Essential divisors, essential components of singular toric varieties. (Diviseurs essentiels, composantes essentielles des variétés toriques singulières.) (French) Zbl 0966.14038

Let \(V\) be a normal affine (irreducible) toric variety and \(\pi:X\to V\) a desingularisation of \(V\). A divisor \(D\) of \(X\) is said to be essential if the image of \(D\) on any desingularisation of \(V\) has codimension one. One says that \(D\) is an essential component if the image of \(D\) on any desingularisation \(\pi':X'\to V\) is an irreducible component (not necessarily of codimension one) of the fibre \({\pi'}^{-1} \pi(D)\). In a previous paper in collaboration with \(G\). Gonzalez-Sprinberg, the authors gave a combinatorical characterisation of divisors which are essential only for equivariant desingularisations of \(V\). The first result states that if a prime divisor of an equivariant desingularisation is essential for equivariant desingularisations, then it is essential. This provides a complete description of essential divisors of \(V\). When \(\pi\) is equivariant, the second result gives a combinatorical condition for a prime equivariant divisor of \(X\) to be an essential component. This condition is written in terms of points in the lattice of one-parameter groups of \(V\). The paper contains two enlightening examples.


14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14C20 Divisors, linear systems, invertible sheaves
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