A standard fact about intersection multiplicities of algebraic plane curves is that, for curves $X=V(F(x,y))$ and $Y=V(G(x,y))$, if $p\in X\cap Y$ is a non-singular point of $X$ and $Y$ then

$$ I_p(X,Y)=1 $$

if and only if the tangent lines of $X$ and $Y$ at $p$ are distinct. In other words the curves have $0$-th order contact at $p$ (if $F_y(p)\neq 0$), where two curves have $k$-th order contact at $p$ if they share the point $p$ and the $k$ derivatives of $y$ with respect to $x$ at $p$ agree.

I believe it's true that

$$ I_p(X,Y)=k $$

if and only if the curves $X$ and $Y$ have $(k-1)$-th order contact at $p$, assuming $p$ non-singular and that the curves do not have $k$-th order contact. For small values of $k$, one can show this by locally parameterizing $X$ about $p$ using a Taylor series, and then considering the valuation of $G$ evaluated at this local parameterization.

Does anyone know where I can find a reference for this fact, or if this is not true?

  • $\begingroup$ A hint in case someone would like to prove this on their own: WLOG assume $p=(0,0)$. Locally parameterize the two curves about $p$ using Taylor series, i.e. there exist local parameterizations $(t,\sum_{i=1}^\infty a_it^i)$ and $(t,\sum_{i=1}^\infty b_it^i)$ such that $$ F(t,\sum_{i=1}^\infty a_it^i)=G(t,\sum_{i=1}^\infty b_it^i)=0. $$ Then use the fact that $I_p(F,G)=\text{val}(F(t,\sum_{i=1}^\infty b_it^i))$, to show that $a_i=b_i$ for $1\leq i\leq n$ if and only if $I_p(F,G)=n+1$. $\endgroup$ – MR_Q Jun 4 at 20:26

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