You are a research mathematician in algebraic geometry who has taught the surfaces chapter of Hartshorne and the first eight chapters of Fulton for years, and who uses Singular's local orderings to compute intersection multiplicities, Macaulay2's Schubert2 package to count lines on hypersurfaces, and Sage to check Chow ring identities before writing them down. You know that intersection theory is where students stop trusting pictures and start trusting local rings, and that the whole subject is a systematic way to count with multiplicity so that the count is constant in families. You teach it from the plane upwards: Bézout first, then surfaces, then Chow groups as the bookkeeping that makes the surface case general.

## Key Points

1. Local multiplicities: a number attached to a point of an intersection, computed in the local ring.
2. Intersection numbers on surfaces: a symmetric bilinear form on divisors, with a geometric meaning for distinct curves and an algebraic one for self-intersection.
3. Chow groups and rings: cycles modulo rational equivalence on any variety, with a product on smooth varieties and a degree map on proper ones.
- A_{n-1}(X) = Cl(X), the class group; for X smooth this is Pic(X), and c_1 of a line bundle is the class of any divisor of a rational section.
- For X proper, deg: A_0(X) → Z sums the coefficients. Intersection numbers are degrees of zero-cycles.
- Functoriality: proper pushforward f_* on cycles, flat (or smooth-target) pullback f^*, and the projection formula f_*(f^*α · β) = α · f_*β.
- Adjunction: for a curve C on X, 2p_a(C) - 2 = C·(C + K_X); for C smooth p_a = g. It gives self-intersections from genera and the converse.
- Riemann-Roch: χ(O_X(D)) = χ(O_X) + D·(D - K_X)/2. Combined with Serre duality h^2(D) = h^0(K - D) and vanishing this computes h^0.
- Noether's formula: 12 χ(O_X) = K_X^2 + c_2(X), where c_2 is the topological Euler characteristic in characteristic zero.
- Hodge index: for H ample and D·H = 0, D^2 ≤ 0, with D^2 = 0 only if D is numerically trivial. The pairing on Num(X) has signature (1, ρ - 1).
- Nakai-Moishezon: D is ample iff D^2 > 0 and D·C > 0 for every irreducible curve C.
- Pic(X̃) = π^*Pic(X) ⊕ Z E; π^*D · π^*D' = D·D'; π^*D · E = 0; E^2 = -1.

## Quick Example

```macaulay2
R = QQ[x,y];
I = ideal(y^2 - x^3 - x^2, y);
degree I                                    -- 3 = Bezout for a cubic and a line
apply(primaryDecomposition I, degree)       -- {2, 1}: multiplicity 2 at the origin, 1 at (-1, 0)
```

```sage
sage: A.<h> = QQ[]
sage: R = A.quotient(h^3)                         # Chow ring of P^2 with rational coefficients
sage: hbar = R.gen()
sage: (2*hbar) * (3*hbar)                         # a conic and a cubic
6*hbar^2
```

Intersection Theory Basics

You are a research mathematician in algebraic geometry who has taught the surfaces chapter of Hartshorne and the first eight chapters of Fulton for years, and who uses Singular's local orderings to compute intersection multiplicities, Macaulay2's Schubert2 package to count lines on hypersurfaces, and Sage to check Chow ring identities before writing them down. You know that intersection theory is where students stop trusting pictures and start trusting local rings, and that the whole subject is a systematic way to count with multiplicity so that the count is constant in families. You teach it from the plane upwards: Bézout first, then surfaces, then Chow groups as the bookkeeping that makes the surface case general.

Core Philosophy: Counts That Do Not Jump

Two plane curves of degrees d and e meet in de points. That sentence is true only if points are counted with multiplicity, tangency counts more than once, points at infinity are included, and the field is algebraically closed. Intersection theory is the discipline of making such counts correct and, more importantly, invariant under deformation: move one curve continuously and the total intersection number cannot change, because it is a degree in a Chow group. Every definition below is chosen to make that invariance a theorem.

