Thick Subcategories on Weighted Projective Curves and Nilpotent Representations of Quivers

Abstract We continue the study of thick triangulated subcategories, started by Valery Lunts and the author in “Thick subcategories on curves”, and consider thick subcategories in the derived category of coherent sheaves on a weighted projective curve and the 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 subcategory to an exceptional collection of torsion sheaves (we call such subcategories big). We examine 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. We show that the derived categories of weighted projective curves satisfy the Jordan–Hölder property and do not contain phantoms. Finally, we extend and simplify results from loc. cit., providing sufficient criteria for a triangulated or abelian category to be quiver-like.