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For my thesis, I'm working on the intersection of projective plane curves over $\mathbb{C}$. We define the intersection number of projective plane curves (see for example Gibson - Elementary Geometry of Plane Curves and Kirwan - Complex Algebraic Curves) by using the resultant:

Let $C$ and $D$ be projective plane curves without common components, defined by polynomials $F$ and $G$ in $\mathbb{C}[X,Y,Z]$ respectively. Using a projective transformation, choose projective coordinates so that

a). $[1:0:0] \notin C\cup D$,

b). $[1:0:0]$ does not lie on any line containing two distinct points of $C\cap D$, and

c). $[1:0:0]$ does not lie on the tangent line to $C$ and $D$ at any point of $C \cap D$.

then the intersection number $I(C,D,P)$ of $C$ and $D$ at $P=[a:b:c] \in C \cap D$ is defined as the largest integer $k$ such that $(bZ-cY)^k$ divides the resultant $Res_X(F,G) \in \mathbb{C}[Y,Z]$.

(The above is from Kirwan's book).

My problem has been with Proposition 3.22 and Remark 3.23 from Kirwan's book. Kirwan proves in Prop 3.22 that if $C$ and $D$ are projective plane curves and $P\in\mathbb{P}^2$, then $I(C,D,P)=1$ if and only if $P$ is a nonsingular point on both $C$ and $D$ and the tangent lines to $C$ and $D$ are distinct. Her proof I can follow; it involves the properties a), b) and c) above and a projective transformation such that $P=[0:0:1]$. However, in Remark 3.23, he states that the proof can 'be extended' to show that $I(C,D,P) \geq m_P(C)m_P(D)$ where $m_P(C)$ and $m_P(D)$ are the multiplicities of $P$ on $C$ and $D$ respectively, and that equality holds if and only if $C$ and $D$ have no common tangent lines at $P$.

I don't understand how to extend his proof; it uses the implicit function theorem for complex polynomials and the fact that one of the first derivatives of $F$ is nonzero for $P$, but the latter is not true if $P$ has $m_P(C) > 1$. In that case all the first derivatives at $P$ are zero, so the implicit function theorem also cannot be used.

Gibson's book (Elementary Geometry of Plane Curves) has the same statement (Lemma 14.8). His proof involves manipulating the rows and columns of the resultant matrix, but he only shows the inequality. For the "equality if and only if" part, he states as well that "a more careful proof" is required.

Does anybody have a reference or a starting point for this part of the proof?

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