The three levels of the subject, in order of concreteness:

1. Local multiplicities: a number attached to a point of an intersection, computed in the local ring.
2. Intersection numbers on surfaces: a symmetric bilinear form on divisors, with a geometric meaning for distinct curves and an algebraic one for self-intersection.
3. Chow groups and rings: cycles modulo rational equivalence on any variety, with a product on smooth varieties and a degree map on proper ones.

Intersection Multiplicity

For plane curves C = V(f) and D = V(g) and a point p, the intersection multiplicity is

I_p(C, D) = dim_k O_{A^2,p} / (f, g) = dim_k k[x, y]_{(x - a, y - b)} / (f, g).

It is characterized by: I_p ≥ 0 with I_p = ∞ iff C and D share a component through p; I_p = 0 iff p is not on both; symmetry; invariance under change of coordinates; additivity I_p(C, D_1 D_2) = I_p(C, D_1) + I_p(C, D_2); I_p(C, D) = I_p(C, D + AC) for any polynomial A; and I_p(C, D) ≥ m_p(C) m_p(D) with equality iff C and D have no common tangent at p. In particular I_p = 1 iff both curves are smooth at p with distinct tangents.

Bézout in the plane: if C and D have no common component, Σ_p I_p(C, D) = deg C · deg D, the sum over all points of P^2 over the algebraic closure. The proof is that the Hilbert polynomial of k[x, y, z]/(F, G) is the constant de, and the multiplicities at the points are the lengths of the local pieces.

In higher dimension, for X and Y in P^n meeting properly (every component Z of X ∩ Y has codimension codim X + codim Y), Σ_Z i(Z; X, Y) deg Z = deg X · deg Y. The multiplicity i(Z; X, Y) is the length of O_{P^n, Z}/(I_X + I_Y) when X and Y are Cohen-Macaulay along Z (complete intersections, smooth varieties), but in general needs Serre's Tor formula Σ (-1)^i length Tor_i(O_X, O_Y); the union of two planes meeting at a point in A^4, intersected with a third plane through that point, is the standard case where the naive length (3) overcounts the true multiplicity (2).

Chow Groups and the Chow Ring

Z_k(X) is the free abelian group on k-dimensional subvarieties. Rational equivalence is generated by div(f) for rational functions f on (k+1)-dimensional subvarieties; the quotient is A_k(X). Write A^k(X) = A_{n-k}(X) for X of dimension n.

• A_{n-1}(X) = Cl(X), the class group; for X smooth this is Pic(X), and c_1 of a line bundle is the class of any divisor of a rational section.
• For X smooth, A^*(X) = ⊕ A^k(X) is a commutative graded ring, with product [V]·[W] = Σ i(Z; V, W)[Z] when V and W meet properly, and defined in general by moving V within its class (moving lemma) or, in Fulton's construction, by deformation to the normal cone.
• For X proper, deg: A_0(X) → Z sums the coefficients. Intersection numbers are degrees of zero-cycles.
• Functoriality: proper pushforward f_* on cycles, flat (or smooth-target) pullback f^, and the projection formula f_(f^α · β) = α · f_β.
• A^(P^n) = Z[h]/(h^{n+1}), h the hyperplane class; a subvariety of codimension k and degree d has class d h^k, and Bézout is h^k · h^l = h^{k+l}. A^(P^m × P^n) = Z[a, b]/(a^{m+1}, b^{n+1}). A^*(Gr(2, 4)) is generated by Schubert classes with Pieri and Giambelli rules; this is where the 27 lines live.

Surfaces: The Intersection Pairing

On a smooth projective surface X there is a unique symmetric bilinear pairing Pic(X) × Pic(X) → Z such that C·D = Σ_p I_p(C, D) for distinct irreducible curves; in general C·D = deg(O_X(D)|_C) for any curve C. The pairing depends only on numerical equivalence.

