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I would like to know some reference (articles, books...) about any kind of moduli spaces of any of the following objects:

  • vector bundles

  • torsion-free sheaves

  • principal bundles

  • parabolic bundles

over singular algebraic curves (reducible or not), in any of the following frameworks:

  • algebraic geometry (in characteristic zero and in positive characteristic)

  • holomorphic geometry

  • integrable systems

  • gauge theory

  • differential geometry

  • topology

  • ...anything you like...

I would be particularly glad to have some reference about torsion-free sheaves in the algebro-geometric setting.


Edit: I should emphasize that my reference request is about some structures over singular curves. The freedom I expect in a typical answer should be on the structure (e.g. bundles, torsion-free sheaves,...) and on the viewpoint (e.g. pure algebraic geometry, trascendental methods, ...), but the base curve must be singular (for the answer not to be offtopic).

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There's an enormous literature on this subject. Torsion-free sheaves, compactified Jacobians, Gieseker bundles, etc. It'd be really helpful if you said more clearly what you were looking for. Are you doing a literature review? Do you need a moduli space with certain properties? –  userN Apr 8 '10 at 16:08
I need that the variety over which we consider these structures is a singular (projective) algebraic curve. –  Qfwfq Apr 8 '10 at 16:37
I'm aware that you want singular curves. But my point still stands: This is a big subject, and it's not exactly unified. I could probably point you towards something useful, if you gave some indication of what you wanted the moduli space for. But without context, the question is too general. –  userN Apr 8 '10 at 16:43
Ok. Let's stick to torsion-free sheaves of rank 2 and fixed determinant (in an algebro-geometric/holomorphic setting), if it can help to "restrict" the literature. –  Qfwfq Apr 8 '10 at 17:15
Edit: should add "semistable". –  Qfwfq Apr 8 '10 at 17:19
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6 Answers

up vote 6 down vote accepted

Some of the many (semi)standard references are below (with no claims to completeness or representativeness, if that's a word -- just the first references that came to mind). My feeling is the subject is still very much in its infancy however, for example one would like to know the standard package of nonabelian Hodge theory results for singular curves (geometry of Higgs bundles and local systems, Hitchin fibration, its self-duality etc) and there are partial results but no complete picture as far as I know.

Caporaso, Lucia A compactification of the universal Picard variety over the moduli space of stable curves. J. Amer. Math. Soc. 7 (1994), no. 3, 589--660.

Pandharipande, Rahul A compactification over $\overline {M}_g$ of the universal moduli space of slope-semistable vector bundles. J. Amer. Math. Soc. 9 (1996), no. 2, 425--471.

Seshadri, C. S. Moduli spaces of torsion free sheaves on nodal curves and generalisations. I. Moduli spaces and vector bundles, 484--505, London Math. Soc. Lecture Note Ser., 359, Cambridge Univ. Press, Cambridge, 2009. (and earlier papers of his)

arXiv:1001.3868 Title: Autoduality of compactified Jacobians for curves with plane singularities Authors: D.Arinkin

--see this reference for refs to the vast literature by Altman-Kleiman and Esteves-Kleiman on compactified Jacobians

Kausz, Ivan A Gieseker type degeneration of moduli stacks of vector bundles on curves. Trans. Amer. Math. Soc. 357 (2005), no. 12, 4897--4955 (electronic).

Schmitt, Alexander H. W. Singular principal $G$-bundles on nodal curves. J. Eur. Math. Soc. (JEMS) 7 (2005), no. 2, 215--251. (and earlier papers of his)

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I think a good illustration of why torsion-free sheaves on singular curves are both interesting and difficult is given by the following. Consider the $GL_n$ case of the Hitchin fibration, i.e., the map from the moduli space of vector bundles of rank $n$ with a twisted endomorphism on a smooth, projective curve to the Hitchin base space of characteristic polynomials.

Then a result of Beauville, Naramsihan, and Ramanan (see this paper http://math.unice.fr/~beauvill/pubs/bnr.pdf) says that for a sufficiently nice characteristic polynomial $a$ in the Hitchin base, the stack of torsion-free coherent sheaves of rank one on the associated spectral curve is isomorphic to the Hitchin fiber associated to $a$. See, for example, the notes on the Hitchin fibration on Drinfeld's geometric Langlands page for a quick introduction to these ideas.

