It is important to note that the notion of isotriviality is up to isomorphism.
In particular it can happen than an affine open subset of an isotrivial
variety is non-isotrivial. The notions of birationally isotrivial and
weakly isotrivial, on the other hand, are birational notions.
If K is the function field of a variety T defined over a finite field
and 
is a variety, then X corresponds to a family 
with generic fibre X. It is well-known that, in the projective case,
X is isotrivial if and only
if 
is generically constant, i.e., there is a non-empty open
subset U of T such that the fibres over all points of U are isomorphic.
(See e.g. [B1] Ch. 1, lemma (1.3)).
In characteristic zero, there is another equivalent notion of isotriviality, that of infinitesimally isotrivial. Although the equivalence of the notions does not hold in characteristic p, it is still very useful to consider infinitesimal isotriviality. To do that we first must define the Kodaira-Spencer class, in a more general setting than usual, following Katz [K].
Let 
be a smooth projective variety over K, D a divisor with normal
crossings on 
and 
. Define a sheaf 
on
of vector fields tangent to D, which is a subsheaf of the tangent
sheaf of 
. Equivalently, 
is the sheaf of K-derivations of
that preserve the ideal sheaf of D. If 
is a derivation on
K we define the Kodaira-Spencer class 
in 
as
follows. Let 
be an open cover of 
fine enough so that we
can lift 
to derivations 
of 
,
preserving the ideal of
. The 1-cocycle 
then defines the class 
.
We define X to be infinitesimally isotrivial if 
for
all 
.
Let X be a variety over a
function field K of transcendence degree 1 over a finite field.
it is convenient to study whether or not X is defined over
using derivations: let t be a separating variable in K, and let
C be an affine model of K over which 
is a regular
vector field. Let 
be a model of X, proper over C. Then
that X can be defined over 
if and only if the
derivation 
lifts to a vector field 
over the inverse
image in U of some open subset of C, satisfying 
. See [V2],
lemma 1 or [Og], lemma 3.5. Note that, obviously, 
lifts if and only if
and, as shown in loc. cit., the condition 
is
automatically satisfied if 
.
Example:
Let 
where E is a supersingular elliptic curve with 
.
Choose a K-rational subspace of the Lie algebra of A not
defined over 
and corresponding to a height one group-scheme G.
Take the quotient 
, which is a non-isotrivial abelian variety over K with
good reduction everywhere. If 
, then we can
base change by any 
, and obtain infinitely many families
of such varieties. (See [Sz]).
For any projective smooth variety X, there is an obvious map
In the case of abelian varieties, this map is an isomorphism and we can talk about the rank of the Kodaira-Spencer class as a linear map between vector spaces. In particular, we talk about the Kodaira-Spencer class having maximal rank, in this context. To justify the assertion, made in section 3, that abelian varieties with sufficiently general moduli satisfy the hypotheses of Theorem 5, we have the following result, proved in [V5].
Acknowledgements: The author would like to thank D. Abramovich for many helpful comments and the NSF (grant DMS-9301157) and the Alfred P. Sloan Foundation for financial support.
