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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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