On open orbits and their complements
Let k be an algebraically closed field, Q be a quiver without oriented cycles, say with vertex set \(\{\) 1,...,n\(\}\), and kQ be the path algebra of Q. For a vector ḏ\(=(d_ 1,...,d_ n)\in N^ n\), denote by R(Q,ḏ) the affine variety of k-representations X of Q with \(X(i)=k^{d_ i}\). Then the group GL(ḏ)\(=\prod^{n}_{i=1}GL(d_ i)\) acts linearly on R(Q,ḏ) such that the GL(ḏ)-orbits in R(Q,ḏ) are just isomorphism classes. Let \(T_ 1,...,T_ r\) be pairwise non-isomorphic representations of Q, \(T=T_ 1^{\lambda_ 1}\oplus...\oplus T_ r^{\lambda_ r}\) with \(\lambda_ i>0\) big enough and such that the support of T is all of Q. Moreover, assume that \(Ext^ 1(T,T)=0\). This implies that T has an open GL(\underbar{dim} T)- orbit in R(Q,\underbar{dim} T). The main result of the paper is the following description of the complement of this orbit. From a construction of \textit{K. Bongartz} [Lect. Notes Math. 903, 26-38 (1981; Zbl 0478.16025)] T admits a canonical completion \(\tilde T\) to a tilting module \(T\oplus \tilde T\), and let \(T_{r+1},...,T_ n\) be pairwise non-isomorphic direct summands of \(\tilde T\) which are not isomorphic to \(T_ 1,...,T_ r\). For \(j=r+1,...,n\), denote by \({\mathcal D}_ j\) the closure of the set of representations in R(Q,\underbar{dim} T) that contain \(T_ j\) as a direct summand. For \(i=1,...,r\), call \(T_ i\) essential if it is either a submodule or a quotient of some representation in the full subcategory of mod kQ whose objects are direct sums of \(T_ k^{\nu_ k}\), \(\nu_ k\geq 0\), \(k\neq i\). If \(T_ i\) is essential, let \({\mathcal C}_ i\) be the closure of the set of representations in R(Q,\underbar{dim} T) which contains \(T_ i^{\lambda_ i+1}\) as a direct summand. The authors prove that the irreducible components of the complement of the GL(\underbar{dim} T)- orbit of T in R(Q,\underbar{dim} T) are the \({\mathcal C}_ i's\) for \(T_ i\) essential and \({\mathcal D}_{r+1},...,{\mathcal D}_ n\). The codimension of \({\mathcal D}_{r+1},...,{\mathcal D}_ n\) in R(Q,\underbar{dim} T) is one, and the codimension of \({\mathcal C}_ i\) is \(\lambda_ i+1>1\). Moreover, every component is the closure of an orbit.
scientific article; zbMATH DE number 4139978
scientific article