We give several results concerning the connected component
of the automorphism
scheme of a proper variety
over a field, such as its behavior with respect to birational modifications, normalization,
restrictions to closed subschemes and deformations. Then, we apply our results to
study the automorphism scheme of not necessarily Jacobian elliptic surfaces
over
algebraically closed fields, generalizing work of Rudakov and Shafarevich, while
giving counterexamples to some of their statements. We bound the dimension
of the space of global vector fields on an elliptic surface
if the generic
fiber of
is
ordinary or if
admits no multiple fibers, and show that, without these assumptions, the number
can be arbitrarily large
for any base curve
and any field of positive characteristic. If
is not isotrivial, we prove
that
and give a bound
on
in terms of the genus
of
and the multiplicity
of multiple fibers of
.
As a corollary, we reprove the nonexistence of global vector fields on K3
surfaces and calculate the connected component of the automorphism
scheme of a generic supersingular Enriques surface in characteristic
.
Finally, we present additional results on horizontal and vertical group
scheme actions on elliptic surfaces which can be applied to determine
explicitly in many concrete cases.
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