A compactification of configuration spaces. (English) Zbl 0820.14037

Fulton, William; MacPherson, Robert

Ann. Math. (2) 139, No. 1, 183-225 (1994).

The authors introduce and study a natural and very nice compactification \(X[n]\) of the configuration space \(F(X,n)\) of \(n\) distinct labeled points in a nonsingular algebraic variety \(X\). \(X[n]\) is nonsingular and may be obtained from the cartesian product \(X^ n\) by a sequence of blow-ups. The locus of the degenerate configurations, \(X[n] - F(X,n)\), is a divisor with normal crossings whose components are explicitly described. Finally the intersection ring (rational cohomology ring in the complex case) of \(X[n]\) as well as those of the components of \(X[n] - F(X,n)\) and their intersections are computed.

Reviewer: E.Casas-Alvero (Barcelona)

MSC:

14M99	Special varieties
14N10	Enumerative problems (combinatorial problems) in algebraic geometry
14C17	Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry

Keywords:

compactification; configuration space; intersection ring