On the modular interpretation of the Nagaraj-Seshadri locus. (English) Zbl 1268.14036
The author constructs a moduli space \({\mathcal M}_0\) for torsion-free sheaves \(\mathcal A\) of rank \(r\) and degree 0 on a complex irreducible curve \(X\) with one node. Each sheaf \(\mathcal A\) is endowed with a homomorphism \({\mathfrak d}: \bigwedge^r {\mathcal A}\to {\mathcal O}_X\) which is an isomorphism away from the node. The curve \(X\) is considered as a degeneration \({\mathcal X}_0\) in a family \({\mathcal X}\to S\) of irreducible projective curves whose generic fibre is smooth curve \({\mathcal X}_{\eta}\) and whose base \(S\) is a spectrum of a discrete valuation ring. The corresponding moduli space \({\mathcal M}_0\) to be constructed is also a degeneration in the family \({\mathcal M}_S \to S\) of moduli spaces (of semistable torsion-free sheaves of rank \(r\) and degree 0 on \(\mathcal X\)) with same base and its generic fibre is moduli \({\mathcal M}_{S,\eta}\) of semistable vector bundles of rank \(r\) and degree 0 on the smooth curve \({\mathcal X}_{\eta}.\)
Let \({\mathcal N}_0\) be the set of all polystable torsion-free sheaves \({\mathcal E}\) of rank \(r\) and degree 0 for which there is an homomorphism \(\bigwedge^r {\mathcal E}/\mathrm{Torsion} \to {\mathcal O}_{{\mathcal X}_0}\) which is an isomorphism away from the node. The problem is to give a modular interpretation to this subset. In particular, this will give it a scheme structure.
The author interprets the theory of vector bundles of rank \(r\) as a theory of principal \(\mathrm{GL}_r(\mathbb C)\)-bundles. There is a closed subvariety \({\mathcal N} \subset {\mathcal M}_{S,\eta}\) which is isomorphic to moduli space of semistable principal \(\mathrm{SL}_r(\mathbb C)\)-bundles. Let \({\mathcal N}_S\) be the closure of \({\mathcal N}\) in \({\mathcal M}_S\), so that \({\mathcal N}_{S,\eta}={\mathcal N}\). He constructs a (relative) moduli space \(\widehat {\mathcal N}_S \to S\) together with a surjective \(S\)-morphism \(\widehat N_S \to \mathcal N_S\) isomorphic on the generic fibre and such that \(\widehat {\mathcal N}_{S,0}\) parametrizes trosion-free sheaves \(\mathcal A\) with a homomorphism \(\mathfrak d\) as required.
A part of the theory is developed for any semisimple linear algebraic group \(G\) instead of \(\mathrm{SL}_r(\mathbb C).\)
Let \({\mathcal N}_0\) be the set of all polystable torsion-free sheaves \({\mathcal E}\) of rank \(r\) and degree 0 for which there is an homomorphism \(\bigwedge^r {\mathcal E}/\mathrm{Torsion} \to {\mathcal O}_{{\mathcal X}_0}\) which is an isomorphism away from the node. The problem is to give a modular interpretation to this subset. In particular, this will give it a scheme structure.
The author interprets the theory of vector bundles of rank \(r\) as a theory of principal \(\mathrm{GL}_r(\mathbb C)\)-bundles. There is a closed subvariety \({\mathcal N} \subset {\mathcal M}_{S,\eta}\) which is isomorphic to moduli space of semistable principal \(\mathrm{SL}_r(\mathbb C)\)-bundles. Let \({\mathcal N}_S\) be the closure of \({\mathcal N}\) in \({\mathcal M}_S\), so that \({\mathcal N}_{S,\eta}={\mathcal N}\). He constructs a (relative) moduli space \(\widehat {\mathcal N}_S \to S\) together with a surjective \(S\)-morphism \(\widehat N_S \to \mathcal N_S\) isomorphic on the generic fibre and such that \(\widehat {\mathcal N}_{S,0}\) parametrizes trosion-free sheaves \(\mathcal A\) with a homomorphism \(\mathfrak d\) as required.
A part of the theory is developed for any semisimple linear algebraic group \(G\) instead of \(\mathrm{SL}_r(\mathbb C).\)
Reviewer: Nadezda Timofeeva (Yaroslavl)
MSC:
| 14H60 | Vector bundles on curves and their moduli |
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
References:
| [1] | DOI: 10.1142/S0129167X09005558 · Zbl 1177.14069 · doi:10.1142/S0129167X09005558 |
| [2] | DOI: 10.1007/BF02384869 · Zbl 0773.14006 · doi:10.1007/BF02384869 |
| [3] | DOI: 10.1155/S1073792804133114 · Zbl 1106.14027 · doi:10.1155/S1073792804133114 |
| [4] | DOI: 10.1007/BF02829806 · Zbl 1096.14027 · doi:10.1007/BF02829806 |
| [5] | DOI: 10.1112/S0024610706022836 · Zbl 1101.14048 · doi:10.1112/S0024610706022836 |
| [6] | Gieseker D., J. Di{\currency}. Geom. 19 pp 173– (2004) |
| [7] | DOI: 10.1016/j.aim.2008.05.015 · Zbl 1163.14008 · doi:10.1016/j.aim.2008.05.015 |
| [8] | Nagaraj D. S., Proc. Indian Acad. Sci. Math. Sci. 107 pp 101– (1997) |
| [9] | DOI: 10.1007/BF01343949 · Zbl 0284.32019 · doi:10.1007/BF01343949 |
| [10] | DOI: 10.1155/S1073792802107069 · Zbl 1034.14017 · doi:10.1155/S1073792802107069 |
| [11] | DOI: 10.1155/S1073792804132984 · Zbl 1093.14507 · doi:10.1155/S1073792804132984 |
| [12] | A. H., J. Di{\currency}. Geom. 66 pp 169– (2004) |
| [13] | A. H., Math. Sci. 115 pp 15– (2005) |
| [14] | DOI: 10.1155/IMRN.2005.1427 · Zbl 1084.14034 · doi:10.1155/IMRN.2005.1427 |
| [15] | Sun X., J. Alg. Geom. 9 pp 459– (2000) |
| [16] | Sun X., Asian J. Math. 7 pp 609– (2003) · Zbl 1055.14039 · doi:10.4310/AJM.2003.v7.n4.a10 |
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