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I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. I have browsed through Hartshone's book but everything is expressed in terms of schemes, and I believe it is still possible to find some toy example to carry with me without having to learn what a scheme is first. Does anyone know any reference or book which has some exercise or example on this? In particular I would like to see examples of linear systems of divisors and how given a linear system of dimension $n$ I can choose a pencil inside it.

Apologies if this is not the place to write this, it is my first post. I have searched through the questions and have not find anything similar.

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Thanks to all of those who replied. Your answers have been very useful –  Jesus Martinez Garcia Dec 14 '09 at 19:26

4 Answers 4

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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 $H\cap C$. 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, $$\phi:\mathbb{P}^2\rightarrow \mathbb{P}^5$$ 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 $\mathbb{P}^2\cong \phi(\mathbb{P}^2)$. 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.

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You want concrete? Then you want curves and surfaces! Check out Chapter V of Miranda's "Algebraic Curves and Riemann Surfaces" to see lots of stuff about divisors, how they're made up of functions, how they can give maps to projective space, etc. As for how to choose a pencil, that's just choosing two divisors that are linearly equivalent, and taking linear combinations.

For surfaces, you'll want Beauville's book "Complex Algebraic Surfaces" which does surface theory without schemes, over $\mathbb{C}$.

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Here's a basic (and often used!) example: the zero locus of a homogeneous polynomial of degree $d$ in $\mathbb{P}^n$. For concreteness, let me spell out the case $n = 1$, $d = 2$. Cover $\mathbb{P}^1$ with the two open subsets $$\mathcal{U}_0 = \lbrace [x_0 : x_1] \; | x_0 \neq 0 \rbrace, \quad\text{and}\quad \mathcal{U}_1 = \lbrace [x_0 : x_1] \; | x_1 \neq 0 \rbrace$$ Local coordinates on these patches are $z:= x_1 / x_0$ and $w := x_0 / x_1$, respectively; on $\mathcal{U}_0 \cap \mathcal{U}_1$, they are related by $w = z^{-1}$. If your quadratic polynomial is given by $F = a_0 x_0^2 + a_1 x_0 x_1 + a_2 x_1^2$, you have the local expressions $$f_0 = a_0 z^2 + a_1 z + a_2, \quad\text{and}\quad f_1 = a_0 + a_1 w + a_2 w^2$$ respectively. You can check that they are related over $\mathcal{U}_0 \cap \mathcal{U}_1$ by multiplication by a unit, and hence define a divisor as a global section of the sheaf $\mathcal{M}^\ast / \mathcal{O}^\ast$. Notice that this divisor just consists of two points (counting multiplicities, of course).

In these terms, a pencil is given by the vanishing loci of linear combinations of the form $\lambda F + \mu G$, where $\lambda, \mu \in \mathbb{C}$ not both zero and $G$ another homogeneous polynomial of degree 2.

As Charlie pointed out, you will want to learn more about curves, where divisors are formal integral linear combinations of points. Other cases with beautiful geometry:

  • Divisors on $\mathbb{P}^2$ give you plane curves, where you can see Bézout's theorem at work.
  • Pencils of homogeneous polynomials of degree 2 in $\mathbb{P}^n$ (i.e., pencils of quadric hypersurfaces) also constitute a very rich case: Griffiths and Harris' last chapter talks about it (although it is a nice case to play with by yourself).
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For divisors on curves, as recommended by Alberto above, I enjoyed Otto Forster's book Lectures on Riemann surfaces. Section 16 of the book has a nice treatment on the way to the proof of Riemann-Roch.

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