  We call a finite-dimensional K-algebra A geometrically irreducible if for all
d, all connected components of the affine scheme of d-dimensional A-modules are
irreducible. We show that the geometrically irreducible algebras without loops
(this includes all algebras of finite global dimension) are the hereditary
algebras. We also show that truncated polynomial rings are the only
geometrically irreducible local algebras. Finally, we formulate a conjectural
classification of all geometrically irreducible algebras.
