Rational curves on Mg and K3 surfaces

Original · EN

Let (S,L) be a smooth primitively polarized K3 surface of genus g and f:X → P¹ the fibration defined by a linear pencil in |L|. For f general and g ≥ 7, we work out the splitting type of the locally free sheaf Ψ*f TMg, where Ψf is the modular morphism associated to f. We show that this splitting type encodes the fundamental geometrical information attached to Mukai's projection map Pg → Mg, where Pg is the stack parameterizing pairs (S,C) with (S,L) as above and C ∈ |L| a stable curve. Moreover, we work out conditions on a fibration f to induce a modular morphism Ψf such that the normal sheaf NΨf is locally free.