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For any effective divisor $D=\sum a_i[Y_i]$, in the language of complex spaces, $\mathcal{O}_D$ is the structure sheaf of the (possibly non-reduced) subspace associated to $D$.

I wonder what the subspace associated to $D$ is? I only know we can associate a line bundle for a divisor, $L(D)\to M$. Is $\mathcal{O}_D$ related to the sheaf $L(D)$, which is the sheaf of sections of the line bundle $L(D)\to M$.

Danny
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  • You have a natural section of $L(D)$, since $D$ is effective. So, we have $\mathcal{O}\subset L(D)$ and tensoring with the inverse of $L(D)$ (which is called $L(-D)$), you get an inclusion $L(-D)\subset\mathcal{O}$ and thus $L(-D)$ can be identified with an ideal sheaf. This ideal sheaf defines a subspace (usually, called a subscheme) of your ambient space. – Mohan Sep 08 '19 at 02:49
  • @Mohan what's the subspace defined by the ideal sheaf? – Danny Sep 08 '19 at 06:02

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