# Philosophy behind effective Cartier Divisors

I learned that effective Cartier Divisors provide a nice tool for anaysing cohomology groups:

If we have a curve $C$ and an invertible sheaf $\mathcal{L} \in Pic(C)$, we can use a Cartier Divisor $D$ with corresponding global section $s \in \Gamma(C, \mathcal{L})$ to get an short exact sequence $$0 \to \mathcal{O}_L \xrightarrow{\text{s}} \mathcal{L} \to \mathcal{O}_D \to 0$$

which induces long exact sequence

$$0 \to H^0(\mathcal{O}_L) \xrightarrow{\text{s}} H^0(\mathcal{L}) \to H^0(\mathcal{O}_D) \to H^1(\mathcal{O}_L) \to ...$$

If there are given some extra informations about cohomology groups of $\mathcal{L}$, we can get new informations about cohomology groups of tzhe struructure sheaf $\mathcal{O}_L$.

But I suppose that there is a much more deeper philosophy about application / understanding of effective cartier divisors. What is it's geometrical interpretation / motivation? Can anybody explain the intuition behind this?

• An irreducible effective cartier divisor should (?) be thought as an integral hypersurface embedded in your scheme. Indeed locally it is defined by a single equation $f_i$ over an open subset $U_i$ and the condition that $f_i$ is regular at every point, mean that locally the codimension of $V(f_i)$ is 1 (by the hauptidealsatz on a loc. noeth. scheme). A general cartier divisor is an integral combination of these, however in the case in which it is effective you can still think about it as a possibily non reduced and non irreducible hypersurface. – Ahr Jan 29 '18 at 12:51
• In fact this is how you should view a "divisor" in general. The fact of the matter is that there are subtleties between the different notions of divisors that you want to consider. On a smooth variety Cartier divisors and Weil divisors coincide but this is not the case in general. – Ahr Jan 29 '18 at 12:52
• In any case there is a strong relationship with line bundles and their rational sections. On an integral scheme (or even a quasi-projective variety) Cartier divisors are the same thing that a line bundle together with a rational section of it. Effective Cartier divisor correspond to the case where the rational section is everywhere defined (you get a bona fide subscheme by considering the zero locus of the section, but with a non effective one you may get the "opposite of a subsceme" or a combination of both.). – Ahr Jan 29 '18 at 12:55
• In any case the intuition stems from curves. For smooth proper curves all the notion of divisors coincide. – Ahr Jan 29 '18 at 12:56