Critical Curvature of Algebraic Surfaces in Three-Space

We study the curvature of a smooth algebraic surface $X\subset {\mathbb{R}}^3$ of degree $d$ from the point of view of algebraic geometry. More precisely, we consider umbilical points and points of critical curvature. We prove that the number of complex critical curvature points is of order $d^3$. For general quadrics, we fully characterize the number of real and complex umbilics and critical curvature points.