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Let $X$ be a variety of arbitrary dimension, let $H$ denote the main component of the Hilbert scheme of points of $X$ (i.e, the closure of locus of reduced subschemes), and let $Z$ be the reduced fiber product $H\times_{Sym^nX}X^n$. $Z$ is (a variant of) the isospectral Hilbert scheme. In section 5.2 of "Hilbert schemes, polygraphs and the Macdonald positivity conjecture", Mark Haiman makes two conjectures which would imply that

  1. $Z$ is normal and Cohen-Macaulay
  2. $Z$ is isomorphic to the blow-up of $X^n$ along the scheme-theoretic union of pairwise diagonals

Both of these are true for $\operatorname{dim}X=2$ (see the same paper.) What is the status of these conjectures/these corollaries?

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