The naive count, and the three things that break it
We close the Varieties rung with a single concrete theorem that pays back everything the rung built: the Zariski topology and Nullstellensatz of Guides 1 and 2, the coordinate rings of Guide 3, and the smooth-versus-singular and blow-up machinery of Guide 4. The question is embarrassingly elementary. A line and a circle in the plane usually meet in two points; a line meets a circle, a conic meets a conic in four. So if a plane curve is the zero set of a degree-d polynomial f(x, y) and another is the zero set of a degree-e polynomial g(x, y), how many points are in their intersection? The honest first guess, suggested by every small example, is d times e. The whole guide is the story of why that guess is wrong as stated, and exactly which three repairs make it true on the nose.
Look at the failures, because each names a repair. First, x^2 + y^2 = 1 (degree 2) and the line y = 2 (degree 1) should meet in 2 points but meet in none over the reals — the equation x^2 = -3 has no real solution. Second, two parallel lines, y = 0 and y = 1 (each degree 1), should meet in 1 point but meet nowhere in the plane: they run off to infinity. Third, the line y = 0 (degree 1) and the parabola y = x^2 (degree 2) should meet in 2 points but meet only at the origin — yet the line is tangent there, and the single point ought to count twice. Three different ways the count of 2 collapses to fewer points, and three different fixes.
The three repairs, in order: (1) work over an algebraically closed field like the complex numbers C, so polynomials always have all their roots and no intersection silently disappears for arithmetic reasons; this is the very setting in which the Nullstellensatz of Guide 2 holds, so the rung's dictionary applies. (2) Work in the projective plane, not the affine plane, so that parallel lines and other escapees meet at well-defined points at infinity. (3) Count each intersection point with an integer weight, its intersection multiplicity, so a tangency counts as 2 and a worse contact counts higher. Make all three corrections and the count becomes exactly d times e, with no exceptions. That clean statement is Bézout's theorem.
Going projective and homogenizing
The projective fix is the one you already met as the home of varieties in Guide 1, so we only need its working form here. A point of the projective plane CP^2 is a triple [X : Y : Z], not all zero, taken up to common scaling, so [X : Y : Z] and [lambda X : lambda Y : lambda Z] are the same point. The affine plane sits inside as the chart Z is not 0, via (x, y) corresponds to [x : y : 1]; everything with Z = 0 is the line at infinity, one extra point at infinity for each direction of affine lines. Parallel affine lines share a direction, so they share their point at infinity — and there they meet. The escapees of the last section are not lost; they were waiting on the line at infinity all along.
To make a polynomial equation define a curve in CP^2 it must be homogeneous: every monomial of the same total degree, so that f(lambda X, lambda Y, lambda Z) = lambda^d f(X, Y, Z) and the zero set is well defined on scaling classes. You turn an affine f(x, y) of degree d into a homogeneous F(X, Y, Z) of the same degree by homogenizing: replace x by X/Z and y by Y/Z, then multiply through by Z^d to clear denominators. The reverse, dehomogenizing, is just setting Z = 1. Working in homogeneous coordinates this way, a degree-d plane curve is the zero locus of a single homogeneous degree-d polynomial, and it is automatically a projective variety.
Affine -> homogenize (x = X/Z, y = Y/Z, clear by Z^d) -> Projective line y = x + 1 -> Y = X + Z parabola y = x^2 -> Y Z = X^2 circle x^2 + y^2 = 1 -> X^2 + Y^2 = Z^2 Points at infinity: set Z = 0. parabola Y Z = X^2 at Z = 0 => X^2 = 0 => [0 : 1 : 0] (one point, doubled) two parallels Y = X+Z, Y = X-Z at Z=0 => Y = X => [1 : 1 : 0] (they MEET here)
Intersection multiplicity, defined honestly
The third repair is the deep one and deserves a real definition, not a slogan. The naive idea — 'count how many times the curves are tangent' — is suggestive but useless for computation and silent about ugly singular cases, like two curves both passing singularly through a point. The right object is local and algebraic. Fix a point p where curves f = 0 and g = 0 meet. Look at the local ring O_p of regular functions defined near p — the localization of the coordinate ring at p, the ring of fractions you can evaluate at p. Inside it sit the two functions f and g. The intersection multiplicity of the two curves at p is the dimension, as a vector space over the field, of the quotient ring O_p / (f, g), where (f, g) is the ideal generated by f and g.
