Enumerative Geometry of Curvature of Algebraic Hypersurfaces

Paul Breiding, Kristian Ranestad, Madeleine Weinstein

Published 2022-06-18Version 1

We study the curvature of a smooth algebraic hypersurface $X\subset \mathbb R^n$ from the point of view of algebraic geometry. We introduce an algebraic variety, the curvature variety, that encodes the second fundamental form of $X$. We apply this framework to study umbilical points and points of critical curvature. We fully characterize the number of real and complex umbilics and critical curvature points for general quadrics in three-space.