Sorry for a maybe stupid long question but I'm reading the paper "*Classical and minimal models of the moduli space of curves of genus two*" by Brendan Hassett and I'm not able to unravel a construction he does at a certain point with families of genus 2 curves. I'm going to a self study of moduli spaces and so I'm not able to ask anyone I know for these types of clarifications.

Let me be more precise:
given a family of genus 2 curves over a base scheme $S$ $$\pi:\mathcal{C} \rightarrow S$$ we can consider the relative dualizing sheaf $\omega_{\pi}$ (which in this case is globally generated), and the map $j:\mathcal{C} \rightarrow \mathbb{P}(\pi_{*}\omega_{\pi})$ such that $\psi:\mathbb{P}:=\mathbb{P}(\pi_{*}\omega_{\pi})\rightarrow S$ is a $\mathbb{P}^1-$bundle commuting with $j$ and $\pi$.
It is clear to me that $j_{*}\mathcal{O}_{\mathcal{C}}$ is a rank 2 vector bundle on $\mathbb{P}$ but in the paper it is stated that using the trace (??) we get the decomposition $j_{*}\mathcal{O}_{\mathcal{C}}=\mathcal{O}_{\mathbb{P}}\oplus \mathcal{L}$ where $\mathcal{L}$ has degree -3 on the fibers of $\psi$.

**Question 1**: Why can we say so easily that $j_{*}\mathcal{O}_{\mathcal{C}}$ splits in that way? I'm not able to figure it out properly.

Next he says that the $\mathcal{O}_{\mathbb{P}}$ algebra structure on $\mathcal{O}_{\mathcal{C}}$ is determined by an isomorphism $\mathcal{L} \otimes \mathcal{L} \rightarrow \mathcal{O}_{\mathbb{P}}$, i.e. by a non vanishing section of $\mathcal{L}^{-2}$ with zeros along the branch locus of $j$.

**Question 2**: What is the meaning of the previous assertion? How can a non vanishing section of a line bundle have zeros along a subscheme?

**Question 3**: I'm not able to see why $\omega_{\psi}=(\psi^{*}\det\pi_{*}\omega_{\pi})(-2)$

Finally, taking the above arguments for granted, I understand that the class of a branch divisor is $$c_1(\mathcal{O}_{\mathbb{P}}(6))-2\psi^{*}c_1(\det \pi_{*}\omega_{\pi})$$

After this he uses the computation above to show that an automorphism of a genus 2 curve $C$ induces a linear isomorphism on $\Gamma(C,\omega_{C})$ that preserves the sextic on $\mathbb{P}^1$ associated to the cover $j$ restricted to $C$ up to a scalar multiple. Finally he checks that this scalar is $(\det(M)^{-2}$ where $M \in GL(2)$ is acting naturally on the space of sextics of $\mathbb{P}^1$. More in particular $(\det(M)^{-2}M\cdot F=F$ if and only if $M$ is induced by an automorphism of $C$. 

**Question 4**: Is easy to see that the computation of the Chern classes above allows me to deduce that $\det(M)^{-2}$ is the right scalar that acts on $\Gamma(C,\omega_{C})$?

Thanks in advance for all the possible answers, suggestions and/or references.