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?


2 Answers 2


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.


Take $Y=\mathbb{P}^1$ and let $X=\mathbb{F}_n=\mathbb{P}(\mathcal{O} \oplus \mathcal{O}(-n))$ be the Hirzebruch surface with a section $C_0$ such that $C_0^2=-n$.

Let $H$ be an ample divisor on $\mathbb{F}_n$, set $t:=HC_0$ and take $$D:=C_0-(t+1)F,$$ where $F \simeq \mathbb{P}^1$ is the fibre of $f \colon \mathbb{F}_n \to \mathbb{P}^1$.

Finally, take $\mathcal{L}=\mathcal{O}(1)$, so that $A$ is a point and $X_A=F$. The divisor $D|_{F}$ has degree $1$ on $F$, hence $D|_{F} = \mathcal{O}_F(1)$ which is ample, in particular big.

On the other hand, we have $$HD=H(C_0-(t+1)F)=t-(t+1)HF <0,$$ and this implies that $D$ is not pseudo-effective.


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