For a divisor \(D\) and an integral curve \(C\) on a smooth projective surface \(X\), \(D\cdot C=\deg_C\OO_X(D)|_C\).
The intersection pairing on a surface (Tag D-SRFINT)
Let \(X\) be a smooth projective surface. There is a unique symmetric bilinear pairing \[ \Div(X) \times \Div(X) \to \ZZ , \qquad (C,D) \mapsto C \cdot D , \] which counts points for transverse smooth curves and depends only on linear equivalence classes. \(C^2 = C \cdot C\) is the self-intersection. Fixing a very ample \(H\), the degree of a curve \(C\) in the embedding determined by \(H\) is \(C \cdot H\).
To compute \(C\cdot D\), replace \(C\) and \(D\) by linearly equivalent divisors that are differences of nonsingular curves meeting transversally, and count intersection points [@Har10a, Theorem V.1.1]. For \(C\) an irreducible nonsingular curve and any divisor \(D\), \(C\cdot D=\deg_C\OO_X(D)\vert_C\); in particular \(C^2 = \deg_C \OO_X(C)\vert_C=\deg_C\mcn_{C/X}\).
On \(\PP^2\), \(\Pic = \ZZ H\) with \(H^2 = 1\), so curves of degrees \(d\) and \(e\) with no common component meet in \(de\) points counted with multiplicity, which is Bézout’s theorem. On \(\PP^1 \times \PP^1\) the two rulings satisfy \(A^2 = B^2 = 0\) and \(A \cdot B = 1\).
For a curve \(C\) on a surface \(S\), the ideal sheaf is \(\mci_C = \OO_S(-C)\) and the conormal sheaf is \(\mci_C/\mci_C^2 = \OO_S(-C)\vert_C\). For \(C\) nonsingular of genus \(g\), the adjunction formula gives \(2g-2=C\cdot(C+K_S)\) [@Har10a, Proposition V.1.5].
For a smooth curve \(C\subset X\), \(C^2=\deg N_{C/X}=\deg\OO_X(C)|_C\). It can be negative: the exceptional curve \(E\) of a blowup at a point has \(E^2=-1\).
The adjunction formula on a surface (Tag T-SRFADJ)
Let \(C\) be a smooth curve on a smooth projective surface \(X\). Then \[ \omega_C = (\omega_X \otimes \OO_X(C))\vert_C , \qquad 2g(C) - 2 = C \cdot (C + K_X) . \] The same numerical formula computes the arithmetic genus of any curve on \(X\).
Adjunction converts genus calculations on surfaces into intersection-number calculations.
On \(\PP^2\) it gives the degree-genus formula. On \(\PP^2\) one has \(K = -3H\) and \(H^2 = 1\), so \(K^2 = 9\); for \(C\) of degree \(d\), \[ 2g-2 = C \cdot (C - 3H) = d^2 - 3d , \] giving \(g = \tfrac{1}{2}(d-1)(d-2)\). On \(\PP^1 \times \PP^1\), \(K = (-2,-2)\) and the same computation gives \(g = (a-1)(b-1)\) for a curve of type \((a,b)\).
In any dimension, for a smooth divisor \(D\) on a smooth \(X\), the sheaf-level form is \(K_D = (K_X + D)\vert_D\).
On \(\PP^2\), with \(K=-3H\), adjunction gives \(g=\tfrac{1}{2}(d-1)(d-2)\) for a smooth curve of degree \(d\). On \(\PP^1\times\PP^1\), with \(K=(-2,-2)\), it gives \(g=(a-1)(b-1)\) for a smooth curve of type \((a,b)\).
Riemann--Roch for surfaces (Tag T-SRFRR)
For a divisor \(D\) on a smooth projective surface \(X\), \[ \chi(\OO_X(D)) = \chi(\OO_X) + \tfrac{1}{2} D \cdot (D - K_X) . \]
(Noether's formula)
\[ \chi(\OO_X) = \tfrac{1}{12}\left( K_X^2 + c_2(X) \right) , \] where \(c_2(X)\) is the topological Euler characteristic over \(\CC\).
The surface formula parallels the curve formula, with \(\deg D\) replaced by \(\tfrac{1}{2}D\cdot(D-K)\) and \(1-g\) replaced by \(\chi(\OO_X)\); adjunction makes these substitutions consistent.
