xxii INTRODUCTION

derivation" we refer to Vojta's preprint [131].) For any non-singular variety X over

K one can then define a projective system of non-singular varieties

... -

Jr(X)

-+

Jr~\X)

- ... -• J°{X) = X

called the (differential algebraic) jet spaces of X. It is easy to review the construc-

tion of these varieties. Assume, for simplicity, that X is affine given by

X = SpecK[T]/(F)

where T is a tuple of indeterminates 7\,..., T/v and F is a tuple of elements in if [T],

Then Jr{X) is, by definition, the spectrum

(0.20)

Jr{X)

:= Spec K[T, T', T",...,

T(r)]/(F,

8F,

52F,..., 5rF)

where

Tf

,T",... are new tuples of variables and

SF,52F,...

are defined by the for-

mulae

f)F

(0.21) (6F)(T,T') := F5(T) + £ — (T) • Tj,

(52F)(T,T',T")

:= (5F)* + £ ^ ( 7 \ T ' ) • Tj + £ ^ ( 7 , 7 " ) • T,

3 J 3 3'

etc., where F6, (JF)^,... are obtained from F, SF,... by applying J to the coefficients.

The polynomials 5F,

52F,...

are the iterated total derivatives of F. There is a

clear analogy between the arithmetic jet spaces in Equation 0.4 and the differential

algebraic jet spaces

Jr(X)

in Equation 0.20. On the other hand it can be shown

([14], p. 1443) that the if—points of the differential algebraic jet space

Jr{X)

can

be identified with the morphisms of if—schemes,

(0.22) Spec K[[T)]eTs/(Tr+1) - X,

where if[[T]]eTs is the ring of power series if [[F]] viewed as a if—algebra via the

map

eTS:K-K[\T}\, eT\x)

:= £ ^ T " .

n=0

(The maps in Equation 0.22 should be viewed as a twisted version of arcs in X

where the twist is given by 5. Recall that usual arcs of order r in X are defined as

morphisms of if—schemes, Spec

K[[T]]/(Fr+1)

— X, where K[[T]} is viewed as a

if—algebra via the inclusion if c if [[T]]; the set of all arcs of order r in X has a

natural structure of algebraic variety over if and is referred to as the arc space of

X of order r. Cf. [45]. Therefore the varieties

Jr(X)

can be viewed as a twisted

analogue of arc spaces. They identify with arc spaces in case X is defined over the

field C = {a G if 15a = 0} of constants of S. But for X not descending to C arc

spaces are different from the varieties Jr(X); cf. [14] for more on this.) Now there

is a clear analogy between the maps in Equation 0.22 and the jets belonging to

Jr(M)

in Equation 0.6. This makes differential algebraic jet spaces an analogue of

jet spaces of smooth manifolds (relative to a submersion). The analogies described

above can be summarized in the following diagram:

(0.23) (1) (2) (3),

(l)=jet spaces of smooth manifolds relative to a submersion; cf. Equation 0.6

(2)=differential algebraic jet spaces; cf. Equation 0.20

(3)=arithmetic jet spaces; cf. Equation 0.4