Enumerative Geometry of Curvature of Algebraic Hypersurfaces

Abstract

We study the curvature of a smooth algebraic hypersurface X ⊂ 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 threespace.

Intersection Bodies of Polytopes: Translations and Convexity

“…We note that proving non-convexity is a much easier task then proving convexity, as the first can be achieved by showing the non-convexity of a small curve on the boundary, while convexity is a global condition. A possible approach to tackle this problem in the case of polytopes might be studying the curvature of the algebraic hypersurfaces defining the boundary of the intersection body, as in [BRW22].…”