Critical Curvature of Algebraic Surfaces in Three-Space

2022/06/18 by Paul Breiding, Kristian Ranestad, Breiding, Paul +3 · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2206.09130

openalex publication_date 2022/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the curvature of a smooth algebraic surface X⊂ \mathbb R³ 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 d³. For general quadrics, we fully characterize the number of real and complex umbilics and critical curvature points.

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