In general, these spectral curves will be singular (which is why I couldn't simply say 'line bundle' in the above correspondence). Given that the Hitchin fibration and Hitchin fibers are some of the most interesting geometric objects currently being studied, I think this gives a flavor for how interesting torsion-free sheaves on singular curves (and these are just rank one) can be.

Also, it's worth mentioning that the curves which arise as spectral curves aren't even that singular (nodal and cuspidal elliptic curves are a couple examples), in the sense that the dimension of the tangent space at any point is at most two. There's an old result (I think from 1979) of Altman, Iarrobino, and Kleiman proving that in this situation, the stack of line bundles is dense in the stack of torsion-free coherent sheaves of rank one. This result has since been generalized to arbitrary reductive groups by Ngo in his paper proving the Fundamental Lemma.

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Thanks for pointing out this. –  Qfwfq Apr 8 '10 at 18:40
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I might as well mention my paper with Frenkel & Teleman, which describes a moduli stack of GL(1)-bundles on semistable marked curves which generalizes many of the existing moduli spaces.

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Reference: Cyril D'souza PHD thesis.(1974). Tata Institute of fundamental research,Mumbai.

This thesis is concerned with constructing a "natural compactifiction" for the generalised jacobian of a singular curve.

can also see Newstead notes.

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Few days ago I started a $n$lab page moduli space of bundles which has almost no real content so far, but has a number of references, mainly about the moduli of stable vector or principal bundles on curves. People are welcome to improve the page with some explanations. Mumford's book Geometric invariant theory should be useful for the background. For the torsion free sheaves part, I would also recommend looking at recent works of Nakajima, Göttsche, Stafford and others.

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Thank you for the reference. But I think most of those articles are in the context of smooth curves, aren't they? Maybe the article of Faltings on bundles on semistable curves... –  Qfwfq Apr 8 '10 at 13:26
Oh, sorry, I was not real careful in reading your question in detail. Good idea to improve mz entry iwth more on singular case (and more input for the related nlab entry on Hitchin fibration) :) –  Zoran Skoda Apr 9 '10 at 0:30
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Carlos Simpson has written a lot on this. Nasatyr & Steer. Biquard Konno All on basic constructions. Verlinde, Thaddeus and others on structural considerations. Anything to do with modular or automorphic forms is also related. There is a lot, can you be more specific?

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As far as I understand, the article by Nasatyr-Steer you linked is about orbifold curves (which are essentially smooth curves together with some additional data on a finite set of points), not to singular curves. –  Qfwfq Apr 8 '10 at 16:19
In the page of Biquard I couldn't find articles about the topic I'm asking for (but perhaps I just didn't search carefully enough). Could you be more specific? There is an article about certain Higgs bundles on curves, but it seems to me that in that case the objects that are allowed to be singular are the Higgs fields (which are meromorphic) and the integrable connections, not the base curve. Plese correct me if I'm mistaken. –  Qfwfq Apr 8 '10 at 16:25
Also, in the article by Konno you linked, the base curve is a "marked Riemann surface", hence a smooth curve. So, it's kind of off topic with respect to this MO question. –  Qfwfq Apr 8 '10 at 16:28
The best understood singularities of curves look like intersections of curves or punctures. The punctures can in the best cases be smoothly completed with a point. It's often convenient to put the point back in and use it as a yardstick for controlling the behaviour of the bundle at the incompleteness which is necessary to form a sensible space. Much of the discussion is therefore about gluing data, either for the extra point or the two points of self intersection. So although much of the analysis is carried out over a smooth curve it is aimed at solving problems over controlled singularities. –  Jack Evans Apr 8 '10 at 17:22
The modular curve is a well studied singular curve. Most likely you are looking at a bunch of curves that either have the same kind of singularities as the modular curve or which are smooth but glued together. Another example is to consider the spectral curves in the Hitchin moduli space construction and look at what happens to the Prym variety or bundles over an elliptic fibration and consider what happens over singular fibres. –  Jack Evans Apr 8 '10 at 17:39
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