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Let $X$ be a smooth projective curve over $\mathbb C$. Let $D$ be a divisor on it. What is known about upper bound on dimension of the cokernel of
$$Sym^2(H^0(X,D)) \to H^0(X,2D)?$$ In my case the divisor $D$ is somewhat larger than half of the canonical divisor, dimension of $H^0(X,D)$ is $O(n)$, while dimension of $H^0(X,2D)$ and genus of $X$ are $O(n^2)$.

I am interested in upper bounds (presumably with some conditions on $X$) of order $O(n)$ or some small constant times genus. Any references will be appreciated.

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  • $\begingroup$ what is $n$ here? $\endgroup$
    – Dan Petersen
    Commented Nov 11, 2013 at 10:14
  • $\begingroup$ I actually have a family of curves (these are modular curves). The point is that the dimension of $H^0(X,D)$ is up to a constant the root of the genus. Also, the map is definitely not surjective, but I would like to bound the defect. $\endgroup$
    – Lev Borisov
    Commented Nov 11, 2013 at 11:51
  • $\begingroup$ Lev, I don't know the answer to your question, but am incidentally interested in what happens when X is a modular curve and D is a half canonical divisor plus a small number of points. Do you have any thoughts or knowledge about this? $\endgroup$
    – David Zureick-Brown
    Commented Nov 15, 2013 at 15:51

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