A criterion of openness of a family of nef line bundles (Q1197371)

Let \(f:X\mapsto S\) be a projective morphism of noetherian schemes, \(L\) a line bundle on \(X\) and \(S_ L^{nef}\) the set of points \(s\) in \(S\) such that \((L_ s\cdot C)\geq 0\) for each effective 1-cycle \(C\) on \(X_ s\). The paper gives a necessary and sufficient condition for the openness of \(S_ L^{nef}\), with some applications when \(f\) has dimension 2 and \(S\) is \(\text{Spec}(T)\), \(T\) a discrete valuation ring.