Curves without a common component meet finitely often

The product $FG$ is a nonzero polynomial in three variables over the infinite domain $k$, so it does not vanish at every triple of elements of $k$: there is $a\in k^3\setminus \{0\}$ with $F(a)G(a)\mathrm{\ne }0$. Choosing a projective change of coordinates $A$ with $A[0:0:1]=[a]$ and replacing $C,D$ by $A^{-1}(C),A^{-1}(D)$, whose intersection is the inverse image of the original one, we may assume $[0:0:1]$ lies on neither curve. The hypothesis of no common component is preserved, so the new defining forms are still nonzero of positive degrees $d,e$ with no common factor.

Any nonzero binary form $\rho \in k[x_0,x_1]$ of degree $N\ge 1$ has a zero in ${\mathbf{P}}^1$: writing $\rho =x_0^s{\rho }_1$ with $s$ maximal and $x_0\nmid {\rho }_1$, the dehomogenisation ${\rho }_1(1,T)$ is a nonzero polynomial of degree $N-s$; if $N-s\ge 1$ it has a root $\lambda \in k$ by algebraic closure and $[1:\lambda ]$ is a zero of $\rho$, while if $N-s=0$ the point $[0:1]$ is a zero.

In the coordinates of step 1.1 the forms $F,G$ satisfy the hypotheses of the resultant detection: ${\mathrm{Res}}_{x_2}(F,G)$ is a nonzero form of degree $de$, and it vanishes at $(a,b)\mathrm{\ne }(0,0)$ exactly when some common point $[a:b:c]$ of $C$ and $D$ exists; over each projective zero of the resultant the fibre of intersection points has at most $\min (d,e)$ elements.

The zeros of the nonzero form ${\mathrm{Res}}_{x_2}(F,G)$ of degree $de$ form a finite nonempty subset $Z\subseteq {\mathbf{P}}^1$: finite because writing $\rho ={\mathrm{Res}}_{x_2}(F,G)=x_0^s{\rho }_1$ with $x_0\nmid {\rho }_1$, the only possible zero with $x_0=0$ is $[0:1]$, while the zeros with $x_0\mathrm{\ne }0$ are the at most $de$ roots of the nonzero polynomial ${\rho }_1(1,T)$; and nonempty by step 1.2. By step 2.1 the projection $C\cap D\to {\mathbf{P}}^1$ has image exactly $Z$ and finite fibres of at most $\min (d,e)$ points, so $C\cap D$ is finite and nonempty, of at most $de\cdot \min (d,e)$ points [step 1.2, step 2.1]. For the scheme description, $X=\mathrm{Proj}(k[x_0,x_1,x_2]\mathrm{/}(F,G))$ is nonempty and zero-dimensional, hence has finitely many points which are the maximal ideals of the standard charts; over the algebraically closed field $k$ these maximal ideals are exactly the evaluation ideals at the points of the corresponding chart lying on both dehomogenised curves, so the points of $X$ correspond bijectively to the points of $C\cap D$ [F5, step 2.1]. This gives both the finite projection description and the equivalent scheme-theoretic description.