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Relative effective Cartier divisor

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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

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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 gives the exact sequence

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 is given by multiplication by f; in particular, 1 corresponds to f. Hence, the zero-locus of is D.

Properties

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  • 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

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Hyperplane bundle

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Effective Cartier divisors on a relative curve

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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 . Let be a finite flat morphism. Then .[1] On the other hand, a base change does not change degree: .[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

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Given an effective Cartier divisor D, there are two equivalent ways to associate Weil divisor to it.

Notes

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  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

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