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We complete a classification of the two-vertex geometrically irreducible algebras. We also classify the algebras in new classes of hom- and ext-irreducible algebras.

Assume that $K$ is an algebraically closed field and denote by $KG(R)$ the Krull-Gabriel dimension of $R$, where $R$ is a locally bounded $K$-category (or a bound quiver $K$-algebra). Assume that $C$ is a tilted $K$-algebra and $\widehat{C},\check{C},\widetilde{C}$ are the associated repetitive category, cluster repetitive category and cluster-tilted algebra, respectively. Our first result states that $KG(\widetilde{C})=KG(\check{C})\leq KG(\widehat{C})$. Since the Krull-Gabriel dimensions of tame locally support-finite repetitive categories are known, we further conclude that $KG(\widetilde{C})=KG(\check{C})=KG(\widehat{C})\in\{0,2,\infty\}$. Finally, in the Appendix Grzegorz Bobi\'nski presents a different way of determining the Krull-Gabriel dimension of the cluster-tilted algebras, by applying results of Geigle., Comment: Acknowledgements added

Let $\Bbbk$ be an algebraically closed field, $Q$ a finite quiver, and denote by $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}$ the affine $\Bbbk$-scheme of representations of $Q$ with a fixed dimension vector ${\mathbf{d}}$. Given a representation $M$ of $Q$ with dimension vector ${\mathbf{d}}$, the set ${\mathcal{O}}_M$ of points in $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}(\Bbbk)$ isomorphic as representations to $M$ is an orbit under an action on $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}(\Bbbk)$ of a product of general linear groups. The orbit ${\mathcal{O}}_M$ and its Zariski closure $\overline{{\mathcal{O}}}_M$, considered as reduced subschemes of $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}$, are contained in an affine scheme ${\mathcal{C}}_M$ defined by rank conditions on suitable matrices associated to $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}$. For all Dynkin and extended Dynkin quivers, the sets of points of $\overline{{\mathcal{O}}}_M$ and ${\mathcal{C}}_M$ coincide, or equivalently, $\overline{{\mathcal{O}}}_M$ is the reduced scheme associated to ${\mathcal{C}}_M$. Moreover, $\overline{{\mathcal{O}}}_M={\mathcal{C}}_M$ provided $Q$ is a Dynkin quiver of type ${\mathbb{A}}$, and this equality is a conjecture for the remaining Dynkin quivers (of type ${\mathbb{D}}$ and ${\mathbb{E}}$). Let $Q$ be a Dynkin quiver of type ${\mathbb{D}}$ and $M$ a finite dimensional representation of $Q$. We show that the equality $T_N\overline{{\mathcal{O}}}_M=T_N{\mathcal{C}}_M$ of Zariski tangent spaces holds for any closed point $N$ of $\overline{{\mathcal{O}}}_M$. As a consequence, we describe the tangent spaces to $\overline{{\mathcal{O}}}_M$ in representation theoretic terms.

We describe, in terms of generators and relations, the derived Hall algebras associated to the one-cycle gentle algebras of infinite global dimension.

We characterize pairs (Q,d) consisting of a quiver Q and a dimension vector d, such that over a given algebraically closed field k there are infinitely many representations of Q of dimension vector d. We also present an application of this result to the study of algebras with finitely many orbits with respect to the action of (the double product) of the group of units.

We study a family of affine varieties arising from a version of an old problem due to Birkhoff asking for the classification of embeddings of finite abelian p-groups. We show that all of these varieties are irreducible and have a dense orbit.

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 prove that a geometrically irreducible algebra with exactly two simple modules has to be of a very special form, which we describe. Based on this result we prove that every minimal geometrically irreducible algebra without shortcuts in its Gabriel quiver has at most two simple modules.

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., Comment: 13 pages

Let A be a concealed canonical algebra and d the dimension vector of an A-module which is periodic respect to the action of the Auslander-Reiten translation In the paper, we investigate the union of the closures of the orbits of the periodic A-modules of dimension vector d. We show that this set is closed and regular in codimension one.

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

We compute the Krull-Gabriel dimension of the category of perfect complexes for finite dimensional algebras which are derived discrete., Comment: 11 pages

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.

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.

Let Delta be an Euclidean quiver. We prove that the closures of the maximal orbits in the varieties of representations of Delta are normal and Cohen--Macaulay (even complete intersections). Moreover, we give a generalization of this result for the tame concealed-canonical algebras.

A cluster tilted algebra is known to be gentle if and only if it is cluster tilted of Dynkin type $\bbA$ or Euclidean type $\tilde{\bbA}$. We classify all finite dimensional algebras which are derived equivalent to gentle cluster tilted algebras.

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.

In the paper we describe the almost split sequences in the homotopy category of perfect complexes over a gentle algebra.

We show that for a class of modules over shod algebras, including the canonical tilting modules, the closures of the corresponding orbits in module varieties are regular in codimension one.

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.

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.

We classify, up to derived (equivalently, tilting-cotilting) equivalence all nondegenerate gentle two-cycle algebras. We also give a partial classification and formulate a conjecture in the degenerate case.

We investigate sets of the common zeros of non-constant semi-invariants for regular modules over canonical algebras. In particular, we show that if the considered algebra is tame then for big enough vectors these sets are complete intersections.

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., Comment: Significant changes to the previous version of the paper

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.

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.

In the paper we characterize, in terms of quivers and relations, the admissible algebras with formal two-ray modules introduced by G. Bobi\'nski and A. Skowro\'nski [Cent. Eur. J.Math.1 (2003), 457--476]., Comment: Mainly correcting typos. Also a new abstract and minor changes in the introduction and subsection 3.1

We show that the orbit closures for directing modules over tame algebras are normal and Cohen-Macaulay. The proof is based on deformations to normal toric varieties., Comment: 13 pages