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<p>We study the curvature of a smooth algebraic surface <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44426_2025_1_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\subset \mathbb R^3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>⊂</mo> <msup> <mi mathvariant="double-struck">R</mi> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> of degree <i>d</i> 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 <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44426_2025_1_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(d^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>d</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>. For general quadrics, we fully characterize the number of real and complex umbilics and critical curvature points.</p>

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Critical curvature of algebraic surfaces in three-space

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月26日

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  • Paul Breiding,
  • Kristian Ranestad,
  • Madeleine Weinstein

摘要

We study the curvature of a smooth algebraic surface \(X\subset \mathbb R^3\) X ⊂ 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 \(d^3\) d 3 . For general quadrics, we fully characterize the number of real and complex umbilics and critical curvature points.