Rational curves on \bar{M}_g and K3 surfaces

Let $(S,L)$ be a smooth primitively polarized K3 surface of genus $g$ and $f:X \rightarrow \mathbb{P}^1$ the fibration defined by a linear pencil in $|L|$. For $f$ general and $g \geq 7$, we work out the splitting type of the locally free sheaf $Ψ^{*}_f T_{\overline{M}_g}$, where $Ψ_f$ is the modular morphism associated to $f$. We show that this splitting type encodes the fundamental geometrical information attached to Mukai's projection map $\mathcal{P}_g \rightarrow \overline{\mathcal{M}}_g$, where $\mathcal{P}_g$ is the stack parameterizing pairs $(S,C)$ with $(S,L)$ as above and $C \in |L|$ a stable curve. Moreover, we work out conditions on a fibration $f$ to induce a modular morphism $Ψ_f$ such that the normal sheaf $N_{Ψ_f}$ is locally free.

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