MINIMAL GENERATOR SYSTEM, ESSENTIAL DIVIS ORS AND G-DESINGULARIZATIONS OF TORIC VARIETIES

We give a geometric interpretation of the minimal generating system of the semi-group defined by a rational polyhedral cone in any dimension , via a natural bijection with the set of essential divisors of equiva riant desingularizations of the toric variety associated to the cone. We prove, for varieties of dimension three, the existence of a desingu larization associated to a regular fan whose edges contain the element s of the minimal generating system, its uniqueness for canonical toric varieties of index at least two, and the uniqueness in general up to flops. We give an example of non-existence of such desingularizations in dimension four.