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.
Dependencies
D-SRFINTThe intersection pairing on a surfaceuses
Backlinks
T-SRFHODGEThe Hodge index theoremuses