Why Bezout needs projectivity, algebraic closure and multiplicity

Assume the Axiom of Choice for the cited intersection results. For plane projective curves $C,D$ of degrees $d,e$ with no common component, the equality ${\sum }_p{I}_p(C,D)=de$ depends on projectivity, algebraic closure and multiplicity The Axiom of Choice.

(a) Projectivity. If the curves are considered only in an affine chart, any intersection points on the line at infinity are missing, so the sum over affine points is at most $de$ and is strictly smaller precisely when there is an intersection at infinity; equivalently one must add the contributions at infinity. The companion counterexample exhibits two affine lines with no common affine point whose projective closures meet at $[0:0:1]$; the missing point at infinity accounts for the deficit, so the affine count $0$ is strictly less than $de=1$ Bezout fails on the affine plane because points at infinity are missing ↗.

(b) Algebraic closure. If the ground field is not algebraically closed, the intersection scheme may have points whose residue field is a nontrivial finite extension and which are invisible to the $k$-points, so no multiplicity-weighted count of $k$-rational points can equal $de$ in general; the published Bezout theorem is stated over an algebraically closed field for exactly this reason Bezout's theorem for plane projective curves, Plane projective curves and their components. The companion counterexample is the imaginary conic $V(x_0^2+x_1^2+x_2^2)$ with no real point on the line $V(x_1)$, while over the algebraic closure the two conjugate points contribute the full degree total Bezout needs algebraic closure: an imaginary conic has no real point ↗.

(c) Multiplicity. If points are counted without multiplicity, a tangency between a line and a conic gives one point rather than two, and a shared tangent or a singular contact reduces the count below $de$; the local multiplicity is what repairs the count. The companion counterexample is the tangent line to a conic meeting it in a single point with $I_p=2$ Counting distinct points is not enough: tangent contact ↗.

Projectivity supplies completeness, algebraic closure makes all intersection points rational over the base field, and multiplicities encode the tangency and singularity defects; the general intersection is nonempty and finite precisely under the no-common-component hypothesis Two plane projective curves meet, Curves without a common component meet finitely often.