## 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.

### MSC:

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

 [1] C. Bouvier, Germes de courbes tracées sur une variété torique singulière et diviseurs essentiels , C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 4, 387-390. · Zbl 0846.14031 [2] C. Bouvier and G. Gonzalez-Sprinberg, Système générateur minimal, diviseurs essentiels et $$G$$-désingularisations de variétés toriques , Tohoku Math. J. (2) 47 (1995), no. 1, 125-149. · Zbl 0823.14006 [3] J. Fine, On varieties isomorphic in codimension one to torus embeddings , Duke Math. J. 58 (1989), no. 1, 79-88. · Zbl 0708.14035 [4] G. Gonzalez-Sprinberg and M. Lejeune-Jalabert, Sur l’espace des courbes tracées sur une singularité , Algebraic geometry and singularities (La Rábida, 1991), Progr. Math., vol. 134, Birkhäuser, Basel, 1996, pp. 9-32. · Zbl 0862.14003 [5] G. Kempf, F. F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I , Lecture Notes in Math., Springer-Verlag, Berlin, 1973. · Zbl 0271.14017 [6] M. Lejeune-Jalabert, Arcs analytiques et resolution minimale des singularités des surfaces quasi homogènes , Séminaire sur les Singularités des Surfaces (Palaiseau, 1976-1977), Lecture Notes in Math., vol. 777, Springer-Verlag, Berlin, 1980, pp. 303-336. · Zbl 0432.14020 [7] J. Nash, Jr., Arc structure of singularities , Duke Math. J. 81 (1995), no. 1, 31-38 (1996). · Zbl 0880.14010 [8] O. Zariski and P. Samuel, Commutative algebra. Vol. II , The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. · Zbl 0121.27801
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