What is known about rational homotopy type of irreducible component of $Hilb^n(\mathbb C^k)$ containing configuration space? I've searched arXiv for a while and found nothing but surmise that Betti numbers are still not computed even for $k=3$ case.
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$\begingroup$ You might know this already but for the case of $k=2$ or surfaces in general the cohomology is intricately related to Vertex algebras arxiv.org/pdf/math/0009132.pdf $\endgroup$– Saal HardaliCommented May 23, 2017 at 22:58
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$\begingroup$ I'm afraid I don't know anything about Hilbert schemes, but if this irreducible component is homotopy equivalent to a configuration space of $\mathbb{C}^k$ then the answer is "a lot" and I can give references. $\endgroup$– Mark GrantCommented May 31, 2017 at 9:35
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$\begingroup$ Inclusion of configuration space is (un)fortunately nullhomotopic; the question is about the fiber over $n$-tuple point. It's possible that they are related nonetheless, because Fulton compactification (which is an easy-going analogue of Hilb) can be used in computation of Conf dga model. $\endgroup$– Denis TCommented May 31, 2017 at 12:02
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