I'm following a course on algebraic geometry. We have seen different kinds of "multiplicity" and I don't understand the difference between them. My course is in dutch, so I will try to translate it as best as possible.
Firstly we have seen the multiplicity of a point $[(\alpha_{i1}, \alpha_{i2})]$ in the algebraic variety $V(f) \subset \mathbb{C}P^1$. Then we can decompose $f(x_0, x_1)$ as $$ f(x_0, x_1) = \prod_{i=1}^k (\alpha_{i1}x_0 + \alpha_{10}x_1)^{m_i} $$ where $m_i$ is called the multiplicity.
Now the second type of multiplicity is the intersection multiplicity. Suppose an algebraic hyperplane $V(f) \subset \mathbb{C}P^n$ where $f$ is a homogeneous polynomial of degree $d$ and suppose $l$ a line in $\mathbb{C}P^n$. Let $P = [p], Q=[q]$ be two points on $l$, then we can write $X=[x] \in l$ as $x=\lambda p + \mu q$. We then define $F(\lambda, \mu)=f(\lambda p + \mu q)$ as the intersection between $l$ and $V(f)$. Now let $X_i = [\lambda_i p + \mu_i q] \in l \cap V(f)$. Then $l$ intersects $V(f)$ with intersection multiplicity $m_i$ if the multiplicity of $[\lambda_i, \mu_i]$ in the algebraic variety $F(\lambda, \mu)=0$ is equal to $m_i$.
The third type of multiplicity is the multiplicity of a line itself. Suppose $C= V(f) \subset \mathbb{C}P^n$ where $f$ has degree $d$. Suppose $P$ a multiple point of $C$ (let's say a k-ple point). Then the tangent conic at $C$ in $P$ can be decomposed in a finite amount of lines. As an example they take $P=E_0$, then the tangent conic is the term in $V(f)$ corresponding to the highest degree term of $x_0$ (it has degree d-k). Let's call this coefficient $\alpha_k(x_1, x_2)$. Since $\alpha_k$ is a homogeneous polynomial in 2 variables we can decompose it in linear fators as $$ \alpha_k(x_1, x_2) = \prod_{i=1}^l (\alpha_{i1}x_1 + \alpha_{i2}x_2)^{m_i} $$ So we have $$ V(\alpha_k(x_1, x_2)) = \prod_{i=1}^l V(\alpha_{i1}x_1 + \alpha_{i2}x_2) $$ So it indeed decomposes in $l$ different lines through $P = E_0$. Now they call $m_i$ the multiplicity of the line $\alpha_{i1}x_1 + \alpha_{i2}x_2=0$.
These are the three types of multiplicity, now I don't understand the connection between these three and I don't understand how I can just calculate them. For example I have written in my notes that for $V(x_1^3-x_0x_2^2) \subset \mathbb{C}P^2$ that $E_2$ is a singular point with tangent $V(x_0)$. This I can calculate, however then it states that this line intersects the curve with multiplicity at least 2 in $E_2$ but the intersection multiplicity is 3. I don't see how I can calculate those multiplicities.
Could someone please explain the difference using my example and also explain how I can caluate those multiplicities. If you've read this far, thank you, and if you can help, please do!