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Some classical formulæ for curves and surfaces

Thomas Dedieu

TL;DR

This work revisits Salmon’s nineteenth-century program of counting hyperplanes with prescribed tangency patterns to a general smooth surface in projective 3-space, focusing on tritangent planes and tacnodal hyperplane sections via projective duality and polar calculus. It develops a comprehensive framework combining dual varieties, polars, Gauss maps, and Plücker formulas to compute degrees and characters of the dual surface S^\vee, its double curves, flecnodal and parabolic loci, and associated developable surfaces. The text also extends to formulae for surfaces projected in $\mathbf{P}^3$, integrating Enriques’ postulation theory and Noether’s formula to relate arithmetic and geometric genera, with extensive appendices on polarity, Hessians, and developables. Together, these tools yield explicit enumerative invariants such as the numbers of tritangent planes $\check{t}$ and flecnodal-related counts, framed within a robust modern exposition and connections to Piene’s duality and generalized Plücker theory.

Abstract

The goal of this text is to present the computation by Salmon, in the second half of the XIXth century, of various numbers enumerating planes with a prescribed tangency pattern with a sufficiently general surface $S$ in $\mathbf{P}^3$ (or, equivalently, of hyperplane sections of $S$ with prescribed singularities). Emblematic among these are the number of tritangent planes, and the number of planes cutting out a curve with a tacnode.

Some classical formulæ for curves and surfaces

TL;DR

This work revisits Salmon’s nineteenth-century program of counting hyperplanes with prescribed tangency patterns to a general smooth surface in projective 3-space, focusing on tritangent planes and tacnodal hyperplane sections via projective duality and polar calculus. It develops a comprehensive framework combining dual varieties, polars, Gauss maps, and Plücker formulas to compute degrees and characters of the dual surface S^\vee, its double curves, flecnodal and parabolic loci, and associated developable surfaces. The text also extends to formulae for surfaces projected in , integrating Enriques’ postulation theory and Noether’s formula to relate arithmetic and geometric genera, with extensive appendices on polarity, Hessians, and developables. Together, these tools yield explicit enumerative invariants such as the numbers of tritangent planes and flecnodal-related counts, framed within a robust modern exposition and connections to Piene’s duality and generalized Plücker theory.

Abstract

The goal of this text is to present the computation by Salmon, in the second half of the XIXth century, of various numbers enumerating planes with a prescribed tangency pattern with a sufficiently general surface in (or, equivalently, of hyperplane sections of with prescribed singularities). Emblematic among these are the number of tritangent planes, and the number of planes cutting out a curve with a tacnode.
Paper Structure (119 sections, 37 theorems, 167 equations, 6 figures)

This paper contains 119 sections, 37 theorems, 167 equations, 6 figures.

Table of Contents

  1. Some classical formulæ for curves and surfaces
  2. Background material
  3. Projective duality
  4. Projective duality for hypersurfaces and polarity
  5. Gauss map and degree of the dual
  6. Plücker formulae for plane curves
  7. Plückerian characters of a plane curve
  8. Plücker formulae for plane curves
  9. Generators for the Plücker relations
  10. Smooth plane curves
  11. The genus formula
  12. Double curves of the dual of a surface in $\mathbf{P}^3$
  13. Local geometry of a surface in $\mathbf{P}^3$ and its dual
  14. Parabolic points
  15. Asymptotic tangent lines
  16. ...and 104 more sections

Key Result

Theorem (1.2)

Let $X$ be a variety in $\mathbf{P}^N$. One has $(X^\vee)^\vee =X$. More precisely, let $p$ and $\varpi$ be smooth points of $X$ and $X^\vee$ respectively: The hyperplane $\varpi^\perp \subseteq \mathbf{P}^N$ is tangent to $X$ at $p$ if and only if the hyperplane $p^\perp \subseteq \check \mathbf{P}

Figures (6)

  • Figure 1: The Gauss map around the parabolic curve
  • Figure 2: Normalization of $S^\flat$ at a triple point
  • Figure 3: Pencil with base locus an ordinary tangent line $\Lambda$
  • Figure 4: Pencil with base locus an asymptotic tangent line $\Lambda$
  • Figure 5: Pencil with base locus the double asymptotic tangent $\Lambda$ at a parabolic point
  • ...and 1 more figures

Theorems & Definitions (72)

  • Definition (1.1)
  • Theorem (1.2)
  • Theorem (1.3)
  • Theorem (1.4)
  • Proposition (1.5)
  • Corollary (2.22)
  • Lemma (2.24)
  • proof : Proof of \ref{['l:proj-flex-tg']} by polarity
  • proof : Proof of \ref{['l:proj-flex-tg']} by duality
  • Lemma (2.25)
  • ...and 62 more