Rational Curves on and K3 Surfaces.

Let (S,L) be a smooth primitively polarized K3 surface of genus g and be the fibration defined by a linear pencil in <INNOPIPE>L<INNOPIPE>. For f general and g≥7, we work out the splitting type of the locally free sheaf , 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 , where is the stack parameterizing pairs (S,C) with (S,L) as above and C∈<INNOPIPE>L<INNOPIPE> 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. [ABSTRACT FROM AUTHOR]