  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.
