intersection multiplicity

When two curves cross, sometimes they cut cleanly through each other and sometimes they merely graze, touching without truly crossing. Counting a tangency as just 'one point' loses information. The intersection multiplicity is the right way to weight each common point so that the count is stable and honest: a transverse crossing counts once, a simple tangency counts twice, a higher-order contact counts more. It is the bookkeeping that makes Bézout's theorem come out to an exact equality.

Let two plane curves C = V(f) and D = V(g) meet at a point p. The intersection multiplicity I_p(C, D) is defined algebraically as the dimension, as a vector space over k, of the local ring quotient O_p / (f, g): you localize the coordinate ring at p and measure how big the algebra cut out by both equations is there. Concretely it is dim_k ( k[x, y]_p / (f, g) ). This number is 1 exactly when the curves meet transversally (distinct tangent lines at a smooth crossing); it is 2 for a simple tangency; it grows with the order of contact and with the multiplicities of any singular points involved. Bézout's setup also requires counting points at infinity, which is why one works in the projective plane.

Intersection multiplicity is the foundation of intersection theory, the part of algebraic geometry that counts how subvarieties meet, and the local input to Bézout's theorem. Two honest caveats. First, the naive dimension formula dim_k O_p / (f, g) gives the correct multiplicity only when the curves share no common component through p (otherwise the quotient is infinite-dimensional and intersection is not a finite count). Second, the symmetric, well-behaved theory really needs the projective and properly-intersecting (no shared components) setting, and the truly general definition — for higher-dimensional subvarieties meeting in excess dimension — requires Serre's Tor-formula, where naive length counting fails and you sum alternating lengths of Tor modules.