6
$\begingroup$

See my two previous questions here: Intuition for thinking about R-module of Kähler differentials, universal receptacles, derivations? and Kähler differentials, define valuation? for background.

If $\omega \in \Omega_{K_C/\mathbb{C}}$, then we define $$\text{div}(\omega) := \sum_{p \in C} v_p(\omega)p,$$so we can associate divisors to differentials.

Suppose $C \subset \mathbb{CP}^2$ is defined by $F(x, y, z) = 0$, and $C \cap \{z = 0\}$ is finite. $C - (C \cap \{z = 0\})$ is the affine curve defined by $f(x, y) = F(x, y, 1)$. What is $\text{div}(dx)$? If $p \in [x_0, y_0, 1]$, then $v_p(dx) = 0$ if $x - x_0$ is a uniformizing parameter at $p$. We have that $x - x_0$ is a uniformizing parameter as long as $f_y(p) \neq 0$.

Any divisor $D$ which is linearly equivalent to $\text{div}(\omega)$, i.e. $D \sim \text{div}(\omega)$, is also the divisor of a differential, trivially. Any divisor of this form is called a canonical divisor. In fact, we see that all divisors of this form are linearly equivalent, so the canonical divisors are an equivalence class in the class group.

(The class group $\text{Cl}(C)$ is defined by the exact sequence $0 \to \mathbb{C}^\times \to K_C^\times \to \text{Div}(C) \to \text{Cl}(C) \to 0.$)

My two questions are as follows.

  1. What is the intuitive way/motivation behind thinking about $\text{div}(\omega)$ (defined as above) in context of algebraic curves, preferably geometrically?
  2. What is the intuitive way/motivation behind thinking about canonical divisors (defined as above) in context of algebraic curves, preferably geometrically?

Thanks in advance.

$\endgroup$
5
  • 1
    $\begingroup$ not a research-level question in my opinion. $\endgroup$
    – Niels
    Jun 6, 2015 at 12:30
  • 1
    $\begingroup$ I suggest reading a good introductory book on algebraic curves -- there are plenty. In the meantime, please use MSE for your questions. $\endgroup$
    – abx
    Jun 6, 2015 at 12:43
  • 1
    $\begingroup$ No existing book on algebraic curves, to the best of my knowledge, has an explanation relating $\text{div}(\omega)$ and canonical divisors to Kähler differentials, much less an intuitive one. $\endgroup$
    – user74565
    Jun 6, 2015 at 12:56
  • 1
    $\begingroup$ Well reading Hartshorne's chapter on curves should be more than enough. You will learn more doing this job by yourself than taking the easy option of asking the question here, where it is not really appropriate. $\endgroup$
    – Niels
    Jun 6, 2015 at 19:47
  • 1
    $\begingroup$ In Riemann's and then Roch's proofs of the Riemann-Roch theorem, Riemann reduced the problem of computing the dimension of the space of sections of a line bundle, to some path integrals. Roch computed those integrals by residue calculus. His computation shows the role played by differential forms vanishing on the given divisor, i.e. the role of the divisor K-D, where K is a canonical divisor. In my opinion this is where the canonical divisor first arose. Then Brill and Noether showed more clearly how to use canonical divisors even more. see: alpha.math.uga.edu/~roy/rrt.pdf $\endgroup$
    – roy smith
    Jun 9, 2015 at 1:58

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.