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Let $X$ be a projective curve over a field $\mathbb{C}$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category of perfect complex $Perf(X)$. It is clear that $Perf(X)$ is a strictly full triangulated subcategory of $D^b_{coh}(X)$. Then following Orlov 2003 we define the triangulated category of singularities of $X$ as the quotient of $D^b_{coh}(X)$ and $Perf(X)$, i.e. $$ D_{sg}(X)=D^b_{coh}(X)/Perf(X). $$

Now is there any study of $D_{sg}(X)$ for the curve $X$? For example does $D_{sg}(X)$ admit a semiorthogonal decomposition?

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  • $\begingroup$ Just a comment: a semi-orthogonal decomposition of D^b_coh induces a semi-orthogonal decomposition of D_sg, according to Corollary 1.12 here. $\endgroup$
    – AAK
    Commented Apr 28, 2015 at 13:40
  • $\begingroup$ @AdeelKhan Sure! Nevertheless for $X$ with arithmetic genus $\geq 1$, $D^b_{coh}(X)$ cannot have semiorthogonal decomposition but I wonder if $D_{sg}(X)$ still has. $\endgroup$
    – Zhaoting Wei
    Commented Apr 28, 2015 at 13:55
  • $\begingroup$ Do you know a reference for that? I only knew that fact in the smooth case. $\endgroup$
    – AAK
    Commented Apr 28, 2015 at 14:52
  • $\begingroup$ @AdeelKhan After some thinking I should say that I only know it in the smooth case also. Sorry for the misleading. Nevertheless if we can compute the Grothendieck group of $D^b_{coh}(X)$ for singular curves $X$, it will be solved. I guess $K_0(D^b_{coh}(X))$ is not far from $K_0(D^b_{coh}(\widetilde{X}))$ where $\widetilde{X}\to X$ is the resolution of singularities but I don't know how to prove it. $\endgroup$
    – Zhaoting Wei
    Commented Apr 28, 2015 at 15:06

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