It is a long-standing question to resolve the singularities of algebraic varieties by Nash-blowups. This is a very natural process associating to smooth points their tangent space and to then take the Zariski-closure of the associated graph. This works for curves and, when combined with normalization, also for surfaces. It was a big surprise when Castillo, Duarte, Leyton-Alvaro and Liendo constructed in 2024 a four- (and then three-) dimensional toric counter-example to resolution. We will briefly describe their construction and then relate it to the concept of algebraic curvatures of varieties. These are invariants of varieties at smooth points transferring and generalizing the differential-geometric notion of curvature to the algebraic geometry setting.