Two coprime projective plane forms meet in total length equal to their degree product

Given: The Axiom of Choice, a field $k$, the polynomial ring $R=k[x_0,x_1,x_2]$, nonzero homogeneous forms $F,G\in R$ of positive degrees $d,e$ without common nonconstant factor, the standard graded quotient $S=R\mathrm{/}(F,G)$, its standard charts $D_{+}(x_i)=\mathrm{Spec}(A_i)$ with $A_i=(S_{x_i}{)}_0$, and $X=\mathrm{Proj}S$.

$X\mathrm{\ne }\mathrm{\varnothing }$, each standard chart $D_{+}(x_i)$ is empty or of Krull dimension $0$ (Krull dimension of a nonzero ring), and $\dim S=1$, so $S$ is not Artinian (A plane intersection with no common component is nonempty and zero-dimensional). The spectrum of a ring is empty exactly for the zero ring: the zero ring has no prime ideal, while every nonzero commutative ring has a maximal ideal, which is prime (Prime ideals and maximal ideals in a commutative ring, In a nonzero commutative ring, every proper ideal is contained in a maximal ideal, Every maximal ideal of a commutative ring is prime); hence the chartwise zero-dimensionality hypothesis "every $A_i$ is zero or of Krull dimension $0$" holds.

Assume AC. For such $X$: $X$ has finitely many points, each local ring ${\mathcal{O}}_{X,x}$ is a finite-dimensional local $k$-algebra with finite length and finite residue degree, and $X$ is the finite disjoint union of the spectra $\mathrm{Spec}({\mathcal{O}}_{X,x})$ of its local rings (A zero-dimensional projective scheme has finitely many closed points with finite-dimensional local rings).

Assume AC. The total length of the zero-dimensional $X$ is ${\mathrm{len}}_k(X)={\sum }_{x\in X}{\mathrm{\ell }}_{{\mathcal{O}}_{X,x}}({\mathcal{O}}_{X,x})[\kappa (x):k]$, a finite sum over the points of $X$, with ${\mathrm{len}}_k(\mathrm{\varnothing })=0$ (Total length of a zero-dimensional projective scheme).

$(F,G)$ is an $R$-regular sequence, because $F,G$ are homogeneous of positive degree and share no nonconstant factor (Coprime positive-degree plane forms form a regular sequence, Regular Sequence On A Module).

For an $R$-regular pair $(F,G)$ of positive degrees $d,e$ the Hilbert function of $S=R\mathrm{/}(F,G)$ is constantly equal to $de$ in every degree $n\ge d+e-2$ (Hilbert series and eventual Hilbert value of a two-form plane complete intersection, The Hilbert function and formal Hilbert series of a graded module with finite-length pieces).

Assume AC. For a homogeneous ideal $I\subseteq R$ whose standard chart rings are zero or of Krull dimension $0$, the eventual value of the Hilbert function of $R\mathrm{/}I$ equals the total length: ${\dim }_k(R\mathrm{/}I{)}_m={\mathrm{len}}_k(\mathrm{Proj}(R\mathrm{/}I))$ for all sufficiently large $m$ (The eventual Hilbert function of a zero-dimensional projective quotient equals its total length).

If $x\in D_{+}(x_i)$ corresponds to the prime ${\mathfrak{p}}_0\subseteq A_i$, then ${\mathcal{O}}_{X,x}\cong (A_i{)}_{{\mathfrak{p}}_0}$ is a localisation of the chart ring $A_i$ (Prime and local-ring correspondence on standard projective charts).

Localisation at a homogeneous element $t$ of degree $\delta \ge 1$ of a nonnegatively graded ring is graded by $(R_t{)}_n=\{r\mathrm{/}{t}^m:r\in R_{n+m\delta }\}$ with degree-preserving localisation map, and for $t$ of degree one and an ideal $J=(a_1,\dots ,a_k)$ generated by homogeneous elements one has $((R\mathrm{/}J{)}_t{)}_0\cong (R_t{)}_0\mathrm{/}(a_1\mathrm{/}{t}^{\mathrm{deg}{a}_1},\dots ,a_k\mathrm{/}{t}^{\mathrm{deg}{a}_k})$ (Localisation at a homogeneous element is graded, with graded kernels and dehomogenised degree-zero parts, Nonnegatively graded rings and modules, homogeneous elements, and twists). Moreover, if $\phi :R\to B$ is a unital ring homomorphism with $\phi (x_i)$ a unit, then $\phi$ extends uniquely to $R_{x_i}$ (Universal property of localisation: maps that invert $S$ factor uniquely through $S^{-1}R$, Multiplicative subsets and the localisation $S^{-1}R$ as equivalence classes of fractions, Principal localisation $R_f=\{1,f,f^2,\dots {\}}^{-1}R$); in the polynomial ring $R$ every element is a finite $k$-linear combination of monomials $x^a$ of total degree $\mathrm{\mid }a\mathrm{\mid }$ (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots ,x_n]$, homogeneous polynomial and homogeneous ideal).

Assume AC (declared for consumers). The Axiom of Choice