In my lecture notes we have the following:
A point $P=\left [x, y, z\right ]$ of an algebraic curve $C_F=V(F)$ is called an inflection point of $C_F$ when
- $P$ is not a singular point of $C_F$.
- The order of the tangent of $C_F$ at the point $P$ is $m_P(F)\geq 3$.
Remark: We consider the curves that don't have lines as components.
My question is : What does "curves that don't have lines as components" mean ???
Edit:
An other remark: So that the origin $P=[0, 0, 1]$ is an inflection point with the tangent at $P$ the line $$bx-ay$$
$F(x, y, z)$ will have the following form:
$$F(x, y, 1)=(bx-ay)+g_2(x, y)+g_3(x, y)+ \dots +g_d(x, y)$$ where $g_i(x, y)$ are homogeneous polynomials of degree $i$.
It should also stand that $$g_2(a \lambda , b \lambda)=\lambda^2 g_2(a, b)=0$$
Can you explain to me why it should stand that $g_2(a \lambda , b \lambda)=\lambda^2 g_2(a, b)=0$ ?
Remember inflexion from Calculus. Inflexion occurs when the second derivative vanishes.
$Y=f(X)=c_1X+c_2X^2+...$ has an inflexion point at $X=0$ (the origin) if $f''(0)=0$, i.e. if $c_2=0$.
See how $c_1X,c_2X^2,...$ are homogeneous and vanish at $X=0$.
A linear change of coordinates $g$ doesn't change these conditions since $(F\circ g)''=F''\circ g\cdot (g'(z))^2$ and $g'(z)\neq0$ and homogeneous composed with linear is still homogeneous.
Now put (if $a\neq0$) $$\begin{align}X&=by-ax\\Y&=y\end{align}$$ if $a=0, b\neq0$ put $$\begin{align}X&=by\\Y&=x\end{align}$$
We never have $a=b=0$ because they are assuming the curve is not singular at the origin.