8
$\begingroup$

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?

$\endgroup$
8
$\begingroup$

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.

$\endgroup$
8
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.