# Finding singular points of a curve in affine space

• Mar 28th 2010, 11:44 PM
Siknature
Finding singular points of a curve in affine space
How do i find the singular points of the following curve in A^2(C) OR prove that there are no singular points? (where A is affine space and C is complex numbers)

The curve is

(x^2)(y^2)-6(x^2)+2xy-6(y^2)+25=0

Taking partial derivatives i get the jacobian matrix

J = (2x(y^2)-12x+2y 2(x^2)y+2x-12y)

I do not know how to proceed from here.

Any help is appreciated.

Thanks
• Mar 29th 2010, 03:35 AM
Opalg
Quote:

Originally Posted by Siknature
How do i find the singular points of the following curve in A^2(C) OR prove that there are no singular points? (where A is affine space and C is complex numbers)

The curve is

$x^2y^2-6x^2+2xy-6y^2+25=0$

Taking partial derivatives i get the jacobian matrix

$J = \begin{pmatrix}2xy^2-12x+2y & 2x^2y+2x-12y\end{pmatrix}
$

I do not know how to proceed from here.

The condition for a singular point is that the Jacobian should have less than full rank. The full rank of a 1×2 matrix is 1. So at a singular point, the matrix must have rank 0, which is the same as saying that both entries should be 0.

Therefore you need to solve the equations
\left.\begin{aligned}2xy^2-12x+2y &= 0, \\ 2x^2y+2x-12y &= 0.\end{aligned}\right\}
An efficient way to do that is to add the equations, getting $(xy-5)(x+y)=0$. So either $xy=5$, leading to the solutions $x=y=\pm\sqrt5$; or $x+y=0$, leading to the solutions $x=-y=\pm\sqrt7$ and also the solution $x=y=0$.

(You then have to check whether any of those points actually lie on the curve. If they don't, they obviously don't yield singular points.)