The typical use is not to compute \(\chi\) but to force a section. Serre duality gives \(h^2(D) = h^0(K-D)\), so \[ h^0(D) \geq \chi(\OO_X) + \tfrac{1}{2}D\cdot(D-K) , \] and if in addition \(h^0(K-D) = 0\) — for instance because \(D \cdot H > K \cdot H\) for an ample \(H\) — then \(D^2\) large forces \(D\) or \(-D\) to be effective. This inequality is a standard mechanism for proving that a divisor on a surface has sections and hence moves in a linear system.
Noether’s formula is the constraint relating \(K^2\) and the topology: on \(\PP^2\), \(K^2 = 9\) and \(c_2 = 3\), and \(\tfrac{1}{12}(9+3) = 1 = \chi(\OO)\).
Since \(h^2(\OO_X(D))=h^0(\OO_X(K-D))\) by Serre duality, \[ h^0(\OO_X(D))+h^0(\OO_X(K-D))\ge\chi(\OO_X)+\tfrac{1}{2}D\cdot(D-K). \]
The Néron–Severi group and the Hodge index theorem
Numerical equivalence and the Néron--Severi group (Tag D-SRFNS)
Divisors \(D\) and \(D'\) on a surface \(X\) are numerically equivalent, \(D \equiv D'\), if \(D \cdot E = D' \cdot E\) for every divisor \(E\). The Néron–Severi group is \(\NS(X) = \Div(X)/\equiv\). Its rank \(\rho(X)\) is the Picard number.
(Theorem of the base)
\(\NS(X)\) is a finitely generated abelian group.
For a smooth projective surface \(X\) over \(\CC\), the connected component \(\Pic^0(X)\) of \(\Pic(X)\) is an abelian variety of dimension \(q = h^1(\OO_X)\), and \(\NS(X)\) as defined here is \(\Pic(X)/\Pic^0(X)\) modulo its torsion subgroup. For \(D\in\Pic^0(X)\) and every divisor \(E\), \(D\cdot E=0\), so the intersection pairing is defined on \(\NS(X)\), where it is a nondegenerate symmetric bilinear form on a free abelian group of finite rank. By the Hodge index theorem, its extension to \(\NS(X)\otimes\RR\) has signature \((1,\rho(X)-1)\).
Examples are \(\rho(\PP^2)=1\) and \(\rho(\PP^1\times\PP^1)=2\), where \(\NS=\Pic\) because \(q=0\); a K3 surface has \(q=0\) and \(1\le\rho\le20\); an abelian surface has \(q=2\), so \(\Pic^0\) is two-dimensional and \(\NS\) is a genuine quotient. Blowing up a point adds one to \(\rho\), adjoining the class of the exceptional curve.
Algebraically equivalent divisors have equal intersection numbers, so the pairing factors through the finitely generated group \(\NS(X)=\Pic X/\Pic^0X\). It is nondegenerate on \(\NS(X)\) modulo torsion.
The Hodge index theorem (Tag T-SRFHODGE)
Let \(X\) be a smooth projective surface and \(H\) an ample divisor. If \(D \cdot H = 0\) and \(D \not\equiv 0\), then \(D^2 < 0\). Equivalently, the intersection form on \(\NS(X) \otimes \RR\) has signature \((1, \rho - 1)\).
One plus direction, everything else negative: the ample cone supplies the single positive direction, and the orthogonal complement of an ample class is negative definite. That is the whole content, and it is why \(\NS\) of a surface is a hyperbolic lattice.
Two consequences follow from the signature statement. If \(D^2 > 0\) then \(D \cdot H \neq 0\) for every ample \(H\), so \(D\) or \(-D\) is on the positive side — combined with Riemann–Roch this is how one proves a divisor with \(D^2 > 0\) and \(D \cdot H > 0\) is effective for large multiples. And the inequality \((D \cdot E)^2 \geq D^2 E^2\) for \(D\) with \(D^2 > 0\) is a signature statement, the reverse Cauchy–Schwarz that holds in a hyperbolic lattice.
Over \(\CC\) this is the Hodge–Riemann bilinear relation on \(H^{1,1}\) specialised to the algebraic classes, which is where the name comes from.
On \(\NS(X)\otimes\RR\) the pairing has signature \((1,\rho-1)\): an ample \(H\) has \(H^2>0\), and \(H^\perp\) is negative definite. For divisors \(D,E\) with \(D^2>0\), it follows that \((D\cdot E)^2\ge D^2E^2\).
The Nakai--Moishezon criterion on a surface (Tag T-SRFNAKAI)
A divisor \(D\) on a smooth projective surface \(X\) is ample if and only if \[ D^2 > 0 \quad\text{and}\quad D \cdot C > 0 \text{ for every irreducible curve } C \subseteq X . \]
This is the surface analogue of “ample means positive degree” on a curve, and the two conditions are exactly the two dimensions of subvariety: positivity against every curve, and positivity against the surface itself.