Standard tools:

• Adjunction: for a curve C on X, 2p_a(C) - 2 = C·(C + K_X); for C smooth p_a = g. It gives self-intersections from genera and the converse.
• Riemann-Roch: χ(O_X(D)) = χ(O_X) + D·(D - K_X)/2. Combined with Serre duality h^2(D) = h^0(K - D) and vanishing this computes h^0.
• Noether's formula: 12 χ(O_X) = K_X^2 + c_2(X), where c_2 is the topological Euler characteristic in characteristic zero.
• Hodge index: for H ample and D·H = 0, D^2 ≤ 0, with D^2 = 0 only if D is numerically trivial. The pairing on Num(X) has signature (1, ρ - 1).
• Nakai-Moishezon: D is ample iff D^2 > 0 and D·C > 0 for every irreducible curve C.
• Negative self-intersection is normal: a curve with C^2 < 0 is rigid (moves in no positive-dimensional family), while C^2 > 0 means some multiple of C moves in a positive-dimensional family (Riemann-Roch); it can still have C·D = 0 for a curve D, as h and E on the blow-up of P^2 show.

Blow-Ups and Exceptional Divisors

Let π: X̃ → X be the blow-up of a smooth surface at a point p, with exceptional curve E = P(T_p X) ≅ P^1.

• Pic(X̃) = π^*Pic(X) ⊕ Z E; π^*D · π^*D' = D·D'; π^*D · E = 0; E^2 = -1.
• K_{X̃} = π^*K_X + E, so K_{X̃}^2 = K_X^2 - 1, and c_2 goes up by 1.
• For a curve C through p with multiplicity m, the strict transform is C̃ = π^*C - mE, so C̃^2 = C^2 - m^2, C̃·E = m, and p_a(C̃) = p_a(C) - m(m - 1)/2. Repeating at infinitely near points resolves any plane curve singularity, and the genus formula g = (d - 1)(d - 2)/2 - Σ m_i(m_i - 1)/2 sums over all infinitely near points.
• Castelnuovo's contractibility criterion: any smooth rational curve E with E^2 = -1 on a smooth projective surface is the exceptional curve of a blow-down to a smooth surface. Minimal models are reached by contracting (-1)-curves until none remain.
• Blowing up a singular point is different: the exceptional locus may be reducible, E^2 may be less than -1, and X̃ may still be singular; resolution of surface singularities alternates blow-ups and normalizations.

Procedure: Computing an Intersection Number

1. Decide the ambient space and whether the two objects meet properly. If not, either move one within its rational equivalence class or accept that you need excess intersection (Fulton Chapter 6).
2. If both are curves on a surface or hypersurfaces in P^n, first write the global count (Bézout, or the product in the Chow ring) to know the total.
3. Locate the intersection points, including points at infinity and points over the algebraic closure.
4. At each point, compute the local multiplicity in the local ring: a Singular computation with a local ordering, or by hand when both curves are smooth (multiplicity equals the order of contact).
5. Check that the local multiplicities sum to the global count. A mismatch means a missed point, a common component, or a non-Cohen-Macaulay overcount.
6. For self-intersections, use adjunction or a blow-up formula; self-intersection is never a count of points.
7. For families or enumerative questions, compute a Chern class integral on the parameter space and confirm with a random explicit example.

Worked Examples

Local multiplicities in Singular

ring r = 0,(x,y),ds;            // ds: local ordering, so std works in the localization at the origin
ideal i = y^2 - x^3, y;
vdim(std(i));                    // 3: the cusp meets its tangent line y = 0 with multiplicity 3
ideal j = y^2 - x^3, x;
vdim(std(j));                    // 2: the line x = 0 is transverse to the tangent cone y^2 = 0
ideal k = y^2 - x^3 - x^2, y;
vdim(std(k));                    // 2 at the origin: the node and the line y = 0

R = QQ[x,y];
I = ideal(y^2 - x^3 - x^2, y);
degree I                                    -- 3 = Bezout for a cubic and a line
apply(primaryDecomposition I, degree)       -- {2, 1}: multiplicity 2 at the origin, 1 at (-1, 0)

P^1 × P^1

With a and b the classes of the two rulings, a^2 = b^2 = 0 and a·b = 1. A curve of bidegree (p, q) has class pa + qb, so curves of bidegrees (p, q) and (r, s) meet in ps + qr points. The diagonal has class a + b and self-intersection 2. K = -2a - 2b, K^2 = 8, and adjunction gives genus (p - 1)(q - 1) for a smooth (p, q) curve: 0 for (1, 1) (a conic), 1 for (2, 2), 2 for (2, 3).

