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A compactification of configuration spaces. (English) Zbl 0820.14037
The authors introduce and study a natural and very nice compactification $X\left[n\right]$ of the configuration space $F\left(X,n\right)$ of $n$ distinct labeled points in a nonsingular algebraic variety $X$. $X\left[n\right]$ is nonsingular and may be obtained from the cartesian product ${X}^{n}$ by a sequence of blow-ups. The locus of the degenerate configurations, $X\left[n\right]-F\left(X,n\right)$, is a divisor with normal crossings whose components are explicitly described. Finally the intersection ring (rational cohomology ring in the complex case) of $X\left[n\right]$ as well as those of the components of $X\left[n\right]-F\left(X,n\right)$ and their intersections are computed.

##### MSC:
 14M99 Special algebraic varieties 14N10 Enumerative problems (algebraic geometry) 14C17 Intersection theory, etc.
##### Keywords:
compactification; configuration space; intersection ring