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

I(D)

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

Ui=\operatorname{Spec}Ai

of X and nonzerodivisors

fi\inAi

such that the intersection

D\capUi

is given by the equation

fi=0

(called local equations) and

A/fiA

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

s:l{O}X\hookrightarrowL

(in other words, s is a

l{O}X(U)

-regular element for any open subset U.)

Choose some open cover

\{Ui\}

of X such that
L|
Ui

\simeql{O}X|

Ui
. For each i, through the isomorphisms, the restriction
s|
Ui
corresponds to a nonzerodivisor

fi

of

l{O}X(Ui)

. Now, define the closed subscheme

\{s=0\}

of X (called the zero-locus of the section s) by

\{s=0\}\capUi=\{fi=0\},

where the right-hand side means the closed subscheme of

Ui

given by the ideal sheaf generated by

fi

. This is well-defined (i.e., they agree on the overlaps) since

fi/fj|

Ui\capUj
is a unit element. For the same reason, the closed subscheme

\{s=0\}

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

s:X\toL

such that s followed by

L\toX

is the identity. Then

\{s=0\}

may be constructed as the fiber product of s and the zero-section embedding

s0:X\toL

.

Finally, when

\{s=0\}

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

I(D)

denote the ideal sheaf of D. Because of locally-freeness, taking

I(D)-1l{OX}-

of

0\toI(D)\tol{O}X\tol{O}D\to0

gives the exact sequence

0\tol{O}X\toI(D)-1\toI(D)-1l{O}D\to0

In particular, 1 in

\Gamma(X,l{O}X)

can be identified with a section in

\Gamma(X,I(D)-1)

, which we denote by

sD

.

Now we can repeat the early argument with

L=I(D)-1

. Since D is an effective Cartier divisor, D is locally of the form

\{f=0\}

on

U=\operatorname{Spec}(A)

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

sD

is D.

Properties

D+D'

is the effective Cartier divisor defined locally as

fg=0

if f, g give local equations for D and D' .

R\toR'

is a ring homomorphism, then

D x RR'

is an effective Cartier divisor in

X x RR'

.

f:X'\toX

a flat morphism over R, then

D'=D x XX'

is an effective Cartier divisor in X' with the ideal sheaf

I(D')=f*(I(D))

.

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

\Gamma(D,l{O}D)

is a locally free R-module of finite rank. This rank is called the degree of D and is denoted by

\degD

. It is a locally constant function on

\operatorname{Spec}R

. If D and D' are proper effective Cartier divisors, then

D+D'

is proper over R and

\deg(D+D')=\deg(D)+\deg(D')

. Let

f:X'\toX

be a finite flat morphism. Then

\deg(f*D)=\deg(f)\deg(D)

. On the other hand, a base change does not change degree:

\deg(D x RR')=\deg(D)

.

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

[D]

to it.

References