I'm reading some introductory material in algebraic geometry. I'm trying to closely follow this http://people.fas.harvard.edu/~amathew/287y.pdf. I keep getting confused with some basic notions. I'd appreciate if someone could help me clarifying some points.

Let $X$ be a projective curve and $D$ be a divisor on $X$. I am comfortable with the classical definition of the Riemann-Roch space $\mathcal{L}(D)$. Here are few questions I have:

(1) I read that $\mathcal{O}_X$ stands for the sheaf of regular functions on $X$. But I learned that every non-constant function has a pole. So is $\mathcal{O}_X$ just the set of constant functions?

(2) What is the meaning of the sheaf $\mathcal{O}_X(D)?$ Is it the same as $\mathcal{L}(D)$ or something more general? What are the line bundles $\mathcal{O}_X(D)?$ which are not of type $\mathcal{L}(D)$?

(3) What should I understand by "global sections" of line bundles?

I apologize for the (perhaps) silly questions. I'd appreciate any help.

  • 1
    $\begingroup$ You should probably look up the definition of sheaf. Global sections means the sheaf is "evaluated" on the whole space. $\mathcal O_X$ is a sheaf, not a set. Is $\mathcal L(D)$ a sheaf in your notation? If not, it's probably the global sections of the sheaf $\mathcal O_X(D)$. $\endgroup$ – Andrew Nov 13 '13 at 14:55
  • $\begingroup$ Ok! so sheaf is a function. So $\mathcal{O}_X$ is the sheaf that associates any open set $U$ of $X$ to a ring of functions that are regular on $U$. Right? $\endgroup$ – user108555 Nov 13 '13 at 15:29
  • $\begingroup$ And a global section is the image of the whole $X$. $\endgroup$ – user108555 Nov 13 '13 at 15:30
  • $\begingroup$ The ring of regular functions that are regular on $U$, not $X$. $\endgroup$ – RghtHndSd Nov 13 '13 at 15:30
  • $\begingroup$ Dear @user108555, that's correct. $\endgroup$ – Andrew Nov 13 '13 at 15:59
  1. The global sections of $\mathcal{O}_X$ (i.e. $\mathcal{O}_X(X)$) are constant functions, if $X$ is a projective variety. However take for example $\mathbb{P}^1$ over $k = \mathbb{C}$. Then $\mathcal{O}_X(\mathbb{P}^1 - \{\infty\}) = \mathcal{O}_X(\mathbb{A}^1)$ is given by $\mathbb{C}[x]$ (because something like $x$ or $x^3 - 3x + 1$ only has a pole at infinity).

  2. $\mathcal{O}_X(D)$ and $\mathcal{L}(D)$ are both used for the line bundle associated to a Cartier divisor. Wikipedia has a decent description of them. However in the notes you linked to, it seems the convention is $\mathcal{L}(D) = \mathcal{L}(D)(X)$, i.e. the global sections.

  3. As mentioned above, sections of a line bundle are with respect to some open subset $U$ of the whole space $X$. A global section is when you choose the open subset $U = X$.

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