Why is $\kappa_1$ ample?

Let $\pi:C_g=\overline M_{g,1}\to \overline M_g$ be the "universal curve". I recall that on $C_g$ there is the line bundle $L$ defined by $L_{(C;P)}=\Omega^1_{C,P}$, giving rise to $\psi:=c_1(L)\in A^1C_g$, and hence to $\kappa_1:=\pi_\ast\psi\in A^1\overline M_g$.

My questions: what does it mean, exactly, and how does one prove, that $\kappa_1$ is ample on $\overline M_g$? Is the same true for the restriction of $\kappa_1$ to $M_g$?

In fact, I only know that something living in the Picard group (invertible sheaves) can be ample. But here the truth is that I do not know (and cannot find in the literature) the relation between Pic $\overline M_g$ and $A^1 \overline M_g$ (I think of both as with $\mathbb Q$ coefficients). For instance, what is known about the morphism Pic $M_g\to A^1 \overline M_g$?

Thank you!

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