups of tautological bundles on symmetric products of curves. Beitrage zur Algebra und Geometrie, 64(2), 493-530. https://doi.org/10.48550/arXiv.2105.13740, https://doi.org/10.1007/s13366-022-00644-0

Krug A. Extension groups of tautological bundles on symmetric products of curves. Beitrage zur Algebra und Geometrie. 2023 Jun;64(2):493-530. Epub 2022 Apr 25. doi: 10.48550/arXiv.2105.13740, 10.1007/s13366-022-00644-0

Krug, Andreas. / Extension groups of tautological bundles on symmetric products of curves. In: Beitrage zur Algebra und Geometrie. 2023 ; Vol. 64, No. 2. pp. 493-530.

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