From the introduction: Let \(\Delta\) be a 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.
Let \(A\) be a finite-dimensional \(k\)-algebra. Given a non-negative integer \(d\) one defines \(\text{mod}_A(d)\) as the set of all \(k\)-algebra homomorphisms from \(A\) to the algebra \(\mathbb M_{d\times d}(k)\) of \(d\times d\)-matrices. This set has a structure of an affine variety and its points represent \(d\)-dimensional \(A\)-modules. Consequently, we call \(\text{mod}_A(d)\) the variety of \(A\)-modules of dimension \(d\). The general linear group \(\text{GL}(d)\) acts on \(\text{mod}_A(d)\) by conjugation: \((g\cdot m)(a):=gm(a)g^{-1}\) for \(g\in\text{GL}(d)\), \(m\in\text{mod}_A(d)\) and \(a\in A\). The orbits with respect to this action are in one-to-one correspondence with the isomorphism classes of the \(d\)-dimensional \(A\)-modules. Given a \(d\)-dimensional \(A\)-module \(M\) we denote the orbit in \(\text{mod}_A(d)\) corresponding to the isomorphism class of \(M\) by \(\mathcal O(M)\) and its Zariski-closure by \(\overline{\mathcal O(M)}\).
The following theorem is the main result of the paper.
Theorem 1. Let \(M\) be a module over a tame hereditary algebra. If \(\mathcal O(M)\) is maximal, then \(\overline{\mathcal O(M)}\) is a normal complete intersection (in particular, Cohen-Macaulay).
Corollary 2. If \(M\) is an indecomposable module over a tame hereditary algebra, then \(\overline{\mathcal O(M)}\) is a normal complete intersection (in particular, Cohen-Macaulay).
Theorem 3. Let \(M\) be a \(\tau\)-periodic module over a tame hereditary algebra. If \(\mathcal O(M)\) is maximal, then \(\overline{\mathcal O(M)}\) is a complete intersection (in particular, Cohen-Macaulay).
We have the following generalization of Theorem 3.
Theorem 4. Let \(M\) be a \(\tau\)-periodic module over a tame concealed-canonical algebra such that \(\mathcal O(M)\) is maximal. Then \(\overline{\mathcal O(M)}\) is a complete intersection (in particular, Cohen-Macaulay). Moreover, \(\overline{\mathcal O(M)}\) is not normal if and only if \(\dim M\) is singular and \(\tau M\simeq M\).
The paper is organized as follows. In Section 1 we recall basic information about quivers and their representations. Next, in Section 2 we gather facts about the categories of modules over the tame concealed-canonical algebras. In Section 3 we introduce varieties of representations of quivers, while in Section 4 we review facts on semi-invariants with particular emphasis on the case of tame concealed-canonical algebras. Next, in Section 5 we present a series of facts, which we later use in Sections 6 and 7 to study orbit closures for the non-singular and singular dimension vectors, respectively. Moreover, in Section 7 we make a remark about relationship between the degenerations and the hom-order for the tame concealed-canonical algebras. Finally, in Section 8 we give the proof of Theorem 4.