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

Abstract

Let $(S,L)$ be a smooth primitively polarized K3 surface of genus $g$ and $f:X\to {\mathbb{P}}^1$ the fibration defined by a linear pencil in $|L|$. For $f$ general and $g\ge 7$, we work out the splitting type of the locally free sheaf ${\Psi }_f^{\ast }{T}_{{\overline{M}}_g}$, 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 ${\mathcal{P}}_g\to {\overline{\mathcal{M}}}_g$, 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.

published version, revisions in the exposition, minor mistakes corrected

Cited by in corpus (1)

• Maximal variation of curves on K3 surfaces