Suppose that $C \in \mathbb P_k^2$ is a projective curve defined by a polynomial $F \in k[X,Y,Z]$. Let $P$ be a point on $C$, and $L$ a tangent line of $C$ at $P$. Suppose furthermore that $T : \mathbb P_k^2 \to \mathbb P_k^2$ is a projective change of coordinates. Is there a constructive proof that $T(L)$ is a tangent line of $T(C)$ at $T(P)$?
I think I've proven this using intersection, however, I don't like this proof as I feel like I don't understand the underlying machinery. The proof goes as follows:
The line $L$ is tangent to $C$ at $P$ if and only if $I(P, F \cap L) > m_P(F)$ (where $I(P, F \cap L)$ of $F$ and $L$ at $P$, and $m_P(F)$ denotes the multiplicity of $F$ at $P$). From the properties of the intersection number, it follows that $I(T(P), T(F) \cap T(L)) = I(P, F \cap L)$. This combined with the fact that $m_{T(P)}(T(F)) = m_P(F)$ shows that $I(T(P), T(F) \cap T(L)) > m_{T(P)}(T(F))$, which concludes the proof.
Can anyone provide a more explicit proof of the statement?
Thanks in advance!
The tangent line at a regular point $(x_0,y_0,z_0)$ to a curve given by $F(x,y,z)=0$ is exactly given by $x\cdot\frac{\partial F}{\partial y}(x_0,y_0,z_0)+x\cdot\frac{\partial F}{\partial y}(x_0,y_0,z_0)+z\cdot\frac{\partial F}{\partial z}(x_0,y_0,z_0)=0$. All you need to do is verify that taking a linear change of coordinates commutes with forming this equation, which you can see by a judicious application of calculus.