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I need help for the following problem : Consider $C_1 = V(F_1)$ and $C_2=V(F_2)$ be algebraic curves in $\mathbb P (\bar K )$ (where $K$ is a field,) without a common component and $F_1, F_2 \in \bar K [X,Y,Z]$ are homogenous with $\deg(F_1 ) \le \deg(F_2)$.

And let $G \in \bar K[X,Y,Z]$ be homogenous with degree $\deg(F_2) - \deg (F_1)$.

How can i show that mult$_P (C_1, C_2) =$ mult$_P (C_1 , V(F_2 + G . F_1))$ .

Thanks for your help .

Theorem
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1 Answers1

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This follows from the definition of the intersection multiplicity.

Let's assume that $P=[a:b:1] \in \Bbb{P}^2$, and denote by $f_1,f_2$ the dehomogenizations of $F_1,F_2$ w.r.t the variable $Z$, i.e. $f_i=F_i(X,Y,1)$, with $i=1,2$.

Then the intersection multiplicity is defined as

$$ \operatorname{mult}_P(C_1,C_2)=\dim_{\overline{K}}(\mathcal{O}_{(a,b)}(\Bbb{A}^2)/(f_1,f_2)) $$

Thus we see that the intersection multiplicity of $C_1=V(F_1)$ and $C_2=V(F_2)$ at $P=[a:b:1]$ depends only on the ideal $(f_1,f_2) \subset \mathcal{O}_{(a,b)}(\Bbb{A}^2)$. But as

$$ (f_1,f_2+g \cdot f_1)=(f_1,f_2) $$

the intersection multiplicities in question are the same.

Nils Matthes
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  • Can you explain the definition of intersection multiplicity ? Thanks . – Theorem May 06 '13 at 09:49
  • Dear @Theorem, what do you mean by "explain"? What exactly troubles you about its definition? Are you maybe accustomed to a different definition of intersection multiplicity? – Nils Matthes May 06 '13 at 10:07
  • i think i have trouble in understanding the notation . Can you tell me what $O$ denotes here ? It would be great if i can understand what the relation precisely telling me . – Theorem May 06 '13 at 10:14
  • Dear @Theorem, $\mathcal{O}{(a,b)}(\Bbb{A}^2)$ denotes the local ring of $\Bbb{A}^2$ at the point $(a,b)$. In this context this means that an element of $\mathcal{O}{(a,b)}(\Bbb{A}^2)$ is a fraction $\frac{f}{g}$, where $f,g \in K[x,y]$ such that $g(a,b) \neq 0$. Think of $\mathcal{O}_{(a,b)}(\Bbb{A}^2)$ as the ring of all rational functions defined at $(a,b)$, i.e. the ring of all quotients of polynomials, whose denominators do not vanish at the point $(a,b)$. – Nils Matthes May 06 '13 at 10:22