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In Section 4 of the paper Kirwan surjectivity for quiver varieties (Inventiones Math. 2018) McGerty and Nevins define a compactification of the moduli space of representations of the preprojective algebra associated to a quiver as the moduli space of representations of another quiver (the "tripling" of the original quiver) with certain relations. This compactification plays a key role in this paper and is defined by an explicit procedure obtaining the vertices and edges for the "tripled" quiver starting with the vertices and edges of the original quiver. My question is: what is the intuition behind the definition of such a compactification? Are there other constructions of (modular) compactifications of quiver varieties?

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    $\begingroup$ If I recall correctly I believe the intuition is roughly this: think of representations of a quiver as an analog of coherent sheaves on a smooth projective curve, modules for its preprojective algebra (having to do with doubling the quiver) as sheaves on the cotangent bundle to the curve, and modules for this tripled version as sheaves on the compactified cotangent bundle (projectivization of cotangent direct sum the trivial line bundle). $\endgroup$ – David Ben-Zvi Feb 1 '20 at 18:33
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    $\begingroup$ I don't know the answers to these questions, but I would expect these compactifications are the same. An example is Hilbert schemes or moduli of torsion-free sheaves on the cotangent to a curve (say on $A^2$) or its noncommutative version (which is what gives CM spaces) - you compactify it by thinking of framed t-f sheaves on a compactification of the cotangent, and it doesn't matter which (eg $P^2$ vs $P^1\times P^1$) since your sheaf is trivialized at $\infty$ anyway.. $\endgroup$ – David Ben-Zvi Feb 2 '20 at 16:45
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    $\begingroup$ (From Hilbert scheme POV you just want to make sure points are not allowed to run off to $\infty$ in a family - if you do you connect $Hilb_n$ together for all $n$, which is what happens in the adelic Grassmannian.) $\endgroup$ – David Ben-Zvi Feb 2 '20 at 16:47
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    $\begingroup$ Yes you can understand Wilson's embedding this way -- the adelic Grassmannian itself is the moduli problem for D-bundles which are unframed (ie torsion-free sheaves on the NC cotangent bundle, not its compactification) which are trivialized generically along the curve (see eg our paper arxiv.org/abs/0807.4992), and ON THE LEVEL OF POINTS there's a natural embedding of the CM spaces here (ie the framing gives you a canonical trivialziation on the locus of the curve where the D-bundle is locally free). $\endgroup$ – David Ben-Zvi Feb 24 '20 at 15:02
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    $\begingroup$ Whether you consider this an algebro-geometric identification of the closures is a different question - IIRC this embedding doesn't make sense in families, ie you don't get a generic trivialization along the curve direction from the CM space, but I could be misremembering. In any case you have to be careful since the adelic Grassmannian is a funny object algebro-geometrically (it's the pushforward to the curve of an ind-scheme over the Ran prestack of the curve). $\endgroup$ – David Ben-Zvi Feb 24 '20 at 15:12

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