This question is related to this MO question.

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?

This question is related to this MO question.

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?