Uma introdução aos quocientes pela ação de grupos algébricos

Abstract

Let G be a finite abelian subgroup of SL(n,C), and suppose there exists a crepant resolution ϕ : X −→ Cn/G of the (Gorenstein) quotient variety Cn/G. In this situation, the exceptional set of ϕ is pure of codimension 1. That is, when we decompose such a set into irreducible components, Exc(ϕ) = E1 ∪ · · · ∪ Es, we can verify that dim(Ei) = n−1, for all i. Furthermore, it is known that Cn/G is a toric variety, and any crepant resolution of such a variety is also toric. Using the tools provided by toric geometry, in this work we prove that, for each irreducible component Ei of Exc(ϕ), there exists an open subset Ui of X such that Ei ⊂ Ui, and Ui is the total space of the canonical bundle ωEi of Ei. Furthermore, X = U1 ∪ · · · ∪ Us. In particular, when s = 1, we obtain that the resolution X itself is a line bundle over Exc(ϕ).