Let $f:X\rightarrow Y$ be a flat morphism of smooth projective varieties, and $\mathcal{L}$ an effective and ample line bundle on $Y$. For a divisor $A\in H^0(Y,\mathcal{L})$ set $X\_A := f^{-1}(A)$.


Let $D\subset X$ be a divisor such that $D\_{|X\_A}$ (the restriction of $D$ to $X\_A$) is big for $A\in H^0(Y,\mathcal{L})$ general. Under these conditions, might $D$ be not pseudo-effective?



Divisors whose restriction is big
Consider a product $X = \mathbb{P}^n\times\mathbb{P}^1$, with projections $g:X\rightarrow\mathbb{P}^n$ and $f:X\rightarrow\mathbb{P}^1$.


Set $H\_1:= g^{\*}\mathcal{O}\_{\mathbb{P}^n}(1)$ and $H\_2:= f^{\*}\mathcal{O}\_{\mathbb{P}^1}(1)$. The effective cone of $X$ is closed and generated by $H\_1,H\_2$.


Now, take a divisor $D = aH\_1+bH\_2$ with $a > 0$ and $b < 0$. Then $D\_{|f^{-1}(p)} = \mathcal{O}\_{\mathbb{P}^n}(a)$, which is ample since $a > 0$, for all $p\in\mathbb{P}^1$. However, since $b < 0$ the divisor $D$ is not pseudo-effective.


