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On the ACC for lengths of extremal rays. (English) Zbl 1271.14075
The authors consider the following conjecture (Conjecture 1.3), which, reformulated, says: For \(X\) an \(n\)-dimensional \(\mathbb Q\)-factorial log canonical Fano variety with Picard number one, let \(l(X):=\min\limits _C (-K_X\cdot C)\), where \(C\) runs over integral curves on \(X\). If \(X_k\) \((k \geq 1)\) are \(n\)-dimensional \(\mathbb Q\)-factorial log canonical Fano varieties with Picard number one such that: \[ l(X_1)\leq l(X_2) \leq \ldots l(X_k) \leq \ldots \] then there is \(\ell\) such that \(\l(X_m)=l(X_\ell)\) for \(m \geq \ell\).
They prove this conjecture for \(\mathbb Q\)- factorial toric Fano varieties with Picard number one.

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14E30 Minimal model program (Mori theory, extremal rays)
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