Two vertex geometrically irreducible algebras - the final step

It has been proved by Schröer and the author that are there are basically two possible classes of geometrically irreducible algebras with exactly two vertices in their Gabriel quiver. The first class is connected with the famous Birkhoff problem and one can interpret the points of the corresponding varieties as homomorphisms between modules over truncated polynomial algebras. It has been confirmed that the algebras appearing in this case are actually geometrically irreducible. In the talk we show that the other class also consists of geometrically irreducible algebras. A crucial observation is a connection between the points of the corresponding varieties with extensions between modules over truncated polynomial algebras. This is a report on a joint work with Grzegorz Zwara.