On quiver Grassmannians and orbit closures for gen-finite modules
Matthew Pressland, Julia Sauter
TL;DR
This work develops a general framework to desingularise orbit closures and quiver Grassmannians for gen-finite modules over finite-dimensional algebras by using cogenerator-tilted algebras and recollements. Central to the approach are the canonical tilting/cotilting structures, the intermediate extension functor $c$, and realizations of relevant varieties as affine quotients $R_B(d,s)/\!\!/\mathbf{Gl}_s$, which yield smooth desingularisations with connected fibres. The paper proves key theorems for both orbit closures and quiver Grassmannians, and provides detailed examples (notably for the $n$-subspace quiver) to illustrate the construction and its computational aspects. Overall, the results generalise prior work of Crawley-Boevey and the second author, extending desingularisation techniques to the broader setting of gen-finite (and dually cogen-finite) modules via tilting theory and derived-category methods, with explicit geometric consequences. The framework offers a versatile toolkit for resolving singularities in representation varieties and related moduli spaces, potentially impacting explicit geometric and combinatorial studies in representation theory.
Abstract
We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.