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 $X^{(N)}$?
