2 Equivariant Resolution of Singularities in Characteristic 00
نویسندگان
چکیده
0. Introduction We work over an algebraically closed eld k of characteristic 0. 0.1. Statement. In this paper, we use techniques of toric geometry to reprove the following theorem: Theorem 0.1. Let X be a projective variety of nite type over k, and let Z X be a proper closed subset. Let G Aut k (Z X) be a nite group. Then there is a G-equivariant modii-cation r : X 1 ! X such that X 1 is nonsingular projective variety, and r ?1 (Z red) is a G-strict divisor of normal crossings. This theorem is a weak version of the equivariant case of Hironaka's well known theorem on resolution of singularities. It was announced by Hironaka, but a complete proof was not easily accessible for a long time. The situation was remedied by E. Bierstone and P. Milman B-M2], and by O. Villamayor V]. They gave constructions of completely canonical resolution of singularities. These constructions are based on a thorough understanding of the eeect of blowing up-one carefully build up an invariant pointing to the next blowup. The proof we give in this paper takes a completely diierent approach. It uses two ingredients: rst, we assume that we know the existence of resolution of singularities without group actions. The method of resolution is not important: any of H], B-M1], V] @-dJ] or B-P] would do. Second, we use equivariant toroidal resolution of singularities. Unfortunately, in KKMS] the authors do not treat the equivariang case. But proving this turns out to be straightforward given the methods of KKMS]. To this end, section 2 of this paper is devoted to proving the following: Theorem 0.2. Let U X be a strict toroidal embedding, and let G Aut(U X) be a nite group acting toroidally. Then there is a G-equivariant toroidal ideal sheaf I such that the normalized blowup of X along I is a nonsingular G-strict toroidal embedding.
منابع مشابه
Equivariant Resolution of Singurlarities in Characteristic 0
0. Introduction We work over an algebraically closed field k of characteristic 0. 0.1. Statement. In this paper, we use techniques of toric geometry to reprove the following theorem: Theorem 0.1. Let X be a projective variety of finite type over k, and let Z ⊂ X be a proper closed subset. Let G ⊂ Aut k (Z ⊂ X) be a finite group. Then there is a G-equivariant modification r : X 1 → X such that X...
متن کاملA ug 2 00 6 Toward resolution of singularities over a field of positive characteristic Dedicated to Professor Heisuke Hironaka Part I . Foundation ; the language of the idealistic
متن کامل
9 M ay 2 00 7 F - THRESHOLDS OF HYPERSURFACES MANUEL
In characteristic zero one can define invariants of singularities using all divisors over the ambient variety. A key result that makes these invariants computable says that they can be determined by the divisors on a resolution of singularities. For example, if a is a sheaf of ideals on a nonsingular variety, then to every nonnegative real number λ one associates the multiplier ideal J (a). The...
متن کاملRings of Singularities
This paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``Advanced School and Conference on homological and geometrical methods in representation theory'' at the International Centre for Theoretical Physics (ICTP), Miramare - Trieste, Italy, 11-29 January 2010. We show how to associate to a triple of posit...
متن کاملar X iv : m at h / 06 04 35 4 v 1 [ m at h . A G ] 1 6 A pr 2 00 6 ENLARGEMENTS OF SCHEMES
In this article we use our constructions from [BS05] to lay down some foundations for the application of A. Robinson's nonstandard methods to modern Algebraic Geometry. The main motivation is the search for another tool to transfer results from characteristic zero to positive characteristic and vice versa. We give applications to the resolution of singularities and weak factorization.
متن کامل