Thick subcategories on weighted projective curves and nilpotent representations of quivers

We continue the study of thick triangulated subcategories, started in arXiv:2007.02134, and consider thick subcategories in the derived category of a weighted projective curve and corresponding abelian thick subcategories. Our main result is that any thick subcategory on a weighted projective curve either is equivalent to the derived category of nilpotent representations of some quiver (we call such categories quiver-like) or is the orthogonal to an exceptional collection of torsion sheaves (we call such subcategories big). We provide several equivalent descriptions of big subcategories and explain that they can be explicitly classified. We give some results clarifying the structure of thick subcategories: in particular, for weighted projective lines we prove that any admissible subcategory is generated by an exceptional collection and any exceptional collection is a part of a full one. Apart from the above, we study general properties of quiver-like categories. In particular, we extend and simplify results from arXiv:2007.02134 providing sufficient criteria for a triangulated or abelian category to be quiver-like.