I will use the notation of Liu's book which is also the standard one. If you don't feel comfortable with it please ask more information in the comments

Let $X$ be a scheme, and consider a Cartier divisor $D\in H^0(X,\mathcal K^\times_X/\mathcal O_X^\times)$ where $\mathcal K_X$ is the sheaf of stalks of meromorphic functions on $X$. Then, since $K^\times_X/\mathcal O_X^\times$ is obtained after a sheafification process, when can write $D$ (in a non unique way) as a family $D=\{(U_i,f_i)_i\}$ where:

  • $X=\bigcup U_i $ is an open cover

  • $f_i\in \mathcal K_X^\times(U_i)$ and $\frac{f_{i|U_i\cap U_j}}{f_{j|U_i\cap U_j}}\in\mathcal O_X^\times$

Now $D$ is said principal if it is in the image of the map $$H^0(X,\mathcal K_X^\times)\to H^0(X,\mathcal K^\times_X/\mathcal O_X^\times)$$ this is equivalent to say that $D=(X,f)$ with $f\in H^0(X,\mathcal K_X^\times)$. Namely, any local data defining $D$ glue to a meromorphic function $f$. I agree completely with this chacterization.

$D$ is said effective if it is in the image of the map: $$H^0(X,\mathcal K_X^\times\cap \mathcal O_X)\to H^0(X,\mathcal K^\times_X/\mathcal O_X^\times)\quad (\ast)$$
and Liu's book says that it is equivalent to say that:

$D=\{(U_i,f_i)\}$ with $f_i\in\mathcal O_X(U_i)$.

Why in this case don't we have that $D=(X,f)$ with $f\in H^0(X,\mathcal K_X^\times\cap \mathcal O_X)$? I mean, as in the case of principal divisors: being image of the morphism in $(\ast)$ should imply that the local data $D=\{(U_i,f_i)\}$ with $f_i\in\mathcal O_X(U_i)$ glue to an element $f\in H^0(X,\mathcal K_X^\times\cap \mathcal O_X)$. Where is my mistake?


Yes, I think this is not quite the correct definition of effectivity. As you say, with this definition, every effective divisor would be principal. The right definition should be something along the lines of requiring $D$ to be in the image of the map

$$H^0(X,(\mathscr{K}_X^\times\cap\mathscr{O}_X)/\mathscr{O}_X^\times)\to H^0(X,\mathscr{K}_X^\times/\mathscr{O}_X^\times)$$

where in the source one has a subsheaf of monoids of $\mathscr{K}_X^\times/\mathscr{O}_X^\times$.

Note that you can also just say that for any defining datum $\{(U_i,f_i)\}$ one has $f_i\in\mathscr{O}_X(U_i)$. If this holds for the given datum, and $\{(V_j,g_j)\}$ is another defining datum for the same divisor, then fixing $j$, we have, for any $i$, $g_j\vert_{U_i\cap V_j}=v_{ij}(f_i\vert_{U_i\cap V_j})$ for some unit $v_{ij}\in\mathscr{O}_X(U_i\cap V_j)^\times$. Thus $g_j\vert_{U_i\cap V_j}\in\mathscr{O}_X(U_i\cap V_j)$. This holds for all $i$, and as $i$ varies, the $U_i\cap V_j$ form an open covering of $V_j$. The sheaf axioms now imply that $g_j\in\mathscr{O}_X(V_j)$. So this datum is also "effective." I think maybe this is the clearest definition of effectivity.


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