Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Polydiagonal compactification of configuration spaces

Author: Alexander P. Ulyanov
Journal: J. Algebraic Geom. 11 (2002), 129-159
Published electronically: November 16, 2001
MathSciNet review: 1865916
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Abstract | References | Additional Information

Abstract: A smooth compactification $X\left\langle n\right\rangle$ of the configuration space of $n$ distinct labeled points in a smooth algebraic variety $X$is constructed by a natural sequence of blowups, with the full symmetry of the permutation group  ${\mathbb S}_n$manifest at each stage. The strata of the normal crossing divisor at infinity are labeled by leveled trees and their structure is studied. This is the maximal wonderful compactification in the sense of De Concini-Procesi, and it has a strata-compatible surjection onto the Fulton-MacPherson compactification. The degenerate configurations added in the compactification are geometrically described by polyscreens similar to the screens of Fulton and MacPherson.

In characteristic 0, isotropy subgroups of the action of  ${\mathbb S}_n$ on  $X\left\langle n\right\rangle$ are abelian, thus $X\left\langle n\right\rangle$ may be a step toward an explicit resolution of singularities of the symmetric products  $X^n/{\mathbb S}_n$.

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Additional Information

Alexander P. Ulyanov
Affiliation: Department of Mathematics, University of Pennsylvania, David Rittenhouse Laboratory, 209 South 33rd Street, Philadelphia, Pennsylvania 19104

Received by editor(s): January 7, 2000
Received by editor(s) in revised form: May 2, 2000
Published electronically: November 16, 2001
Additional Notes: Research partially supported by NSF grant DMS–9803593