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I'm reading the paper "A classification of two-dimensional integrable mappings and rational elliptic surfaces". I have two questions:

Let $X$ be a generalized Halphen surface (e.g. an elliptic surface), $D = -K_X$, $\omega$ a 2-form on $X$ with $Div(\omega) = -D_{red}$ and let $Q$ the root lattice defined as the orthogonal complement of $D$ with respect to intersection. On page 5 the period mapping from $Q$ to $\mathbb{C}$ is defined as $\chi(\alpha) := \int_\alpha \omega $. Later on page 12 the authors use the 2-form $\omega = \frac{1}{2\pi i}\frac{dx\wedge dy}{xy}$ to compute the period mapping. I do not understand why it is possible to use this 2-form (on $\mathbb{P} \times \mathbb{P}$, defining the divisor $D'_{red}=-H_x-H_y$ on $\mathbb{P} \times \mathbb{P}$) instead of a 2-form on $X$ defining $D_{red}$. Later in the paper (p.17 almost at the bottom) the authors even say "The anti-canonical divisor $xy=0$" which confuses me even more. I guess this has something to do with the fact that those forms behave well under blow ups (Since $X$ is a blow up of $\mathbb{P} \times \mathbb{P}$)?

On p. 7 the mapping $\Phi :\mathbb{P} \times \mathbb{P}\rightarrow\mathbb{P} \times \mathbb{P}$ is defined via $\Phi(x,y) = (y,-x\frac{(y-a)(y-1/a)}{(y+a)(y+1/a)})$. The authors then describe how the "induced bundle mapping $\Phi_*$" acts on $Pic(X)$. What is this $\Phi_*$ and how do I calculate it? $\Phi$ induces an automorphism on $X$ and thus an automorphism on $Pic(X)$, but that's obviously not it (since this would act differently).