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Difference between revisions of "Hessian (algebraic curve)"

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''of an [[Algebraic curve|algebraic curve]] of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471501.png" />''
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''of an [[Algebraic curve|algebraic curve]] of degree $n$''
  
The set of points whose conic polars can be split into two straight lines, as well as the set of double points of the first polars. The Hessian of a non-singular curve of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471502.png" /> is a curve of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471503.png" /> and class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471504.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471505.png" /> is the equation of a curve of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471506.png" /> in homogeneous coordinates and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471507.png" />, then
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The set of points whose conic polars can be split into two straight lines, as well as the set of double points of the first polars. The Hessian of a non-singular curve of degree $n$ is a curve of degree $3(n-2)$ and class $3(n-2)(3n-7)$. If $f=0$ is the equation of a curve of degree $n$ in homogeneous coordinates and if $f_{ik}=\partial^2f/\partial x_i\partial x_k$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471508.png" /></td> </tr></table>
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$$\begin{vmatrix}f_{11}&f_{12}&f_{13}\\f_{21}&f_{22}&f_{23}\\f_{31}&f_{32}&f_{33}\end{vmatrix}=0$$
  
 
is the equation of the Hessian. The Hessian of a non-singular curve of degree 3 in characteristic not equal to three intersects the curve at nine ordinary points of inflection. Named after O. Hesse (1844).
 
is the equation of the Hessian. The Hessian of a non-singular curve of degree 3 in characteristic not equal to three intersects the curve at nine ordinary points of inflection. Named after O. Hesse (1844).

Revision as of 12:03, 29 June 2014

of an algebraic curve of degree $n$

The set of points whose conic polars can be split into two straight lines, as well as the set of double points of the first polars. The Hessian of a non-singular curve of degree $n$ is a curve of degree $3(n-2)$ and class $3(n-2)(3n-7)$. If $f=0$ is the equation of a curve of degree $n$ in homogeneous coordinates and if $f_{ik}=\partial^2f/\partial x_i\partial x_k$, then

$$\begin{vmatrix}f_{11}&f_{12}&f_{13}\\f_{21}&f_{22}&f_{23}\\f_{31}&f_{32}&f_{33}\end{vmatrix}=0$$

is the equation of the Hessian. The Hessian of a non-singular curve of degree 3 in characteristic not equal to three intersects the curve at nine ordinary points of inflection. Named after O. Hesse (1844).


Comments

References

[a1] J.L. Coolidge, "A treatise on algebraic plane curves" , Dover, reprint (1959)
How to Cite This Entry:
Hessian (algebraic curve). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hessian_(algebraic_curve)&oldid=18385
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article