I've been following the works of Totaro, Pereira, and Bogomolov/Pirutka/Silberstein about algebraic varieties over complex numbers with families of disjoint divisors. The last one generalizes results of the previous two for the normal case. I'm interested in the smooth projective (complete) case. Roughly speaking the authors prove that if a variety $X$ has a collection $\{D_i\}_{i\in I}$ of disjoint codimension one connected subvarieties with $|I|\gg 0$, then there exists a surjective morphism $f:X\to C$ to a smooth projective curve $C$ such that each $D_i$ is contained in a fiber. These results fail if $|I|=2$. In order to have a greater understanding and intuition about the previous works:
I'm wondering about non-trivial families of examples of projective smooth varieties having disjoint prime divisors in particular of any dimension $\geq 2$. By non-trivial I mean varieties distinct from ruled surfaces, products or projective bundles over curves.
For example, in 1 was constructed a smooth threefold $X = (B_1\times B_2 \times \mathbb{P}^1)/(\mathbb{Z}/2\mathbb{Z})^2$ where $B_i$ are double covers over curves of genus $\geq 1$, and where the action on $\mathbb{P}^1$ is given by the group generated by $x\mapsto -x$ and $x\mapsto \frac{1}{x}$. In this way, $X$ has two disjoint divisors $D_1$ and $D_2$ each one given by the images of $B_1\times B_2\times 0$ and $B_1\times B_2\times 1$.
An example that I think that can be generalized is the following: Take $C$ a curve of genus $\geq 1$, and a vector bundle $\mathcal{E} = \mathcal{O}_C \otimes \mathcal{L}$ on $C$ such that $\deg \mathcal{L} = 0$ and $\mathcal{L} \neq \mathcal{O}_C$. Then the ruled surface $\mathbb{P}(\mathcal{E}) \to C$ has two disjoint sections $C_0\sim C_{\infty} \sim \mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ with $C_i^2 = 0$ [See AG-Hartshorne – Example.V.2.11.2]. I'm wondering if this generalizes as follows: Take a vector bundle $\mathcal{E}$ of rank $2$ on a variety $Z$ of dimension $d-1$, then $X = \mathbb{P}(\mathcal{E})$ is of dimension $d$. Is there a condition on $\mathcal{E}$ such that $X$ has disjoint sections $Z_1,…,Z_r \sim \mathcal{O}_X(1)$ (maybe r=d) ? What about $Z^d_i = 0$ ?
Finally, an extra question: There are smooth varieties of general type with disjoint divisors? I'm wondering this since the main examples that I bear in mind are projective bundles which clearly are not of general type by having rational curves.
Thanks for all.
Best Answer
Take a smooth curve $C$ of genus $\geq 2$ and consider a smooth divisor $D \subset C \times C$ which is $2$-divisible in $\operatorname{Pic}(C \times C)$ and which is transverse to both factors (by Bertini-type arguments one see that there is plenty of them).
Then there exists a double cover $f \colon X \to C \times C$, branched precisely over $D$. Since $C \times C$ is a smooth surface of general type, the same is true for $X$. Moreover, the pull-back of the two natural fibrations on $C \times C$ provide two distinct fibrations on $X$.
Summing up, $X$ is a surface of general type containing two distinct families of disjoint prime divisors, both parametrized by the curve $C$.