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.

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 through 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.