[Paper Review] A Generalization of Fulton-MacPherson Configuration Spaces

Bumsig Kim, Fumitoshi Sato|ArXiv.org|Jun 24, 2008

Advanced Numerical Analysis Techniques5 references7 citations

TL;DR

This paper introduces a generalized wonderful compactification of configuration spaces for $ n $ distinct labeled points in a nonsingular variety $ X $, excluding a nonsingular subvariety $ D $. By iteratively blowing up diagonals and subvarieties $ D_{c,S} $, the authors construct $ X_D^{[n]} $ and $ X_D[n] $, proving both are nonsingular with transversally intersecting boundary divisors, generalizing the Fulton-MacPherson space when $ D = \emptyset $. The construction provides universal families of stable degenerations, crucial for moduli of relative maps.

ABSTRACT

Presented is a wonderful compactification of n distinct labeled points in X away from D, where X is a nonsingular variety and D is a nonsingular proper subvariety. When D is empty, it is the Fulton-MacPherson configuration space.

Motivation & Objective

• To construct a compactification $ X_D^{[n]} $ of $ n $ labeled points in $ X \setminus D $, where points are not allowed to approach $ D $.
• To construct a compactification $ X_D[n] $ of $ n $ distinct labeled points in $ X \setminus D $, where points are not allowed to collide or approach $ D $.
• To generalize the Fulton-MacPherson configuration space by incorporating a divisor $ D $, extending its structure to relative settings.
• To establish that both $ X_D^{[n]} $ and $ X_D[n] $ are nonsingular with transversal boundary divisors, and to define universal families of stable degenerations.

Proposed method

• Define $ X_D^{[n]} $ as the closure in a product of $ X^n $ and iterated blowups along $ D_{c,S} $, the loci where points labeled by $ S $ lie in component $ D_c $.
• Define $ X_D[n] $ as the closure in a product of $ X_D^{[n]} $ and blowups along proper transforms of diagonals $ \widetilde{\Delta}_I $ for $ |I| \geq 2 $.
• Use iterated blowups in a specific order: first along $ D_{c,S} $, then along $ \widetilde{\Delta}_I $, ensuring transversality of boundary divisors.
• Apply L. Li’s general theory of wonderful compactifications to prove nonsingularity and transversality of boundary components.
• Utilize the notion of nested collections of index sets $ S_i $ and $ I_j $ to characterize nonempty intersections of boundary divisors.
• Leverage the universal family $ X_D^{[n]+} \to X_D^{[n]} $ and $ X_D[n]^+ \to X_D[n] $ to describe stable degenerations of $ X $ with labeled points.

Experimental results

Research questions

• RQ1 How can one construct a compactification of the configuration space of $ n $ labeled points in $ X \setminus D $, where points are forbidden from approaching $ D $?
• RQ2 What conditions ensure that the boundary divisors of the compactification intersect transversally?
• RQ3 How does the construction generalize the Fulton-MacPherson space when $ D = \emptyset $?
• RQ4 What is the geometric structure of the universal family over $ X_D[n] $, and how does it parametrize stable degenerations?
• RQ5 Can the moduli space of stable relative maps be constructed using fibers of $ X_D[n]^+ $?

Key findings

• The space $ X_D^{[n]} $ is nonsingular, with a universal flat family of stable degenerations of $ X $, where $ n $ labeled points are away from $ D $.
• The boundary of $ X_D^{[n]} $ is a union of divisors $ \widetilde{D}_{c,S} $, and their intersections are transverse if and only if the corresponding index sets are nested.
• The space $ X_D[n] $ is nonsingular, with a universal family $ X_D[n]^+ \to X_D[n] $, parametrizing stable degenerations with $ n $ distinct labeled points away from $ D $.
• The boundary of $ X_D[n] $ consists of divisors $ \widetilde{D}_{c,S} $ and $ \widetilde{\Delta}_I $, and their intersections are nonempty and transverse precisely when the entire collection is nested.
• When $ D = \emptyset $, $ X_D[n] $ recovers the classical Fulton-MacPherson compactification $ X[n] $.
• The construction provides a natural framework for studying stable relative maps and stable (un)ramified maps, with applications in moduli theory.