Are there any steps in finding the multiplicity of a point in a projective curve?
For example, I wish to find the singular points of the curve $\Gamma=V_{+}(F)$ on $P^2(\mathbb{C})$ where $F=X^2Y^2+Y^2Z^2+X^2Z^2$ and determine their multiplicities. I get that to find the singular points, I need to find the points in $P^2(\mathbb{C})$ such that the partial derivatives with respect to $x,y$ and $z$ are zero. However, I am stuck at finding their multiplicities.
Usually when I need to calculate the multiplicity of a point on a projective curve I do the following. Say $C$ is your projective curve.
1) Knowing what $x\in C$ is, $x$ is in a chart of $\mathbb{P}^2$ (Define $E_i = \{(x_0:x_1:x_2)\in \mathbb{P}^2|x_i\neq 0\}$, then $E_0, E_1, E_2$ is an atlas of $\mathbb{P}^2$ and each $E_i$ I call a chart). Multiplicity being a local quantity of the curve it only depends on a neighborhood of $x\in C$ in $\mathbb{P}^2$. Since $E_i$ are open, assuming $x\in E_0$ (or any other $E_i$) then you can study the multiplicity of $x$ in $E_0$ (the neighborhood containing it).
2) Dehomogenize the homogeneous polynomial with respect to this chart, meaning $f(x_1, x_2) = P(1, x_1, x_2)$ where $P$ is the homogeneous polynomial. Now multiplicity of $x=(1:x_1:x_2)$ on $C$ is the same as multiplicity of $(x_1, x_2)$ in affine curve $C'\subset \mathbb{A}^2$ where $C'$ is determined by $f$.
3) Shift $(x_1, x_2)$ to $(0,0)$ by a coordinate transformation. This doesn't change the multiplicity. Denote $\tilde{f}$ as your transformed polynomial.
4) The multiplicity of $(0,0)$ for a curve in $\mathbb{A}^2$ is simply the degree of the leading homogeneous polynomial (leading means least degree) in $\tilde{f}$; i.e. if $\tilde{f}=\tilde{f}_n + \cdots + \tilde{f}_r$ (decreasing order) where $\tilde{f}_d$ is homogeneous of degree $d$, then multiplicity is simply $r$.
As a side note one can prove that the multiplicity of $x\in C$ is independent of what chart you choose for doing this procedure.
