# zbMATH — the first resource for mathematics

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.

##### MSC:
 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14E30 Minimal model program (Mori theory, extremal rays)
Full Text: