Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field). Let $C$ and $D$ be either affine plane curves in $\mathbb A_k^2$ or projective plane curves in $\mathbb P_k^2$, and let $p \in C \cap D$. Assume that $C$ and $D$ have no common irreducible component passing through $p$.

In the affine case, assume that $C$ and $D$ are nonsingular at $p$ and that their tangent lines in $T_p\mathbb A_k^2$ are distinct. In the projective case, assume that for some, equivalently any, affine chart containing $p$, the corresponding affine curve germs are nonsingular at $p$ and have distinct tangent lines. Then the local intersection multiplicity satisfies

\begin{align*} I_p(C,D)=1. \end{align*}