Polydiagonal compactification of configuration spaces

Abstract

A smooth compactification <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X mathematical left-angle n mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mspace width="negativethinmathspace"/> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>n</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">X\!\left \langle n\right \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the configuration space of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> distinct labeled points in a smooth algebraic variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is constructed by a natural sequence of blowups, with the full symmetry of the permutation group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Subscript n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">{\mathbb S}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> manifest at each stage. The strata of the normal crossing divisor at infinity are labeled by <italic>leveled</italic> 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 <italic>polyscreens</italic> similar to the screens of Fulton and MacPherson. In characteristic 0, isotropy subgroups of the action of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Subscript n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">{\mathbb S}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X mathematical left-angle n mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mspace width="negativethinmathspace"/> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>n</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">X\!\left \langle n\right \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are abelian, thus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X mathematical left-angle n mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mspace width="negativethinmathspace"/> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>n</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">X\!\left \langle n\right \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> may be a step toward an explicit resolution of singularities of the symmetric products <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Superscript n Baseline slash double-struck upper S Subscript n"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mspace width="negativethinmathspace"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">X^n\!/{\mathbb S}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .