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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 proof we use knowledge of the representation theory of the tame quasi-tilted algebras and a construction of semi-invariants as determinants.

We show that the types of singularities of Schubert varieties in the flag varieties Flag n , 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 the orbit closures in representation varieties of Dynkin quivers of type D are normal and Cohen-Macaulay varieties.be a parabolic subgroup of Gl n . We may assume that B n ⊆ P . Using the canonical fibre bundle Gl n /B n → Gl n /P with smooth fibre P/B n , one can show that Sing B n (Gl n /P ) ⊆ Sing B n (Flag n ).Let Grass(n, a) denote the Grassmannian variety of the a-dimensional subspaces of k n . The product Grass(n, a) × Grass(n, b) equipped with the diagonal action of Gl n is also a spherical variety, for any nonnegative integers a and b with a, b ≤ n. If a + b ≤ n we will denote by O(n, a, b) the maximal Gl n -orbit of Grass(n, a) × Grass(n, b) consisting of the pairs (U, V ) of subspaces of k n such that U ∩ V = {0}. Obviously, O(n, a, b) is also a spherical variety. We put Sing(Grass 2 ) = {Sing B n (Grass(n, a) × Grass(n, b)); a, b, n ∈ N, a, b ≤ n},Let Q = (Q 0 , Q 1 , s, e) be a finite quiver. Here Q 0 is the set of vertices, Q 1 is the set of arrows and s, e : Q 1 → Q 0 are functions such that any arrow α ∈ Q 1 has the starting vertex s(α) and the ending vertex e(α). We denote by rep(Q) the category of representations of the quiver Q. The objects of rep(Q) are the tuples V = (V i , f α ) i∈Q 0 ,α∈Q 1 , where V i are finitedimensional vector spaces over k and f α : V s(α) → V e(α) are k-linear maps. A homomorphism between two representations V = (V i , f α ) i∈Q 0 ,α∈Q 1 and W = (W i , g α ) i∈Q 0 ,α∈Q 1 is a collection (h i ) i∈Q 0 of linear maps h i : V i → W i satisfying h e(α) f α = g α h s(α) for any arrow α ∈ Q 1 . Furthermore, the sequence dim V = (dim k V i ) i∈Q 0 is called a dimension vector of V .Let d = (d i ) i∈Q 0 ∈ N Q 0 be a dimension vector. We define the affine space rep Q (d) as the set of the tuples V = (f α ) α∈Q 1 , where f α is a d e(α) × d s(α)matrix with coefficients in k for any α ∈ Q 1 . The product Gl(d) = i∈Q 0 Gl d i of general linear groups acts on rep Q (d) by conjugations g V = (g e(α) f α g −1 s(α) ) α∈Q 1 for any g = (g i ) i∈Q 0 ∈ Gl(d) and V = (f α ) α∈Q 1 ∈ rep Q (d). The orbits of this action correspond to the isomorphism classes of the representations of Q with dimension vector d. We denote by Sing(A) and Sing(D) the set of types of singularities of Gl(d)-orbit closures in rep Q (d) for all dimension vectors d ∈ N Q 0 and Dynkin quivers Q of type A n , n ≥ 1, and D n , n ≥ 4, respectively. If Q is a Dynkin quiver of type A n and d ∈ N Q 0 , then the Gl(d)-orbit closures in rep Q (d) are varieties with rational singularities (see [9] and [3]). The idea of the proof is t...

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a b s t r a c tThroughout the paper k denotes a fixed field. All vector spaces and linear maps are k-vector spaces and k-linear maps, respectively. By Z, N, and N + , we denote the sets of integers, nonnegative integers, and positive integers, respectively. For i, j ∈ Z, [i, j] IntroductionGiven an abelian category A one defines its bounded derived category D b (A) [24] (of bounded complexes of objects of A) having a structure of a triangulated category, which is an important homological invariant of A. In particular, given a finite dimensional algebra A one may study the bounded derived category D b (mod A) of the category mod A of finite dimensional A-modules, which one shortly calls the derived category of A and denotes D b (A). Since the observation of Happel [15] (generalized by Cline et al. [11]), which states that derived category is invariant under tilting process, an importance of derived categories in the representation theory of finite dimensional algebras became clear. This observation was supported by results connecting derived categories of finite dimensional algebras with derived categories of coherent sheaves over projective schemes [6,12]. Since that time a lot of results concerning derived categories of finite dimensional algebras were obtained (see for example [1,8,9,13,20]). In particular, Rickard [21] developed the Morita theory for derived categories of finite dimensional algebras. One of the consequences is that the derived categories of two finite dimensional algebras are equivalent as triangulated categories if and only if the subcategories of perfect complexes are equivalent as triangulated categories. Recall that if A is a finite dimensional algebra, then the subcategory of D b (A) formed by perfect complexes can be identified with the bounded homotopy category K b (proj A) of (bounded complexes of) projective A-modules.A class of finite dimensional algebras whose derived categories attract a lot of interest is the class of gentle algebras introduced by Assem and Skowroński [4]. An important feature of this class of algebras is that it is closed under derived equivalence, i.e. if A is a gentle algebra and D b (A) is equivalent as a triangulated category to D b (B) for a finite dimensional algebra B, then B is also gentle [23]. Next, this class of algebras appears naturally in many classification problems. Namely, the tree gentle algebras are precisely the piecewise hereditary algebras of type A [2] (i.e. the algebras derived equivalent to hereditary algebras of type A). Further, if A is a derived discrete algebra, then either A is piecewise hereditary of Dynkin type or A is a one-cycle gentle algebra which does not satisfy the clock condition [25]. Moreover, the one-cycle gentle algebras coincide with the piecewise hereditary algebras of typeà [4].If A is a gentle algebra, then it is possible to investigate D b (A) by means of the stable category mod of the module category mod over the repetitive algebra [22] (which is no longer finite dimensional) and the Happel functor [16] D b (...