# Nef and generically ample line bundles

Let $$f:X\rightarrow Y$$ be a proper and surjective morphism of projective $$k$$-varieties, where $$k$$ is a field of characteristic $$0$$. Let $$L$$ be a line bundle over $$X$$. Assume that $$L$$ is nef and generically ample ( here generically ample means $$L$$ is ample on the generic fiber of $$f$$). Does there exists an line bundle $$A$$ on $$Y$$ such that $$L\otimes f^*A$$ is big?

This holds for any ample line bundle $$A$$ on $$Y$$. Indeed, then $$f^*A$$ is nef [Laz, Ex. 1.4.4(i)], hence so is $$D= L + f^*A$$. Setting $$n = \dim X$$ and $$m = \dim Y$$, we see that $$D^n \geq 0$$ [Laz, Thm. 1.4.9] and $$D$$ is big if and only if $$D^n > 0$$ [Laz, Thm. 2.2.16].
For positivity of $$D^n$$, we may replace $$(L,A)$$ by $$(dL,dA)$$ for $$d \gg 0$$ to assume $$A$$ is very ample. There is a dense open $$U \subseteq Y$$ with preimage $$V = f^{-1}(U)$$ such that $$f \colon V \to U$$ is flat (and proper) of relative dimension $$r = n-m$$ and $$L|_V$$ is relatively $$f$$-ample [Laz, Thm. 1.2.17]. General hyperplanes $$H_1,\ldots,H_m \in \lvert A\rvert$$ intersect transversally in a finite (reduced) scheme $$S \subseteq U$$. We find $$L^r \cdot (f^*A)^m = L^r \cdot f^*S = \sum_{s \in S} \big(L\big|_{X_s}\big)^r > 0.$$ Since $$f^*A$$ and $$L$$ are nef, we get $$L^i \cdot (f^*A)^{n-i} \geq 0$$ for all $$i$$ [Laz, Ex. 1.4.16], hence $$D^n = \big(L + f^*A\big)^n = \sum_{i=0}^n {n \choose i}L^i \cdot \big(f^*A\big)^{n-i} \geq {n \choose r} L^r \cdot \big(f^* A\big)^m > 0.\tag*{\square}$$