Relative effective Cartier divisor explained
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
of
X and nonzerodivisors
such that the intersection
is given by the equation
(called local equations) and
is flat over
R and such that they are compatible.
An effective Cartier divisor as the zero-locus of a section of a line bundle
Let L be a line bundle on X and s a section of it such that
(in other words,
s is a
-
regular element for any open subset
U.)
Choose some open cover
of
X such that
. For each
i, through the isomorphisms, the restriction
corresponds to a nonzerodivisor
of
. 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
is a unit element. 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
such that
s followed by
is the identity. Then
may be constructed as the fiber product of
s and the zero-section embedding
.
Finally, when
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
of
0\toI(D)\tol{O}X\tol{O}D\to0
gives the exact sequence
0\tol{O}X\toI(D)-1\toI(D)-1 ⊗ l{O}D\to0
In particular, 1 in
can be identified with a section in
, which we denote by
.
Now we can repeat the early argument with
. Since
D is an effective Cartier divisor,
D is locally of the form
on
for some nonzerodivisor
f in
A. The trivialization
L|U=Af-1\overset{\sim}\toA
is given by multiplication by
f; in particular, 1 corresponds to
f. Hence, the zero-locus of
is
D.
Properties
- If D and D' are effective Cartier divisors, then the sum
is the effective Cartier divisor defined locally as
if
f,
g give local equations for
D and
D' .
- If D is an effective Cartier divisor and
is a ring homomorphism, then
is an effective Cartier divisor in
.
- If D is an effective Cartier divisor and
a flat morphism over
R, then
is an effective Cartier divisor in
X' with the ideal sheaf
.
Examples
Effective Cartier divisors on a relative curve
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.) Then
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
is proper over
R and
\deg(D+D')=\deg(D)+\deg(D')
. Let
be a finite flat morphism. Then
. On the other hand, a base change does not change degree:
.
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.
Weil divisors associated to effective Cartier divisors
Given an effective Cartier divisor D, there are two equivalent ways to associate Weil divisor
to it.
References
- Book: Katz . Nicholas M . Nick Katz . Mazur . Barry . Barry Mazur . Arithmetic Moduli of Elliptic Curves . . 1985 . 0-691-08352-5 .