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Let $D$ be a big but not nef divisor on a normal $\mathbb{Q}$-factorial projective variety. Assume that the section ring $$R(D) = \bigoplus_{n\in\mathbb{N}}H^0(X,nD)$$ is finitely generated and consider the rational map $$\phi_{D}:X\dashrightarrow X_{D}\subseteq Proj(R(D))$$ Could it happen that for any effective irreducible curve $C\subseteq X$ such that $C\cdot D=0$ (zero intersection product) we have that $C$ is contained in the indeterminacy locus of $\phi_{D}$ ?

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  • $\begingroup$ It can happen yes, e.g., if $\phi_D$ is an embedding. $\endgroup$ – byu Dec 16 '17 at 17:58
  • $\begingroup$ I was implicitly excluding that case, $D$ is not nef. I edited the question. $\endgroup$ – J. Ross Dec 16 '17 at 18:04
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Consider the Hirzebruch surface $\mathbb{F}_n$, $n>1$. Let $\sigma$ be the negative section and $\ell$ be the rulling. Consider $D=\ell+\sigma$, which is a big but not nef divisor. This $D$ satisfies your requirement, because there is no such curve $C$ such that $D\cdot C=0$.

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  • $\begingroup$ $D$ is not big. $\endgroup$ – abx Dec 17 '17 at 5:20
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    $\begingroup$ @abx sure it is. $(n+1)D\geq (n+1)\ell+\sigma$. The latter is ample. $\endgroup$ – Chen Jiang Dec 17 '17 at 6:11

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