I have a question about Example II.7.8.6 in Hartshorne's Algebraic Geometry.
Let $k$ algebraically closed field,
$V=(t^4,t^3u,tu^3,u^4)$ and $V'=(t^4,t^3u+at^2u^2,tu^3,u^4)(a\in k^*)$ are two quartic curves in $\mathbb{P}^3_k$.
How can I show these are not equivalent by an automorphism of $\mathbb{P}^3_k$.
In Joe Harris's Algebraic Geometry, he defines ration quartic curves as curves defined by the parameterization
$[X_0^4-\beta X_0^3X_1,X_0^3X_1-\beta X_0^2X_2^2,\alpha X_0^2X_1^2-X_0X_1^3,\alpha X_0X_1^3-X_1^4]$.
So it is slightly different from one in Hartshorne's book and it doesn't help.
Also Maccaulay2 showed the homogeneous ideal of $V'$ is
$(z^4-xw^3,4y^3z^2-y^4w+4xyz^2w-6xy^2w^2-x^2w^3+x^2s^2,y^5+20xy^2z^2-10xy^3w+4x^2z^2w-15x^2yw^2-4x^2z^2-x^2yw)$
It is so complicated so that it doesn't help.
The homogeneous ideal of $V$ contains a quadratic polynomium whereas the homogeneous ideal of $V'$ does not. This property whould be preserved by an automorphism of $\Bbb{P}^3$ since an automorphism is given by linear forms.
Edit: This argument is flawed. As KReiser pointed out in the comments, the homogeneous ideal of any quartic rational curve in $\Bbb{P}^3$ will always contain a quadratic polynomial.