Relative effective Cartier divisor

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 ${\displaystyle I(D)}$ 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 ${\displaystyle U_{i}=\operatorname {Spec} A_{i}}$ of X and nonzerodivisors ${\displaystyle f_{i}\in A_{i}}$ such that the intersection ${\displaystyle D\cap U_{i}}$ is given by the equation ${\displaystyle f_{i}=0}$ (called local equations) and ${\displaystyle A/f_{i}A}$ 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 ${\displaystyle s:{\mathcal {O}}_{X}\hookrightarrow L}$ (in other words, s is a ${\displaystyle {\mathcal {O}}_{X}(U)}$-regular element for any open subset U.)

Choose some open cover ${\displaystyle \{U_{i}\}}$ of X such that ${\displaystyle L|_{U_{i}}\simeq {\mathcal {O}}_{X}|_{U_{i}}}$. For each i, through the isomorphisms, the restriction ${\displaystyle s|_{U_{i}}}$ corresponds to a nonzerodivisor ${\displaystyle f_{i}}$ of ${\displaystyle {\mathcal {O}}_{X}(U_{i})}$. Now, define the closed subscheme ${\displaystyle \{s=0\}}$ of X (called the zero-locus of the sections) by

${\displaystyle \{s=0\}\cap U_{i}=\{f_{i}=0\},}$

where the right-hand side means the closed subscheme of ${\displaystyle U_{i}}$ given by the ideal sheaf generated by ${\displaystyle f_{i}}$. This is well-defined (i.e., they agree on the overlaps) since ${\displaystyle f_{i}/f_{j}|_{U_{i}\cap U_{j}}}$ is a unit element. For the same reason, the closed subscheme ${\displaystyle \{s=0\}}$ 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 ${\displaystyle s:X\to L}$ such that s followed by ${\displaystyle L\to X}$ is the identity. Then ${\displaystyle \{s=0\}}$ may be constructed as the fiber product of s and the zero-section embedding ${\displaystyle s_{0}:X\to L}$.

Finally, when ${\displaystyle \{s=0\}}$ 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 ${\displaystyle I(D)}$ denote the ideal sheaf of D. Because of locally-freeness, taking ${\displaystyle I(D)^{-1}\otimes _{{\mathcal {O}}_{X}}-}$ of ${\displaystyle 0\to I(D)\to {\mathcal {O}}_{X}\to {\mathcal {O}}_{D}\to 0}$ gives the exact sequence

${\displaystyle 0\to {\mathcal {O}}_{X}\to I(D)^{-1}\to I(D)^{-1}\otimes {\mathcal {O}}_{D}\to 0}$

In particular, 1 in ${\displaystyle \Gamma (X,{\mathcal {O}}_{X})}$ can be identified with a section in ${\displaystyle \Gamma (X,I(D)^{-1})}$, which we denote by ${\displaystyle s_{D}}$.

Now we can repeat the early argument with ${\displaystyle L=I(D)^{-1}}$. Since D is an effective Cartier divisor, D is locally of the form ${\displaystyle \{f=0\}}$ on ${\displaystyle U=\operatorname {Spec} (A)}$ for some nonzerodivisor f in A. The trivialization ${\displaystyle L|_{U}=Af^{-1}{\overset {\sim }{\to }}A}$ is given by multiplication by f; in particular, 1 corresponds to f. Hence, the zero-locus of ${\displaystyle s_{D}}$ is D.

Properties

• If D and D' are effective Cartier divisors, then the sum ${\displaystyle D+D'}$ is the effective Cartier divisor defined locally as ${\displaystyle fg=0}$ if f, g give local equations for D and D' .
• If D is an effective Cartier divisor and ${\displaystyle R\to R'}$ is a ring homomorphism, then ${\displaystyle D\times _{R}R'}$ is an effective Cartier divisor in ${\displaystyle X\times _{R}R'}$.
• If D is an effective Cartier divisor and ${\displaystyle f:X'\to X}$ a flat morphism over R, then ${\displaystyle D'=D\times _{X}X'}$ is an effective Cartier divisor in X' with the ideal sheaf ${\displaystyle I(D')=f^{*}(I(D))}$.

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 ${\displaystyle \Gamma (D,{\mathcal {O}}_{D})}$ is a locally free R-module of finite rank. This rank is called the degree of D and is denoted by ${\displaystyle \deg D}$. It is a locally constant function on ${\displaystyle \operatorname {Spec} R}$. If D and D' are proper effective Cartier divisors, then ${\displaystyle D+D'}$ is proper over R and ${\displaystyle \deg(D+D')=\deg(D)+\deg(D')}$. Let ${\displaystyle f:X'\to X}$ be a finite flat morphism. Then ${\displaystyle \deg(f^{*}D)=\deg(f)\deg(D)}$. ^[1] On the other hand, a base change does not change degree: ${\displaystyle \deg(D\times _{R}R')=\deg(D)}$. ^[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]

Weil divisors associated to effective Cartier divisors

Given an effective Cartier divisor D, there are two equivalent ways to associate Weil divisor ${\displaystyle [D]}$ to it.

Notes

1. Katz & Mazur 1985 , Lemma 1.2.8.
2. Katz & Mazur 1985 , Lemma 1.2.9.
3. Katz & Mazur 1985 , Lemma 1.2.3.

References

• Katz, Nicholas M; Mazur, Barry (1985). Arithmetic Moduli of Elliptic Curves. Princeton University Press. ISBN   0-691-08352-5.