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?
Best Answer
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.