# Kodaira dimension of a ruled surface

I am trying to solve 21.5.J in Vakil’s FOAG.

The exercise is about showing that $$X:=C\times_{\mathbb C} \mathbb P_{\mathbb C}^1$$ is neither Fano, Calabi-Yau, nor general type where $$C$$ is a smooth complex curve of genus $$>1$$.

Since $$\Omega_X = \text{pr}_1^\ast\Omega_C\oplus\text{pr}_2^\ast\Omega_{\mathbb P^1}$$, we have $$K_X = \text{pr}_1^\ast K_C\otimes \text{pr}_2^\ast K_{\mathbb P^1}$$ and I can show that $$X$$ is not Fano nor CY.

To show that $$X$$ is not general type, I need to show that the Kodaira dimension of $$X$$ is not $$2$$. The Kodaira dimension is defined as the least integer $$m$$ such that $$h^0(X,K_X^j)/j^m$$ is bounded for $$j>0$$. Some google search tells me $$H^0(X, K_X^j) = 0$$ for all $$j>0$$, so I tried to show this as follows.

Let $$s$$ be a section of $$K_X^j$$. Then, for any point $$p\in C$$, pulling back $$s$$ to $$p\times \mathbb P_{\mathbb C}^1$$, we have $$s=0$$ since we can see that $$K_X=\text{pr}_1^\ast K_C\otimes \text{pr}_2^\ast K_{\mathbb P^1}$$ pulls back to $$\mathcal O_{\mathbb P^1}\otimes K_{\mathbb P^1}=\mathcal O(-2)$$ by considering the compositions
$$p\times \mathbb P^1\to C\times\mathbb P^1 \to \mathbb P^1\\ p\times \mathbb P^1\to C\times\mathbb P^1\to C$$
Therefore, $$s$$ pulls back to zero for any choice of $$p\in C$$.

Considering affine open subsets $$\text{Spec} A, \text{Spec}B$$ of $$C,\mathbb P^1$$, respectively, the above translates to algebra as the following: given a $$A\otimes B$$ module $$M$$ and $$s\in M$$, we have $$s\in \mathfrak m M$$ for all maximal ideals $$\mathfrak m$$ of $$A$$.

(i) Can we conclude that $$s=0$$ from here easily in general? Or since $$M$$ corresponds to an invertible sheaf, should I only consider the form $$M=A\otimes \mathbb C[x]$$ (or possibly something like $$M=A\otimes \mathbb C(x)$$)?

(ii) Is there a simpler way to solve this problem?

(More General Question) (iii) Some google search also tells me that for two irreducible complex projective varieties $$X,Y$$, the Kodaira dimension of $$X\times Y$$ is the sum of those of $$X$$ and $$Y$$. How do I show this?

It’s not too hard to check $$\mathscr{K}_{C \times_{\mathbb{C}} \mathbb{P}_{\mathbb{C}}^1} \cong \mathscr{K}_{C} \boxtimes \mathscr{K}_{\mathbb{P}^1_{\mathbb{C}}}$$. Then, a Künneth formula computation (18.2.8) yields $$h^0(C \times_{\mathbb{C}} \mathbb{P}_{\mathbb{C}}^1, \mathscr{K}_{C}^{\otimes j} \boxtimes \mathscr{K}_{\mathbb{P}^1_{\mathbb{C}}}^{\otimes j}) = h^0(C, \mathscr{K}_C^{\otimes j})h^0(\mathbb{P}_{\mathbb{C}}^1, \mathscr{K}_{\mathbb{P}_\mathbb{C}^1}^{\otimes j}) = 0$$
for $$j > 0$$ because $$\mathscr{K}_{\mathbb{P}_\mathbb{C}^1} \cong \mathcal{O}_{\mathbb{P}_{\mathbb{C}}^1}(-2)$$.