Abstract
This paper gives a complete answer of the following question: which (singular, projective) curves have a categorical resolution of singularities which admits a full exceptional collection? We prove that such full exceptional collection exists if and only if the geometric genus of the curve equals to 0. Moreover we can also prove that a curve with geometric genus equal or greater than 1 cannot have a categorical resolution of singularities which has a tilting object. The proofs of both results are given by a careful study of the Grothendieck group and the Picard group of that curve.
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