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].
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D-VARDEGDegree as an intersection countuses
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D-SRFNSNumerical equivalence and the Néron--Severi groupusesT-SRFRRRiemann--Roch for surfacesusesT-SRFNAKAIThe Nakai--Moishezon criterion on a surfaceusesT-SRFADJThe adjunction formula on a surfaceusesFE-SRFBLOWWhat a blowup does to $\Pic$, to $K$, and to the intersection formuses