Intersection with a smooth curve is a vanishing order

Given: AC, a plane projective curve $C=V(F)$ smooth at $p$, a plane projective curve $D=V(G)$ with local equation $g$ at $p$ whose restriction $g{\mathrm{\mid }}_C$ is nonzero, and a local equation $f$ of $C$ at $p$.

The local ring ${\mathcal{O}}_{C,p}={\mathcal{O}}_{{\mathbf{P}}^2,p}\mathrm{/}(f)$ is a discrete valuation ring with valuation ${\mathrm{ord}}_p$, and the class of $g$ in it is the restriction $g{\mathrm{\mid }}_C$ Uniformising parameters at smooth points of a plane curve, Discrete valuation rings, Uniformising parameters.

Length is unchanged on passing to the quotient by the equation of the curve, because an $O\mathrm{/}(f)$-module has exactly the same submodules over $O$ and over $O\mathrm{/}(f)$: ${\mathcal{O}}_{{\mathbf{P}}^2,p}\mathrm{/}(f,g)\cong {\mathcal{O}}_{C,p}\mathrm{/}(g{\mathrm{\mid }}_C)$ by localisation commuting with quotients Localisation commutes with quotient rings: $S^{-1}R\mathrm{/}{S}^{-1}I\cong {\bar{S}}^{-1}(R\mathrm{/}I)$, so $I_p(C,D)={\mathrm{\ell }}_{{\mathcal{O}}_{C,p}}({\mathcal{O}}_{C,p}\mathrm{/}(g{\mathrm{\mid }}_C))$ Local intersection multiplicity of two plane curves.

In a discrete valuation ring $V$ with uniformiser $\pi$, a nonzero element $x=u{\pi }^n$ with $u$ a unit has ${\mathrm{\ell }}_V(V\mathrm{/}(x))=n$; a nonzero element of the DVR is a nonzerodivisor, so the quotient is a finite-length module exactly of this length Every nonzero fraction is a unit times a power of a uniformiser, Length and valuation in a DVR, Composition series and length of a module.

Since $g$ does not vanish identically on $C$, its class in the DVR is nonzero, so the finiteness hypothesis of the definition is satisfied and [F3] applies Finite local length exactly when no common local branch.