Critical curvature of algebraic surfaces in three-space

Abstract We study the curvature of a smooth algebraic surface $$X\subset \mathbb R^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> 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 $$d^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>d</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> . For general quadrics, we fully characterize the number of real and complex umbilics and critical curvature points.