When two curves of degrees m and n meet, count mn points — if you look in the right place
A line meets a conic twice, two conics meet four times: curves of degrees d and e cross in d · e points. Making that true every time takes some careful counting.
Pick two curves and compare the Bézout count d1×d2 with the points you can see. The missing ones are complex or lie at infinity.
Bézout's theorem predicts d1×d2 intersections in projective space over an algebraically closed field. The visible count may be fewer because some intersections are complex, at infinity, or have multiplicity > 1.
Bézout's theorem is one of the most elegant results in algebraic geometry: two projective plane curves C1 and C2 of degrees d and e, with no common component, intersect in exactly d · e points — provided we count correctly. "Counting correctly" means working in projective space CP2, over an algebraically closed field, and assigning each intersection point its proper multiplicity.
Bézout's theorem requires three conditions to give exact counts. First, work in projective space: in the affine plane, a line and a parabola might meet in 0, 1, or 2 points, but in CP2 the "missing" intersections appear at infinity. Second, work over an algebraically closed field: the real curves x² + y² = 1 and x² + y² = -1 don't meet over R, but they do over C. Third,count with multiplicity: a tangent line meets a conic in one geometric point, but with intersection multiplicity 2.
The intersection multiplicity at a point p can be defined as the dimension of the local ring Op/(f, g), where f and g define the two curves. For transverse intersections this gives 1; for tangencies, cusps, or higher-order contacts, it gives larger values. The sum of all multiplicities always equals d · e.
This theorem unifies many classical results. A line meets a conic in 2 points (1 · 2 = 2). Two conics meet in 4 points (2 · 2 = 4). A line meets a cubic in 3 points (1 · 3 = 3) — the very fact that makes the group law on elliptic curves possible. Bézout's theorem is the starting point of intersection theory, which generalizes these counts to higher dimensions.
Bézout's theorem generalizes: n hypersurfaces of degrees d1, …, dn in CPn meet in d1 · d2· … · dn points (counted with multiplicity), provided they intersect in finitely many points. Modern intersection theory, developed by Fulton and MacPherson, extends these ideas to arbitrary subvarieties using the Chow ring — a graded ring whose multiplication encodes intersection products. The degree of the product class gives the intersection number.
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