# Divisor Class group for (reducible) varieties

Good Afternoon,

I'm running into something confusing in a book, and am not sure if its a definition, or if I am missing something significant.

Consider the following 3 copies of $\mathbb{P}^1$, $C_1,C_2,C_3$ sitting inside $\mathbb{P}^2$ given by the equation $X_i=0$ (here, I'm writing $\mathbb{P}^2$ as a collection of triples $[X_1,X_2,X_3]$.

In section 3.4 of fulton's book on toric varieties, he writes $Cl(C_1 \cup C_2 \cup C_3) \cong \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$, where $Cl$ denotes the group of Weil divisors modulo linear equivalence. (This isn't exactly what he writes but rather a special case of that). I'm wondering what exactly is meant by the class group of a reducible variety - to talk about linear equivalence of Weil divisors, we need rational functions, and rational functions don't make any sense on a reducible variety (since the coordinate ring isn't a domain).

How do we define $Cl(C_1 \cup C_2 \cup C_3)$? Is it just defined as $Cl(C_1) \oplus Cl(C_2) \oplus Cl(C_3)$?

Thanks, Robert

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Does he mean the ring of fractions? i.e. invert everything that isn't a zero divisor? (sorry about thinking in terms of rings rather than schemes) – Hurkyl Sep 15 '11 at 21:51