For any such closed subscheme *Z* of *T* we have a vector space scheme
given by
(_{a})_{Z} extended by zero on the rest of *T*. We denote
this vector space scheme by (*P*) when *Z* is the subscheme associated
to the a closed point *P*. The vector space scheme associated with the
``thickened'' closed points is equivalent, in the *K*-group, to *n*(*P*)
for some integer *n*. This can be shown by a ``composition series
argument''. A similar Jordan-Hölder composition series can be used
to show that the *K*-group of *T* is generated by
(_{a})_{T} and the
elements (*P*). Moreover, if we consider an element *D* of the form
*n*_{i}(*P*_{i}) of the *K*-group then the number
deg(*D*) = *n*_{i}deg(*P*_{i}) can be shown to be well-defined (independent of the
representation of *D*). Thus the important group becomes the group of
``divisors of degree 0'' of the subgroup of the *K*-group consisting
of elements of the form
*n*_{i}(*P*_{i}) where
*n*_{i}deg(*P*_{i}) = 0. This group is denoted
Pic^{0}(*T*). An important
theorem of Weil states that there is a group scheme *J* (called the
Jacobian variety of *T*) such that
Pic^{0}(*T*) can be naturally
identified with
*J*(). There is also a natural analogy of this with
the divisor class group for quadratic number fields that we will
consider in the next subsection.

To compute the group
Pic^{0}(*T*) of divisors of degree 0, it enough to
work modulo () which is of degree 1, since any divisor can be
converted to one of degree 0 by subtracting a suitable multiple of
(). Thus, we see that this group is generated by the elements
[*P*] = (*P*) - deg(*P*)(). For a divisor *D* of degree *d* we
introduce the notation
[*D*] = *D* - *d* () to denote the corresponding
element in
Pic^{0}(*T*).