Thick subcategories on weighted projective curves and nilpotent representations of quivers

TL;DR

This paper classifies thick subcategories in derived categories of weighted projective curves, showing they are either quiver-like or related to exceptional collections, and explores their structural properties.

Contribution

It provides a complete classification of thick subcategories on weighted projective curves, extending previous results and establishing criteria for quiver-like categories.

Findings

Thick subcategories are either quiver-like or big, related to exceptional collections.

Any admissible subcategory on weighted projective lines is generated by an exceptional collection.

Derived categories satisfy Jordan-Holder property and contain no phantoms.

Abstract

We continue the study of thick triangulated subcategories, started by Valery Lunts and the author in arXiv:2007.02134, 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 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 Jordan-Holder property and do not contain phantoms. Finally, we extend and simplify results from arXiv:2007.02134, providing sufficient criteria for a triangulated or abelian category to be quiver-like.