# Geometrically Characterizing Representation Type of Finite-dimensional Algebras

# Geometrically Characterizing Representation Type of Finite-dimensional Algebras

Given a finite-dimensional algebra $A$, the set of $A$-modules of a fixed dimension d can be viewed as a variety. This variety carries a group action whose orbits correspond to isomorphism classes of $A$-modules. A natural problem is to characterize various properties of an algebra $A$ in terms of its module varieties. For example, if $A$ is assumed to have global dimension one, then it is not difficult to show that $A$ has finitely many indecomposable modules (up to isomorphism) if and only if all of its module varieties have a dense orbit, which is also if and only if all weight spaces of semi-invariants in the coordinate rings of its module varieties have dimension one. Our goal is to generalize these statements (with modification) to higher global dimension. After explaining the background, we present counterexamples to the naive generalizations, along with plausible modifications and cases where these modifications are correct. (Joint work with Calin Chindris, Piotr Dowbor, and Jerzy Weyman)