Both are needed. On the blowup of \(\PP^2\) at a point, \(D = \pi^*H\) has \(D^2 = 1 > 0\) but \(D \cdot E = 0\) on the exceptional curve, so it is not ample — it is only the pullback of an ample class, and it contracts \(E\). Conversely a class can meet every curve positively on a surface while failing \(D^2 > 0\), which is why the first condition is not implied by the second.
The criterion is purely numerical, so ampleness on a surface depends only on the class in \(\NS(X)\). That is what lets one draw the ample cone inside \(\NS(X) \otimes \RR\) and use the Hodge index theorem to describe it.
By the Nakai–Moishezon criterion, ampleness of \(D\) depends only on its class in \(\NS(X)\). The condition on curves is not implied by \(D^2>0\): on the blowup \(\pi\) of \(\PP^2\) at a point, \((\pi^*H)^2=1\) and \(\pi^*H\cdot E=0\).
Chow rings in any dimension
Chow groups, the Chow ring and the cycle class map (Tag D-CHOWRING)
Let \(X\) be a variety. The group \(Z_k(X)\) of \(k\)-cycles is the free abelian group on \(k\)-dimensional subvarieties. For a \((k+1)\)-dimensional subvariety \(W\) and \(f \in k(W)^\times\), the cycle \(\div(f) \in Z_k(X)\) is rationally equivalent to zero. The Chow group \(A_k(X)\) is \(Z_k(X)\) modulo these cycles, and for \(X\) of pure dimension \(n\) one writes \(A^k(X) = A_{n-k}(X)\).
(Chow ring)
For \(X\) smooth and quasiprojective, the Chow ring is \(A^\bullet(X) = \bigoplus_k A^k(X)\) with the intersection product: for subvarieties \(V, W\) meeting properly, \([V] \cdot [W] = \sum_i m_i [Z_i]\) over the components \(Z_i\) of \(V \cap W\) with intersection multiplicities \(m_i\). The moving lemma makes this well defined on classes.
(Cycle class map)
For \(X\) smooth projective over \(\CC\), the cycle class map \(\operatorname{cl} \colon A^k(X) \to H^{2k}(X, \ZZ)\) sends a subvariety \(V\) of codimension \(k\) to the Poincaré dual of its fundamental class. It is a ring homomorphism and lands in the Hodge classes \(H^{2k}(X, \ZZ) \cap H^{k,k}(X)\).
\(A^\bullet(\PP^n) = \ZZ[H]/(H^{n+1})\), where \(H\) is the class of a hyperplane, a subvariety of degree \(d\) and codimension \(k\) has class \(d H^k\), and the product formula recovers Bézout’s theorem. Here \(\operatorname{cl}\) is an isomorphism onto \(H^{\mathrm{even}}(\PP^n, \ZZ)\). For an elliptic curve \(E\), \(A^1(E) = \Pic(E)\) is uncountable while \(H^2(E, \ZZ) = \ZZ\), so \(\operatorname{cl}\) is not injective: on \(A^1(E)\) it is the degree map, with kernel \(\Pic^0(E)\cong E(\CC)\).
The Todd genus
The Todd class and the Todd genus (Tag D-VARTODD)
Let \(X\) be a smooth projective variety of dimension \(n\) over an algebraically closed field, and let \(\alpha_1, \ldots, \alpha_n\) be the Chern roots of \(T_X\). The Todd class of \(X\) is \[\operatorname{td}(X) = \prod_{i=1}^n \frac{\alpha_i}{1 - e^{-\alpha_i}} = 1 + \frac{c_1}{2} + \frac{c_1^2 + c_2}{12} + \frac{c_1 c_2}{24} + \cdots,\] with \(c_i = c_i(T_X)\). The Todd genus of \(X\) is the degree of its component \(\operatorname{td}_n(X)\) in dimension \(0\).
(Hirzebruch--Riemann--Roch for the structure sheaf)
The Todd genus of \(X\) equals \(\chi(X, \OO_X) = \sum_i (-1)^i h^i(X, \OO_X)\); in particular it is an integer.
For a curve of genus \(g\), \(\operatorname{td}_1 = c_1/2\) has degree \((2 - 2g)/2 = 1 - g = \chi(\OO_C)\). For a surface, \(c_1^2 = K_X^2\) and \(c_2 = e(X)\) is the topological Euler characteristic, and the theorem is Noether’s formula \(\chi(\OO_X) = \frac{1}{12}(K_X^2 + e(X))\).