Write I_p(f, g) = dim_k O_p / (f, g). Read it slowly: you are measuring how much algebraic room is left near p after you impose BOTH equations f = 0 and g = 0 at once. If the curves cross cleanly and transversally, imposing both equations pins down the point completely, the quotient is just the field k itself, and I_p = 1 — one honest crossing. If the curves are tangent, f and g fail to be 'independent enough' at first order; the ideal (f, g) is smaller, the quotient is bigger, and I_p jumps to 2 or more. This definition needs nothing about derivatives or limits; it is pure commutative algebra, and it handles singular contacts the geometric picture cannot describe.
- Take the line y = 0 and the parabola y = x^2, meeting at the origin p = (0, 0). Set f = y and g = y - x^2. The local ring O_p is rational functions in x, y regular at the origin.
- Form the ideal (f, g) = (y, y - x^2). Since y is in the ideal, y - x^2 minus y = -x^2 is too, so the ideal equals (y, x^2). The quotient O_p / (y, x^2) sets y = 0 and x^2 = 0.
- What survives is spanned by the classes of 1 and x: every higher power of x dies (x^2 = 0) and y is gone. So O_p / (f, g) has dimension 2 over k.
- Therefore I_p(f, g) = 2: the tangent line meets the parabola with multiplicity 2 at the origin, exactly the 'doubled point' the naive count was missing.
Bézout's theorem and how it is proved
Now we can state the theorem cleanly. Bézout's theorem: let F and G be homogeneous polynomials in CP^2 of degrees d and e, defining plane curves with NO common component (no shared irreducible factor — they do not literally overlap along a whole curve). Then the sum over all intersection points p of the intersection multiplicities I_p(F, G) equals d times e. Every correction from the first section now does its job: working over C makes all the roots present, working in CP^2 captures the points at infinity, and weighting by intersection multiplicity makes tangencies and singular contacts pay their full share. The slogan 'two curves meet in d times e points' is true exactly when read with these three caveats in force.
Where does the product d times e actually come from? The cleanest proof uses the resultant, an old idea from elimination theory. Treat F and G as polynomials in one variable, say Y, with coefficients that are polynomials in X and Z. Their resultant Res_Y(F, G) is a single polynomial in X and Z that vanishes exactly when F and G share a common Y-root — that is, exactly over the X, Z-projections of the intersection points. A degree count shows this resultant is homogeneous of degree d times e in X and Z. Over the algebraically closed C such a form factors into exactly d times e linear factors counted with multiplicity, and each factor corresponds, with the correct weight, to an intersection point. The product d times e is the degree of the resultant — that is the engine.
What Bézout buys you, and where the rung leads next
A theorem this clean is a counting machine, and the payoffs are immediate. Two distinct lines (d = e = 1) meet in exactly 1 point — the projective version of 'two lines meet unless parallel', with the parallel case rescued at infinity. A line meets a degree-d curve in d points counted with multiplicity, which is how you define and count the order of contact of a tangent line. Two conics (d = e = 2) meet in 4 points, the classical fact behind the pencil of conics and behind the power of a point you may recall from circle geometry. And a smooth plane cubic met by a line in its 3 points is the geometric seed of the group law on an elliptic curve: three collinear points sum to zero. Bézout turns 'how many?' into arithmetic.
Be honest about scope, the way Volume I urged. Bézout as stated is a statement about TWO plane curves in CP^2; it is the dimension-one, codimension-one face of a far larger intersection theory. The general theorem counts how many points cut out by hypersurfaces of degrees d_1, ..., d_n in CP^n, giving the product d_1 times ... times d_n — but already there one must be careful that the pieces meet in the expected dimension (a 'proper' intersection), or the naive product can lie. The full machinery, where intersection multiplicities live in a Chow ring or are defined by Serre's Tor formula, is a separate course; this guide gives you the honest n = 2 case and the right local definition, not the general theory in maximum generality.
Step back and see the arc. Guide 1 gave us affine and projective varieties and the Zariski topology; Guide 2 made the ideal-variety dictionary precise through the Nullstellensatz; Guide 3 packaged functions on a variety as its coordinate ring and function field; Guide 4 separated smooth from singular points and resolved singularities by blow-ups. This guide cashed all four in on a single answer: Bézout's theorem, where two curves of degrees d and e meet in exactly d times e points once you count over C, projectively, and with multiplicity. From here the road forks toward schemes — where Spec of a ring replaces the variety and the same intersection ideas become Tor and sheaf cohomology — and toward the curve theory of Riemann-Roch. Every fork starts where this one ended: with a local ring measuring how two equations meet.