Wonderful Compactification of an Arrangement of Subvarieties

Fix a nonsingular algebraic variety Y over an algebraically closed field (of arbitrary characteristic). An arrangement of subvarieties S is a finite collection of nonsingular subvarieties such that all nonempty scheme-theoretic intersections of subvarieties in S are again in S, or equivalently, such that any two subvarieties intersect cleanly and the intersection is either empty or a subvariety in this collection (see Definition 2.1). Let S be an arrangement of subvarieties of Y . A subset G ⊆ S is called a building set of S if ∀S ∈ S \ G, the minimal elements in {G ∈ G : G ⊇ S} intersect transversally and the intersection is S. A set of subvarieties G is called a building set if all the possible intersections of subvarieties in G form an arrangement S (called the induced arrangement of G) and G is a building set of S (see Definition 2.2). For any building set G, the wonderful compactification of G is defined as follows.