Suppose \(k\) is an algebraically closed field, \(Q\) is a finite quiver, \(I\) is an admissible ideal of the quiver algebra \(kQ\), and \(A=kQ/I\). For each dimension vector \(\mathbf{d}\) of \(A\), there is a representation variety \(\mathrm{rep}_A(\mathbf{d})\), defined as a product of matrix varieties on which a product of general linear groups \(\mathrm{GL}(\mathbf{d})\) acts. In this paper, the authors relate the representation varieties for algebras that are related by splitting nodes. The technique of node splitting was introduced by R. Martinez-Villa [Lect. Notes Math. 832, 396–431 (1980; Zbl 0443.16019)]. A node \(x\) of \(A\) can be split by a certain local operation around \(x\), resulting in a new algebra \(A^x\) with one fewer node. The authors relate the representation varieties of \(A\) and \(A^x\) by a homogeneous fibre bundle construction and collapsing maps, in the sense of G. R. Kempf [Invent. Math. 37, 229–239 (1976; Zbl 0338.14015)]. This allows them to relate their singularities and to relate defining equations of the prime ideals of equivariant closed subvarieties. By splitting one node at a time, the authors demonstrate that there are many nonhereditary algebras \(A\) such that the irreducible components of \(\mathrm{rep}_A(\mathbf{d})\) are all normal with rational singularities. Moreover, they obtain explicit generators of the prime defining ideals of these irreducible components. In particular, this class of algebras contains all those whose radical square is zero.