Rational curves on \(\bar{M}_g\) and \(K3\) surfaces (Q2929646)

Let \(X\) be a nonsingular projective surface and let \(B\) be a nonsingular projective connected curve. A nonisotrivial fibration \(f:X\to B\) induces a nonconstant modular map \(\Psi_f: B\to \bar{M}_g\), where \(\bar{M}_g\) is the coarse moduli space of stable curves of genus \(g\). Let \(N_{\Psi_f}\) be the sheaf, which is defined as the cokernel of the map \(0\to T_B\to \mathrm{Ext}^1_f(\Omega^1_{X/B}, O_X)\). The map \(\kappa_f: T_B\to \mathrm{Ext}^1_f(\Omega^1_{X/B},O_X)\) is fiberwise, over the points \(b\) such that the fiber \(X(b)\) is reduced, the Kodaira-Spencer map of the family \(f\) at \(b\). If the fibration is stable and the image of \(\Psi_f\) is contained in the smooth locus of \(\bar{M}_g\), then \(\mathrm{Ext}^1_f(\Omega_{X/B}^1, O_X)\cong \Psi_f^*T_{\bar{M}_g}\), and \(\kappa_f\) is exactly the inclusion in the target sequence, and hence \(N_{\Psi_f}\) can be regarded as the normal sheaf to the map \(\Psi_f\). The main result of the first part of the paper is formulated in the following theorem:NEWLINENEWLINETheorem 1. Let \(f: X\to B\) be a nonisotrivial fibration with reduced fibers such that \(\mathrm{Ext}^1_f(\Omega^1_{X/B},O_X)\) is locally free and \(h^0(T_{X/F})=0\) for all smooth fibers of \(f\). Then \(N_{\Psi_f}\) is locally free.NEWLINENEWLINEIn the second part of the paper, \(K3\) type fibrations are introduced. These are fibrations \(f\) defined by a linear pencil \(\Lambda\subseteq |L|\) where \(L\) is an ample globally generated and primitive line bundle on a smooth \(K3\) surface. Let \({\mathcal{P}}_g\) be the stack parameterizing pairs \((S,C)\) such that \((S,L)\) is a smooth primitively polarized \(K3\) surface of genus \(g\) and \(C\in|L|\) is a stable curve. Let \(c_g: {\mathcal{P}}_g\to {\bar{\mathcal{M}}}_g\) be the projection, where \({\bar{\mathcal{M}}}_g\) is the moduli stack of stable curves of genus \(f\). Let \({\mathcal{K}}_g\subset {\mathcal{M}}_g\) be the image of \(c_g\) restricted to pairs \((S,C)\) such that \(C\) is smooth. The main result of the second part of the paper is to determine the structure of \(\Psi^*_f T_{\bar{M}_g}\). The geometrically interpreted results are formulated in the following theorem:NEWLINENEWLINETheorem 2. Let \(f\) be a general \(K3\) type fibration of genus \(g\geq 7\). Let \(a_g\) be the dimension of the general fiber of the morphism \(c_g\) and let \(b_g\) be the codimension of \({\mathcal{K}}_g\) in \({\mathcal{M}}_g\). Then NEWLINE\[NEWLINE\Phi^*_fT_{\bar{M}_g}\simeq O_{{\mathbb{P}}^1}(2)\oplus O_{{\mathbb{P}}^1}(1)^{\oplus g-1} \oplus O_{{\mathbb{P}}^1}(1)^{\oplus a_g}\oplus O_{{\mathbb{P}}^1}^{\oplus 2g-3-a_g-b_g} \oplus O_{{\mathbb{P}}^1}(-1)^{\oplus b_g}.NEWLINE\]