Deform a divisor from a fiber in a fibration

Suppose $$X\rightarrow Z$$ is a projective smooth morphism. Let $$0\in Z$$ be a closed point, $$X_0$$ the corresponding fiber. Suppose $$H^1(X_0,\mathcal{O})=H^2(X_0,\mathcal{O})=0$$, then a line bundle $$L$$ on $$X_0$$ can be extended (after an 'etale cover) to a line bundle $$\mathcal{L}$$ on $$X$$ by Professor Totaro's result in "Jumping of the nef cone for Fano varieties".

My question is, if $$L=\mathcal{O}_{X_0}(D_0)$$ for a divisor $$D_0$$ on $$X_0$$, then can we extend the divisor $$D_0$$ (after an 'etale cover) to a divisor $$D$$ on $$X$$? And if $$D_0$$ is effective, can we choose $$D$$ to be also effective?

• Certainly Totaro uses that result, but it goes back (at least) to Kodaira and Spencer. Dec 8, 2023 at 13:07
• Thank you for pointing out this issue, I didn't know about it before. Dec 10, 2023 at 4:00

The statement for divisors is true. Indeed, up to an étale cover, we may suppose that $$Z$$ is affine and that $$\mathscr L$$ is a lift of $$L$$ to $$X$$. Take $$\mathscr M$$ on $$X$$ very ample relative to $$Z$$, and set $$M = \mathscr M|_{X_0}$$. By Serre vanishing, there exists $$n \gg 0$$ such that $$H^1\big(X,\mathscr I \otimes \mathscr L^{\otimes i} \otimes \mathscr M^{\otimes n}\big) = 0$$ for $$i \in \{0,1\}$$, where $$\mathscr I$$ is the ideal sheaf of $$X_0 \hookrightarrow X$$. In particular, the maps $$H^0(X,\mathscr L^{\otimes i} \otimes \mathscr M^{\otimes n}) \to H^0(X_0,L^{\otimes i} \otimes M^{\otimes n})$$ are surjective for $$i \in \{0,1\}$$, so if $$D_0 \in \lvert L\rvert$$ and $$E_0 \in \lvert M^{\otimes n} \rvert$$, then both $$D_0 + E_0$$ and $$E_0$$ can be lifted to divisors on $$X$$.

The statement for effective divisors is not true, by the following (well-known) example.

Example. Let $$Z = \mathbf A^1$$ with variable $$t$$ and set $$Y = \mathbf P^1 \times Z$$, with its structure map $$f \colon Y \to Z$$. Consider the group $$\operatorname{Ext}^1_{\mathbf P^1}(\mathcal O_{\mathbf P^1}(1),\mathcal O_{\mathbf P^1}(-1)) \cong k$$, and let $$\alpha$$ be a generator. Then the element $$\alpha t \in \operatorname{Ext}^1_Y(\mathcal O_f(1),\mathcal O_f(-1))$$ gives a family of extensions $$0 \to \mathcal O_{\mathbf P^1}(-1) \to \mathscr E_t \to \mathcal O_{\mathbf P^1}(1) \to 0$$ over $$Z$$ that is non-split for $$t \neq 0$$ and split for $$t = 0$$. In particular, $$\mathscr E_t \cong \begin{cases} \mathcal O_{\mathbf P^1}(-1) \oplus \mathcal O_{\mathbf P^1}(1), & t = 0, \\ \mathcal O_{\mathbf P^1} \oplus \mathcal O_{\mathbf P^1}, & t \neq 0. \end{cases}$$ This gives a family $$X := \mathbf P_Y(\mathscr E) \to Z$$ of Hirzebruch surfaces such that $$X_t \cong \begin{cases} F_2, & t = 0, \\ F_0 = \mathbf P^1 \times \mathbf P^1, & t \neq 0. \end{cases}$$ (There are also other ways to construct this degeneration of $$F_n$$ to $$F_{n-2}$$.)

Then all fibres satisfy $$H^1(X_t,\mathcal O_{X_t}) = H^2(X_t,\mathcal O_{X_t}) = 0$$, and the Picard groups are all free of rank $$2$$, generated by $$g^*\mathcal O_f(1)$$ and $$\mathcal O_g(1)$$, where $$g \colon X \to Y$$ is the projection onto the constant family $$Y = \mathbf P^1 \times Z$$.

But $$X_0 \cong F_2$$ contains an irreducible effective divisor $$C$$ with $$C^2 = -2$$, whereas all effective divisors $$D$$ in $$X_t$$ for $$t \neq 0$$ have $$D^2 \geq 0$$. So in particular, $$C$$ cannot lift to an effective divisor on $$X$$ (up to an étale cover $$Z' \to Z$$ around $$0$$).

(Footnote: in the smooth case, the theorem of Totaro is a well-known result from deformation theory, going back many decades. The contribution is the specific generalisation to a certain mildly singular setup.)

• Thank you for your kind reply! May I ask under what circumstances does the statement about effective divisors hold? Dec 10, 2023 at 4:09
• There is no easy criterion. You need to know that $H^0(X,\mathscr L) \to H^0(X_0,\mathscr L|_{X_0})$ is surjective for every line bundle $\mathscr L$ on $X$. Without any hypothesis on $\mathscr L$, it seems pointless to try to obtain this surjectivity from vanishing theorems. Dec 10, 2023 at 16:44