Abstract: Quantum discord is arguably a more sensitive measure of quantum correlations than quantum entanglement, and may be able to serve as a resource for quantum computation. All quantum correlations are subject to destruction by external noise. The route by which this destruction takes place depends on the shape of the hypersurface of zero discord in the space of generalized Bloch vectors. In the case of 2 qubits, we show that, except at the origin, this hypersurface is a 9-dimensional manifold with boundary embedded in a 15-dimensional background space. This is done by computing the tangent vectors explicitly and verifying that there are no self-intersections. We discuss the implications for the time evolution of discord in physical models, which contrasts sharply with the evolution of entanglement.