9.3 Divisors

Let Z be a proper closed subscheme of T and W be its image in $\mathbb {P}$¹. As before W is defined as the vanishing locus of a homogeneous polynomial F(U, V). Let F = V^kF₁^{k₁ ...} F_r^n_r be a factorisation of F with F_i irreducible and distinct from each other and V. Clearly W is the disjoint union of closed subschemes W_i each defined by the vanishing of F_i(U, V)^k_i and the scheme W₀ defined by V^k = 0. As before, we write f_i(x) = F_i(U, V)/V^deg(F_i), where x = U/V; let Q_i denote the closed point in $\mathbb {A}$¹ defined by f_i and Q₀ be the point in $\mathbb {P}$¹ defined by V = 0. We can decompose Z into the components Z_i that lie over the component W_i of W. We can then classify Z_i according to the classification of the polynomials f_i as above. In cases (1), (2) and (3) above there is exactly one closed point that lies over Q_i, thus the schemes Z_i are ``thickenings'' of the corresponding closed points P<sub>i</sub>. In case (4) there are two closed points corresponding to the distinct roots; we denote these by P<sub>i, 1</sub> and P<sub>i, 2</sub>. Let P<sub>i, 1</sub> correspond to the solution y = g(x) or y<sup>2</sup> + a(x)y + b(x) = 0 in $\mathbb {F}$[x]/(f<sub>i</sub>(x)). By Hensel's lemma we can find g<sub>ki</sub>(x) in $\mathbb {F}$[x]/(f<sub>i</sub>(x)<sup>k<sub>i</sub></sup>) which is a ``lift'' of the solution g(x). Thus we have the closed subscheme Z_{i, 1} of Z_i defined by the solution y = g_ki(x). Similarly, we have Z_{i, 2} and it is clear that Z_i is the union of these two schemes. Thus each proper closed subscheme of T is the disjoint union of ``thickened'' closed points.

For any such closed subscheme Z of T we have a vector space scheme given by ($\mathbb {G}$_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 ($\mathbb {G}$_a)_T and the elements (P). Moreover, if we consider an element D of the form $\sum_{i}^{}$n_i(P_i) of the K-group then the number deg(D) = $\sum$n_ideg(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 $\sum_{i}^{}$n_i(P_i) where $\sum_{i}^{}$n_ideg(P_i) = 0. This group is denoted Pic⁰(T). An important theorem of Weil states that there is a group scheme J (called the Jacobian variety of T) such that Pic⁰(T) can be naturally identified with J($\mathbb {F}$). 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 Pic⁰(T) of divisors of degree 0, it enough to work modulo ($\infty$) which is of degree 1, since any divisor can be converted to one of degree 0 by subtracting a suitable multiple of ($\infty$). Thus, we see that this group is generated by the elements [P] = (P) - deg(P)($\infty$). For a divisor D of degree d we introduce the notation [D] = D - d ($\infty$) to denote the corresponding element in Pic⁰(T).