Abstract:
We characterize the canonical algebras such that for all dimension vectors of homogeneous modules the corresponding module varieties are complete intersections (respectively, normal). We also investigate the sets of common zeros of semi-invariants of non-zero degree in important cases. In particular, we show that for sufficiently big vectors they are complete intersections and calculate the number of their irreducible components.

Abstract:
We show that the orbit closure of a directing module is regular in codimension one. In particular, this result gives information about a distinguished irreducible component of a module variety.

Abstract:
Let A be a tame quasi-tilted algebra and d the dimension vector of an indecomposable A-module. In the paper we prove that each irreducible component of the variety of A-modules of dimension vector d is regular in codimension one.

Abstract:
We describe in the paper the graded centers of the derived categories of the derived discrete algebras. In particular, we prove that if $A$ is a derived discrete algebra, then the reduced part of the graded center of the derived category of $A$ is nontrivial if and only if $A$ has infinite global dimension. Moreover, we show that the nilpotent part of the graded center is controlled by the objects for which the Auslander--Reiten translation coincides with a power of the suspension functor.

Abstract:
In the paper is we generalize known descriptions of rings of semi-invariants for regular modules over Euclidean and canonical algebras to arbitrary concealed-canonical algebras.

Abstract:
We prove that if a quasi-tilted algebra is tame, then the associated moduli spaces are products of projective spaces. Together with an earlier result of Chindris this gives a geometric characterization of the tame quasi-tilted algebras.

Abstract:
We complete the derived equivalence classification of the gentle two-cycle algebras initiated in earlier papers by Avella-Alaminos and Bobinski-Malicki.

Abstract:
We classify canonical algebras such that for every dimension vector of a regular module the corresponding module variety is normal (respectively, a complete intersection). We also prove that for the dimension vectors of regular modules normality is equivalent to irreducibility.

Abstract:
The aim of the paper is to classify the indecomposable modules and describe the Auslander--Reiten sequences for admissible algebras with formal two-ray modules.