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Let $X$ be a smooth projective algebraic variety and $D^b(X)$ be the derived category of coherent sheaves on $X$. Denote by $Sym^nX$ the $n$-th symmetric product of $X$. Can we describe the derived category $D^b(Sym^nX)$ in terms of $D^b(X)$. If so, how are they related? Is there any reference?

This question is intrigued by the question Hilbert schemes of points and exceptional collections asked by Cat.

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There is a category closely related to $D^b(Sym^n X)$ which can be described. Namely, the $S_n$-equivariant derived category of coherent sheaves on $X^n$. This category can be considered as a noncommutative resolution of singularities of $D^b(Sym^n X)$ (the latter category is singular when $\dim X > 1$). The description of the $S_n$-equivariant derived category I have in mind is the following. First, one can consider the $n$-th tensor power'' of $D^b(X)$ --- if one fixes a DG-enhancement for $D^b(X)$ then it is the derived category of DG-modules over the $n$-th tensor power of the underlying DG-category. The symmetric group acts naturally on this category, so one can consider the corresponding equivariant category. That's it.

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  • $\begingroup$ Thanks a lot for the answer. Can you help me to understand that how to view the $S_n$-equivariant category as a noncommutative resolution of singularities of $D^b(Sym^nX)$. What does the singular locus of $D^(Sym^nX)$ look like? $\endgroup$ – Fei YE Sep 30 '10 at 16:55

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