HC-equivalence vs numerical equivalence for ample line bundles on surfaces

Given a smooth complex projective variety X and an ample line bundle L on it, one can associate the Hilbert curve with (X,L). This is a plane affine algebraic curve, whose equation depends only on the numerical characters of X and L. In particular, two numerically equivalent ample line bundles on X lead to the sameHilbert curve. Focusing on the case of surfaces, in this paper we investigate how far the converse is from being true.