Intersection with a line is the order of vanishing of the restricted equation

Given: AC, an algebraically closed field $k$, a line $L=V(\mathrm{\ell })$ with $\mathrm{\ell }$ a nonzero linear form, a plane projective curve $C=V(F)$ of degree $d\ge 1$ not containing $L$, and $p\in L\cap C$.

$L$ is a plane projective curve of degree one, isomorphic to ${\mathbf{P}}^1$ by a linear parametrisation; under such a parametrisation the restriction of a degree-$d$ form is a binary form of degree $d$ in the two parameters, which is nonzero because $L\subseteq ̸C$ Plane projective curves and their components, projective space points, morphism to projective space homogeneous coordinates, standard projective opens are affine spaces.

Choose linear coordinates on the plane chart taking $L$ to $y=0$ and $p$ to the origin. Then ${\mathcal{O}}_{L,p}=k[x{]}_{(x)}$ by localisation commuting with quotients. Its only primes are $(0)$ and $(x)$ (a nonzero prime below $(x)$ contains an irreducible divisor, necessarily associate to $x$), so its dimension is one; its cotangent space has basis the class of $x$, so it is regular. Thus the local ring ${\mathcal{O}}_{L,p}$ is a discrete valuation ring with maximal ideal generated by the image of any local parameter $t$ vanishing at $p$; this is the one-dimensional regular local ring case, and the image of $\mathrm{\ell }$ is a local equation of $L$ in ${\mathcal{O}}_{{\mathbf{P}}^2,p}$ one dimensional regular local rings are dvrs, Discrete valuation rings, Uniformising parameters, A local ring is a nonzero commutative ring with a unique maximal ideal.

Localisation commutes with quotients, and $I_p(C,L)={\mathrm{\ell }}_{{\mathcal{O}}_{{\mathbf{P}}^2,p}}({\mathcal{O}}_{{\mathbf{P}}^2,p}\mathrm{/}(f,\mathrm{\ell }))$ for local equations $f,\mathrm{\ell }$ Local intersection multiplicity of two plane curves, Localisation commutes with quotient rings: $S^{-1}R\mathrm{/}{S}^{-1}I\cong {\bar{S}}^{-1}(R\mathrm{/}I)$, Finite local length exactly when no common local branch.

In a discrete valuation ring $V$ with uniformiser $\pi$, every nonzero $x$ has a normal form $x=u{\pi }^{v(x)}$ with $u$ a unit and $v$ the valuation, and ${\mathrm{\ell }}_V(V\mathrm{/}(x))=v(x)$ 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. The order of vanishing of a nonzero element of the function field of ${\mathbf{P}}^1$ at a point is the valuation of the corresponding discrete valuation ring Discrete valuation rings, Uniformising parameters.