Some Technical Points

So, I’ve been a bad math blogger.  I’ve been identifying a bunch of different classes of things that we can really only identify on nice algebraic schemes.  Things like smooth varieties (where I’ve grabbed all of my examples).  There are actually three different classes of “codimension one gadgets” that I’ve been treating as interchangeable.  So today I’m going to talk about them, and why they aren’t quite the same thing.

The first class is the most basic.  We say that a Weil Divisor on X is an element of Z_{n-1}X, where \dim X=n.  That’s it.

Next up: Cartier divisors.  These are collections (U_\alpha,f_\alpha) such that the U_\alpha cover X, f_\alpha is a nonzero function on U_\alpha, and on the overlaps the ratios are nowhere vanishing.  So if V is a subvariety of X of codimension one, we define \mathrm{ord}_V(D) to be \mathrm{org}_V(f_\alpha) for any local equation on an open set containing some point of V.

Now, every Cartier divisor defines a Weil divisor by [D]=\sum \mathrm{ord}_V(D)[V].  This is NOT an isomorphism in general, however.  It just gives a map from \mathrm{Div}(X)\to Z_{n-1}(X), where the first is the group of Cartier divisors.

Example: Let X be a quadric cone in \mathbb{C}^3, and \ell a line through the singular point.  Then \ell is a Weil divisor (it’s definitely a cycle of codimension 1) but it is NOT a Cartier divisor.

The relation between Cartier and Weil divisors is part of the basic theory of algebraic geometry, as is the relation between Cartier divisors and line bundles.

The last gadget is the Pseudo-Divisor.  These are triples (L,Z,s) where L is a line bundle, Z is a closed subset of X, and s is a nowhere vanishing section on L on X\setminus Z.  We call the terms the line bundle, the support and the section.  Two such objects are the same if they have the same support and there is an isomorphism of line bundles taking one section to the other.

Any Cartier divisor gives a Pseudo-Divisor by D\mapsto (\mathscr{O}(D),|D|,s_D), and we say that D represents this Pseudo-Divisor.  Now, note that we can have Z larger than |D|.  So if Z=X, then all linearly equivalent Cartier divisors represent the same Pseudo-divisor.

Now, any Pseudo-divisor D gives a Weil divisor in A_{n-1}(|D|) on its support by taking a Cartier divisor that represents D (there always is one) and taking its associated Weil divisor.

Now, all these maps actually give us objects in A_{n-1} of things, not just in Z_{n-1}.  So everything is a Weil divisor, and some things are Cartier divisors or Pseudo-divisors, which are just rigidified Cartier divisors.  So going forward, we’re going to have all of these objects floating around, but most of the time, it will be an exercise to distinguish them (when it’s not terribly important, we’ll just refer to a “divisor”).

About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
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