Hypersurface Singularities

Hypersurface singularities are points where a mathematical shape defined by a single equation, called a hypersurface, behaves irregularly or "breaks down." These points are where the usual properties of smoothness or well-behaved geometry fail, such as having sharp edges, cusps, or intersections. They are important in fields like algebraic geometry and topology because they reveal deep structural characteristics of the shape. Studying these singularities helps mathematicians understand the shape's underlying complexity and how it can be smoothed or classified.