ag.algebraic geometry - Examples of divisors on an analytical manifold

Hartshorne is the reference where you can find the following example which might be useful.
I what follow everything is with multiplicity. Now Alberto pointed out above the case of the divisor over $mathbb{P}^1$ associated to its "tangent bundle": Two points over the sphere counted with multiplicity (from here though, it is not hard to believe that the Chern class of such a bundle is going to be 2). Notice that these two points are given by zeros of polynomials of degree two defined over the sphere. I think nothing stops you taking now polynomial of degree 3, 4 and so on. Then what we get are nothing but 3, 4 points over the sphere: Divisors of degree 3, 4 and so on. We can do something similar over all the curves (Riemann Surfaces) and what we get are divisors: points with labels. Such labels are the multiplicities. Chapter IV Hartshorne. or Klaus-Hulek: Elementary Algebraic Geometry.

Now, let's take a look at divisors over the surface $mathbb{P}^2$: they are algebraic curves (Riemann Surfaces). Do not get confused please by the name Surface here. Applying the same argument as before, a divisor of degree two is going to be the zero locus of polynomials of degree 2: conics. Same for degree three (cubics), four (quartics), and so on and so forth. For instance, in degree two we might have the divisor $C=([x:y:z]in mathbb{P}^2| x^2+y^2=z^2)$. Deshomogenizing with $H=[z=1]$ you get a perfect polynomial $x^2+y^2=1$ which defines the intersection $Hcap C$. This is how your global divisor $C$ looks like locally.

Now taking a family of divisors of degree two, the conics, it is well known that the space of embeddings of conics in $mathbb{P}^2$ is (the linear system) $mathbb{P}^5$. We get this by considering the coefficients in the equation $ax^2+by^2+cz^2+dxy+exz+fyz=0$ as coordinates in $mathbb{P}^5$. Notice that we get the following map out of the previous considerations, $$phi:mathbb{P}^2rightarrow mathbb{P}^5$$ given by $[x:y:z]mapsto [x^2:y^2:z^2:xy:xz:yz]$. Here pencils are a subfamily of conics in the complete linear system given above with a certain property (find out which one). However, we can consider the following subfamily of conics: all those conics passing through a fixed point in $mathbb{P}^2$. This is nothing but a hyperplane $H$ in $mathbb{P}^5$. We can even consider $phi(mathbb{P}^2)cap H$. This is going to be a divisor on $mathbb{P}^2cong phi(mathbb{P}^2)$. Guess which one?. Hartshorne II section 7.

One can apply the the ideas with zero locus of polynomials of degree three: Divisors of degree 3 in $mathbb{P}^2$. These were given the name of elliptic curves. (did someone say that in considering such curves, we find the divisor associated to the canonical bundle of $mathbb{P}^2$?). We can go on with the degree and getting divisors on the projective plane of higher degree. These were only examples of divisors on $mathbb{P}^2$. Notice that all of them have a nontrivial topology and geometry. This fact is not a coincidence and the book of HG argues in this direction in Chapter zero.