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.
Dependencies
D-SRFNSNumerical equivalence and the Néron--Severi groupuses