In my lecture notes, after the Bezout theorem there is the following collary:
If the plane projective curves, $x=V(F), y=V(G)$, intersect at exactly $m \cdot n$ discrete points, then the intersection multiplicity is at each point one. (the points are simple)
The proof is the following: $I(P, x \cap y)=1, \forall P \in x \cap y$, so $m_P(x) \cdot m_P(y) \leq I(P, x \cap y)=1$
$\Rightarrow m_P(x)=1, m_P(y)=1, \forall P \in x \cap y$ .
Can you explain to me this collary??
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Edit:
An other collary is the following:
Let $C_F=V(F)$ and $C_G=V(G)$ two algebraic curves of degree $n$ and that the two curves intersect in $n^2$ points. We suppose that exactly $m \cdot n$ from these belong to an irreducible curve of degree $m<n$. Then the remaining $n(n-m)$ belong to a curve of degree $n-m$.
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What does it mean that "$m \cdot n$ from these belong to an irreducible " ???
The proof in your notes looks a little muddled. I'm assuming, although it's not explicitly stated in the question, that $m$ is the degree of $F$ and $n$ is the degree of $G$. The point is that you know that the sum of all the intersection multiplicities is $mn$, and that each intersection point contributes intersection multiplicity $\geq 1$. If you have $mn$ integers each $\geq 1$, and they add up to $mn$, then clearly each integer was in fact exactly $1$.