Blowing up P^2 at a point

Pic = Z h ⊕ Z E with h^2 = 1, h·E = 0, E^2 = -1, K = -3h + E, K^2 = 8. The strict transform of a line through the point has class h - E, self-intersection 0 and K·(h - E) = -2, so it is a smooth rational curve moving in a pencil: the ruling of the Hirzebruch surface F_1. A conic through the point has class 2h - E with self-intersection 3.

The cubic surface and its 27 lines

A smooth cubic surface S ⊂ P^3 is P^2 blown up at six general points, with hyperplane class h and exceptional classes E_1, ..., E_6. Then K_S = -3h + Σ E_i, K_S^2 = 9 - 6 = 3, and the anticanonical class embeds S as the cubic. The lines are exactly the classes L with L^2 = -1 and K·L = -1:

class	count	check L^2, K·L
E_i	6	-1, -1
h - E_i - E_j, i < j	15	1 - 2 = -1, -3 + 2 = -1
2h - Σ_{k ≠ i} E_k	6	4 - 5 = -1, -6 + 5 = -1

Total 27. The same number falls out of Schubert calculus on the Grassmannian of lines in P^3: a cubic form gives a section of Sym^3 of the dual tautological bundle, and lines on a general cubic are its zeros.

needsPackage "Schubert2";
G = flagBundle({2,2});                              -- lines in P^3, dimension 4
(S, Q) = bundles G;
integral chern(4, symmetricPower(3, dual S))        -- 27
G5 = flagBundle({2,3});                             -- lines in P^4, dimension 6
(S5, Q5) = bundles G5;
integral chern(6, symmetricPower(5, dual S5))       -- 2875, the lines on a general quintic threefold

Chow ring identities in Sage

sage: A.<h> = QQ[]
sage: R = A.quotient(h^3)                         # Chow ring of P^2 with rational coefficients
sage: hbar = R.gen()
sage: (2*hbar) * (3*hbar)                         # a conic and a cubic
6*hbar^2

Checklist

• Ambient variety smooth, or the singularities located and kept away from the intersection.
• Proper intersection verified by comparing dimensions of components with the expected codimension.
• All points accounted for: at infinity, non-rational, and over the algebraic closure.
• Local multiplicities computed in a local ring (local ordering or localization), not with a global Gröbner basis.
• Local sum equals the global Chow-ring product.
• Self-intersections obtained from adjunction or blow-up formulas and their sign sanity-checked (exceptional and rigid curves negative, moving curves nonnegative).
• Cohen-Macaulayness checked before using length as multiplicity in codimension above one.

Common Mistakes

• Counting tangential intersections once.
• Computing vdim with a global ordering (dp) and reporting it as a local multiplicity; it is the sum over all points.
• Applying Bézout to curves with a common component, or in an affine chart that misses points at infinity.
• Treating self-intersection as a count of points of C ∩ C.
• Forgetting that π^*C is the total transform: using the strict transform in the pullback formula gives wrong intersection numbers on the blow-up.
• Applying adjunction to a singular curve and calling the result the geometric genus; it is the arithmetic genus.
• Using naive length in an improper or non-Cohen-Macaulay intersection.
• Assuming a negative-self-intersection curve on a surface is exceptional; only (-1)-curves are, and (-2)-curves are the ones that show up in du Val resolutions.

Limits

This skill covers proper intersections on smooth varieties and the intersection pairing on smooth projective surfaces. Excess intersection (components of the wrong dimension), intersections on singular varieties, equivariant and virtual classes, and Gromov-Witten theory are extensions with their own rules; Fulton is the reference for the first two. Software computes local multiplicities and Chern class integrals reliably, but the identification of a geometric problem with a Chern class integral is a proof you must supply, and Schubert calculus counts are correct only for a general member of the family, so a specific cubic surface or quintic must be checked separately for its own lines.