### Degenerations for modules over representation-finite selfinjective algebras

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We show that a homomorphism of algebras is a categorical epimorphism if and only if all induced morphisms of the associated module varieties are immersions. This enables us to classify all minimal singularities in the subvarieties of modules from homogeneous standard tubes.

We give an example of a representation of the Kronecker quiver for which the closure of the corresponding orbit contains a singularity smoothly equivalent to the isolated singularity of two planes crossing at a point. Therefore this orbit closure is neither Cohen-Macaulay nor unibranch.

We show that the types of singularities of Schubert varieties in the flag varieties Flagₙ, n ∈ ℕ, are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type 𝔸. Similarly, we prove that the types of singularities of Schubert varieties in products of Grassmannians Grass(n,a) × Grass(n,b), a, b, n ∈ ℕ, a, b ≤ n, are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type 𝔻. We also show that...

Let A be a finitely generated associative algebra over an algebraically closed field. We characterize the finite-dimensional A-modules whose orbit closures are local hypersurfaces. The result is reduced to an analogous characterization for orbit closures of quiver representations obtained in Section 3.

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