## Minimal generating system, essential divisors and $$G$$-desingularisation of toric varieties. (Système générateur minimal, diviseurs essentiels et $$G$$- désingularisations de variétés toriques.)(French)Zbl 0823.14006

Let $$V$$ be the toric variety associated to a rational polyhedral cone $$\sigma$$ over a field $$k$$, and $$G$$ the minimal generating system of the semi-group defined by $$\sigma$$. We give a geometric interpretation of $$G$$ in any dimension, via a natural bijection with the set of essential divisors of equivariant desingularizations of $$V$$. A $$G$$- desingularization of $$V$$ corresponds to a regular subdivision of $$\sigma$$ whose edges contain the elements of $$G$$. We prove the existence of $$G$$- desingularizations in dimension 3 by a construction from a minimal terminal model, its uniqueness for canonical varieties $$V$$ of index $$>1$$, and the uniqueness in general up to flops. We give an example of non existence in dimension 4.

### MSC:

 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14J30 $$3$$-folds 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14L30 Group actions on varieties or schemes (quotients)
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