Let $C$ be a smooth, non-hyperelliptic curve of genus $3$ and $X:= \mathrm{Sym}^2{C}$ its symmetric square. Then $X$ is a smooth, minimal surface of general type with $p_g=q=3, \, K^2=6$.

Calling $\Omega_X$ the cotangent bundle of $X$, we see that $\Omega_X$ is globally generated and $h^0(X, \, \Omega_X)=3$; in fact, since $C$ is non-hyperelliptic, the Albanese map of $X$ is an embedding into a principally polarized abelian threefold.

For all $a \in C$, denote by $C_a$ the curve in $X$ given by $$C_a:=\{ a+x \; | \; x \in C \}.$$ Then $\mathscr{C}:=\{C_a \}$ is an irreducible, $1$-dimensional family of smooth curves on $X$, parametrized by $C$; every $C_a \in \mathscr{C}$ is isomorphic to $C$ and satisfies $(C_a)^2=1$.

Question 1.For some undirect reason, I know that the restriction of $\Omega_X$ to $C_a$ splits and contains a trivial summand. What is a direct proof of this fact? Is there any reference?

Consider now the following diagram: $\require{AMScd}$ \begin{CD} \mathbb{P}(\Omega_X) @>{\psi}>> \mathbb{P}H^0(X, \, \Omega_X) \simeq \mathbb{P}^2\\ @V{\pi}VV\\ X {} \end{CD}

where $\pi \colon \mathbb{P}(\Omega_X) \to X$ is the standard projection and $\psi \colon \mathbb{P}(\Omega_X) \to \mathbb{P}^2$ is induced by the complete linear system $|\mathcal{O}_{\mathbb{P}(\Omega_X)}(1)|$.

By the fact mentioned in Question 1, the map $\psi$, when restricted to the ruled surface $\pi^{-1}(C_a)$, contracts a section (corresponding to the trivial summand of the restriction of $\Omega_X$ to $C_a$) to a point $p_a \in \mathbb{P}^2$. Call $\Delta$ the locus described in $\mathbb{P}^2$ by the points $p_a$, when $C_a$ varies in $\mathscr{C}$.

Question 2.Does $\Delta$ coincide with the canonical image of $C$?

Finally note that, if $p_a \in \Delta$, then $\require{enclose} \enclose{horizontalstrike}{C_a = \pi(\psi^{-1}(p_a))}$ $C_a$ is a component of $\pi(\psi^{-1}(p_a))$.

Question 3.Take any point $q \in \mathbb{P}^2$ such that $q \notin \Delta$. What is $\pi(\psi^{-1}(q))$?