For the special case of divisors (*i.e.*, (*X*)) much of the picture is
unchanged from that for curves. To begin with, as we saw above we have
an isomorphism
(*X*) (*X*).

When *X* is projective one can define an equivalence relation on (*X*)
as follows. Let *C* be any smooth curve and
*D* *X*×*C* be a divisor
which does not contain any fibre of
*X*×*C**C*. For any pair of
points *p*,*q* on *C* the divisor
*D* *X*×{*p*} - *D* *X*×{*q*}
can be considered as a divisor on *X*, which is said to be *algebraically equivalent* to 0. The quotient of (*X*) by this
equivalence relation is a finitely generated Abelian group--the
*Néron-Severi group* (*X*). This gives us a generalisation of the
degree homomorphism for curves, namely the quotient map
*cl* : (*X*)(*X*).

Upto torsion this equivalence relation can also be defined using
intersection theory. We define a divisor *D* to be *numerically
equivalent* to zero if the intersection number
(*D*^{ . }*C*) = 0 for every
curve *C* contained in *X*. Then one knows that some multiple of *D* is in
fact algebraically equivalent to 0. Conversely, if a divisor *D* is
algebraically equivalent to 0 then it is also numerically equivalent to 0.
In case the ground field is *C* then we can also identify algebraic
equivalence with homological equivalence: *i.e.*, a divisor is algebraically
equivalent to 0 precisely if it lies in the kernel of the cycle class map
(*X*)(*X*,*Z*).

In particular,
deg(*ch*(*L*)^{ . }*td* (*X*))_{n} depends only on the class *cl* (*L*)
of *L* in (*X*). The Grothendieck-Hirzebruch-Riemann-Roch theorem then
actually gives a method for computing (*X*, *L*) in terms of the class
*cl* (*L*) in (*X*). However, the exact formula
dim_{k}(*X*,_{X}(*D*)) = deg *D* + 1 - *g*, valid for divisors of large degree
on a curve of genus *g*, has only a partial generalisation to higher
dimensions: if *D* is an *ample* divisor, then the
Grothendieck-Riemann-Roch theorem gives a formula for
dim_{k}(*X*,_{X}(*mD*)) for large *m*, since
(*X*,_{X}(*mD*)) = 0 for
*i* > 0 (by Serre's vanishing theorem), and so

dim_{k}(*X*,_{X}(*mD*)) = (_{X}(*mD*)) = *deg*(*ch*(*L*)*td* (*X*))_{n} .

For effective divisors
The collection of all effective divisors on *X* corresponding to a fixed
class *c* in (*X*) form a projective scheme *Hilb*_{c}(*X*). Also, in
analogy with the case for curves, the kernel *A*^{1}(*X*) of the
morphism
(*X*)(*X*)
is also naturally isomorphic to (the group of *k*-rational points
of) an Abelian variety, the *Picard variety* (*X*). Fixing one
divisor *C* in the class *c* we obtain a natural morphism
*Hilb*_{c}(*X*)(*X*), the *Abel-Jacobi morphism*. The fibres of this
morphism precisely consist of effective divisors corresponding to a
fixed class in (*X*). As in the case of curves, one can show that for
a ``sufficiently large'' multiple of an ample class *c* the morphism
*Hilb*_{m . c}(*X*) is surjective.