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Let a smooth projective curve $X$ over $\mathbb{C}$. Let a pair $(x, D)$ a pair xith a closed point $x$ and $D$ an effective divisor on $X$, such that $d_{x}:=m_{x}(D)\neq 0$. Let $N=\deg (D)$ and $X^{(N)}$ the scheme of effective divisors of degeree $N$.

We consider a integer $d'< d_{x}$ and consider the constructible set :

$\{D'\in X^{(N)}\vert ~ d'\leq m_{x}(D')\leq d_{x}\}$

Does it contains a subscheme $U$ open dense in $X^{(N)}$?

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    $\begingroup$ I'm confused, $x$ is fixed? I don't a dense open subset of divisors will vanish to degree at least $d'$ at $x$. $\endgroup$ Sep 7, 2013 at 22:46
  • $\begingroup$ That should have read "... I don't believe a dense open subset..." $\endgroup$ Sep 7, 2013 at 23:50

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