Minimal generating system, essential divisors and \(G\)-desingularisation of toric varieties (Q1893808)

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.