In algebraic geometry, a relative effective Cartier divisor is roughly a family of effective Cartier divisors.
Precisely, an effective Cartier divisor in a scheme X over a ring R is a closed subscheme D of X that (1) is flat over R and (2) the ideal sheaf
of D is locally free of rank one (i.e., invertible sheaf).
Equivalently, a closed subscheme D of X is an effective Cartier divisor if there is an open affine cover
(called local equations) and
Let L be a line bundle on X and s a section of it such that
-regular element for any open subset U.)
Choose some open cover
Now, define the closed subscheme
of X (called the zero-locus of the section s) by where the right-hand side means the closed subscheme of
given by the ideal sheaf generated by
This is well-defined (i.e., they agree on the overlaps) since
For the same reason, the closed subscheme
is independent of the choice of local trivializations.
Equivalently, the zero locus of s can be constructed as a fiber of a morphism; namely, viewing L as the total space of it, the section s is a X-morphism of L: a morphism
may be constructed as the fiber product of s and the zero-section embedding
is flat over the base scheme S, it is an effective Cartier divisor on X over S. Furthermore, this construction exhausts all effective Cartier divisors on X as follows.
Let D be an effective Cartier divisor and
denote the ideal sheaf of D. Because of locally-freeness, taking
Now we can repeat the early argument with
Since D is an effective Cartier divisor, D is locally of the form
is given by multiplication by f; in particular, 1 corresponds to f. Hence, the zero-locus of
is D. From now on suppose X is a smooth curve (still over R).
Let D be an effective Cartier divisor in X and assume it is proper over R (which is immediate if X is proper.)
is a locally free R-module of finite rank.
This rank is called the degree of D and is denoted by
It is a locally constant function on
If D and D' are proper effective Cartier divisors, then
be a finite flat morphism.
[2] A closed subscheme D of X is finite, flat and of finite presentation if and only if it is an effective Cartier divisor that is proper over R.[3] Given an effective Cartier divisor D, there are two equivalent ways to associate Weil divisor