Abstract
Let
(S,L)
be a smooth primitively polarized K3 surface of genus
g
and
f:X→
P
1
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
T
M
¯
¯
¯
¯
¯
¯
g
, where
Ψ
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
P
g
→
M
¯
¯
¯
¯
¯
¯
g
, where
P
g
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
Ψ
f
such that the normal sheaf
N
Ψ
f
is locally free.