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Consider a smooth curve $C$ with a base point free linear system $|D|$ on it.

Let $\pi_1: C \times \mathbb{P}^2 \to C$ and $\pi_2: C \times \mathbb{P}^2 \to \mathbb{P}^2$ be the projections.

*What do elements of the linear system $|\pi_1^*\mathcal{O}_C \otimes \pi_2^*\mathcal{O}_{\mathbb{P}^2}(3)|$ look like geometrically? These are surfaces, but I would like to know how to interpret this complicated tensor product geometrically.

*The adjunction formula should imply that $K_S = \pi_1^*(K_C + D)$. I don't see how...

*If $\deg(D) > 2 - 2 \cdot g(C)$, then $\deg(K_C + D) > 0$. This should imply that $\kappa(S) \geq 1$.

Is the following attempt at a proof correct? If we have $\deg(K_C + D) > 0$, then $K_C + D$ is an ample divisor. Hence some multiple $\ell(K_C + D)$ is very ample and hence linearly equivalent to a non-zero effective divisor. Hence $\ell K_S = \pi_1^*(\ell(K_C + D))$ is linearly equivalent to the pullback of an effective divisor, therefore itself linearly equivalent to an non-zero effective divisor (not sure about this), so that the linear system $|\ell K_S|$ is non-empty and at least one-dimensional. This means by definition of Kodaira dimension that $\kappa(S) \geq 1$.

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