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I saw the following in an article:

Let $C$ be an irreducible smooth projective curve over an algebraically closed field $K$ and let $g$ be its genus. By Riemann-Roch, if N is large enough for every line bundle $\mathcal{L}$ on $C$ of degree $\leq g+1$, $\mathcal{O}(N)|_C\otimes\mathcal{L}^{-1}$ is generated by its global sections. It follows that, for $N$ large enough any function on $C$ with at most $g+1$ poles is the quotient of two homogeneous polynomials of degree $N$.

Why is this true? What is the connection two the fact that it is generated by global sections?

(crossposted from: Understanding an application...)

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  • $\begingroup$ It follows from RR that a line bundle on a smooth projective curve is ample if and only if its degree is positive. This implies what you need. If you want a proof of this fact, it is somewhere in Section 3 of Chapter IV in [Hartshorne77]. $\endgroup$
    – Sándor Kovács
    Jun 9, 2016 at 16:13
  • $\begingroup$ Actually, I've just realized that if your $\mathscr O(1)$ means an ample line bundle, then the statement is a consequence of the definition of ample. If it only means a line bundle of positive degree, then you need the above. :) $\endgroup$
    – Sándor Kovács
    Jun 9, 2016 at 16:18
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    $\begingroup$ I think the OP's concerns begin with "it follows that..." $\endgroup$
    – Hoot
    Jun 9, 2016 at 16:41
  • $\begingroup$ @Hoot1-you're right. They want to show that every function on the curve is a product of quotients of hom. polynomials a of degree N. They first show that every function is a product of functions with at most g+1 poles (Riemann-Roch and induction). The next part, which I've quotes is what I don't understand. $\endgroup$
    – user4231
    Jun 9, 2016 at 18:22
  • $\begingroup$ Oh, I see. I'm sorry I misunderstood the issue. In my defense, the title says "understanding an application of RR" and you don't need RR for that part. Let me add an answer for that then. $\endgroup$
    – Sándor Kovács
    Jun 9, 2016 at 21:42

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The poles of your function determine an effective divisor of degree at most $g+1$ which in turn determines a line bundle $\mathscr L$ of degree at most $g+1$ and with an appropriate embedding of $\mathscr L$ into $\mathscr K_C$, the constant sheaf determined by $K(C)$ the original function corresponds to a global section of this $\mathscr L$.

The statement you would like follows if you can embed $\mathscr L$ into $\mathscr O(N)|_C$, since the global sections of the latter correspond to quotients of homogenous polynomials of degree $N$. (Here I am assuming that $C$ is embedded into a projective space $\mathbb P^n$ and $\mathscr O(1)=\mathscr O_{\mathbb P^n}(1)$, which you never said, but you should have!). Well, actually you need a little more, because you want to do this as subsheaves of $\mathscr K_C$.

An embedding of of $\mathscr L$ into $\mathscr O(N)|_C$ is the same as a global section of $\mathscr O(N)|_C\otimes \mathscr L^{-1}$. In other words, every global section of $\mathscr O(N)|_C\otimes \mathscr L^{-1}$ gives you an embedding of of $\mathscr L$ into $\mathscr O(N)|_C$. You need global generation to guarantee that you can do this inside $\mathscr K_C$.

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  • $\begingroup$ Yes I should have! Aren't the global sections of O(n) polynomials? Can you elaborate more on the last paragraph? Why does this give an embedding? $\endgroup$
    – user4231
    Jun 13, 2016 at 16:03
  • $\begingroup$ The global sections of $\mathscr O(N)$ correspond to polynomials. Polynomials don't give you functions on the projective space. The way this correspondence works is that you have to pick one degree $N$ polynomial, say $x_0^N$ and use its inverse to identify $\mathscr O(N)$ with a subsheaf of $\mathscr K$. Then every hom poly of degree $N$ gives you a section. If you choose a different embedding, you get a different correspondence. This is the idea behind everything in the last paragraph. $\endgroup$
    – Sándor Kovács
    Jun 14, 2016 at 4:52
  • $\begingroup$ Thanks. Now I understand that it gives an embedding, it follows since $\mathcal{L}\otimes \mathcal{L}^{-1}$ is isomorphic to the structure sheaf and $\mathcal{O}(N)|_C$ is an $\mathcal{O}_C$-module. I still don't understand where I'm using global generation. I take $\mathcal{L}$ embedded in $\mathscr{K}_C$ and so is $\mathcal{O}(N)|_C$. $\endgroup$
    – user4231
    Jun 14, 2016 at 7:37
  • $\begingroup$ And another - does it mean that given a point $a$ on the curve I can always write $f$ as a quotient of hom. pol where the denominator does not vanish on $a$? $\endgroup$
    – user4231
    Jun 14, 2016 at 7:44

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