Geometry of decomposable directing modules over tame algebras. (Q1396376)

Denote by \(\text{mod}_A(d)\) the affine variety of \(A\)-modules with dimension vector \(d\), where \(A\) is a finite dimensional basic algebra over an algebraically closed field. Assume that \(A\) is of tame representation type, and that \(d\) is the dimension vector of a directing module \(M\). Several geometric properties of \(\text{mod}_A(d)\) are proved in the paper, generalizing results of \textit{G. Bobiński} and \textit{A. Skowroński} [J. Algebra 215, No. 2, 603-643 (1999; Zbl 0965.16009)], who dealt with the special case when \(M\) is indecomposable. In particular, it is shown that \(\text{mod}_A(d)\) is a complete intersection, and its dimension is expressed in terms of the Tits form. The closure of the orbit \(O(M)\) of \(M\) is an irreducible component, and \(O(M)\) is the unique orbit of maximal dimension in \(\text{mod}_A(d)\). The maximal orbits in \(\text{mod}_A(d)\) consist of non-singular points. There are only a finite number of orbits of codimension one, they are all contained in the closure of \(O(M)\). A module \(N\) corresponds to a non-singular point in \(\text{mod}_A(d)\) if and only if \(\text{Ext}^2_A(N,N)=0\). Moreover, if \(M\) is a tilting module, then \(\text{mod}_A(d)\) is irreducible and normal.