Let $F$ be a projective plane curve on $P_2(k)$ where $k$ is a field. Then, we introduce a system of local co-ordinates to convert this projective curve into an affine curve i.e. we take a point $(x_0,y_0,z_0)$ in $P_2(k)$ and take a projective transformation that takes this point to the point $(0,0,1)$ and use this transformation to convert $F$ to an affine curve. Does the singularity of $F$ depend upon $(x_0,y_0,z_0)$?
Does the singularity of a projective plane curve depend upon the system of local co-ordinates?
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algebraic-curves