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Let $C$ be a smooth projective curve over an algebraically closed field. If $D$ is an effective divisor on $C$ (let's say reduced to make things easier) of degree $m$ and $d>m$, is the dimension of (maybe the closure of) $X_D:=\{L\in\mbox{Pic}^d(C):\mbox{supp}(D)\subseteq\mbox{Bs}(|L|)\}$ known? Or is there maybe a bound?

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    $\begingroup$ I recommend that you look at Theorem 5.37 of Harris-Morrison, "Moduli of Curves". For a $g^r_d$ on a smooth curve $X$ of genus $g$, containing a point $p_i$ in the base locus is equivalent to having vanishing sequence $(1,2,\dots,r+1)$, rather than $(0,1,\dots,r)$. If you let $D$ be the divisor with support $\{p_1,\dots,p_m\}$, then Theorem 5.37 describes existence and non-existence results. $\endgroup$ Nov 17, 2014 at 16:53
  • $\begingroup$ Dear Jason, thank you for your response. I think this gives me what I'm looking for if $C$ is general; unfortunately in the case I'm looking at I'm not sure I can assume that. $\endgroup$
    – rfauffar
    Nov 19, 2014 at 12:22

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