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

TL;DR

This paper investigates the geometry of rational curves on the moduli space of curves and K3 surfaces, analyzing the splitting types of certain sheaves and their relation to Mukai's projection map for genus g ≥ 7.

Contribution

It explicitly computes the splitting type of the pullback of the tangent bundle of the moduli space via the modular morphism associated to a K3 surface fibration, revealing geometric information about Mukai's projection.

Findings

Splitting type encodes geometric data of Mukai's projection map.

Conditions identified for fibrations to induce modular morphisms with locally free normal sheaves.

Provides new insights into the structure of the moduli space of curves via K3 surface fibrations.

Abstract

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 $\Psi^{*}_f T_{{\overline{M}}_g}$, where $\Psi_f$ 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 $\Psi_f$ such that the normal sheaf $N_{\Psi_f}$ is locally free.