Enriched finite element methods have gained traction in recent years for modeling problems with material interfaces and cracks. By means of enrichment functions that incorporate a priori behavior about the solution, these methods decouple the finite element (FE) discretization from the geometric configuration of such discontinuities. Taking advantage of this greater flexibility, recent studies have proposed the adoption of Non-Uniform Rational B-Splines (NURBS) to preserve the interfaces' exact geometries throughout the analysis. In this article, we investigate NURBS-based geometries in the context of the Discontinuity-Enriched Finite Element Method (DE-FEM) based on linear field approximations. While optimal convergence is retained for problems with weak discontinuities without singularities, representing exact geometry via NURBS does not yield noticeable improvements when extracting stress intensity factors of cracked specimens. For low-order elements, we conclude that the benefits of exact geometry representation do not outweigh the increased complexity in formulation and implementation. The choice of linear FEs hinders the accuracy of the proposed formulation, suggesting that its full potential may only be unleashed by increasing the field representation order.