Logarithmic Fulton--MacPherson configuration spaces

TL;DR

The paper develops a logarithmic analogue of Fulton–MacPherson configuration spaces by constructing $(X|D)^{[n]}$, the moduli of stable $n$-pointed grid expansions, and the logarithmic FM compactification $ ext{FM}_n(X|D)$, as an iterated blow-up of expansions. It then builds a logarithmically smooth degeneration $ ext{FM}_n(W/B) o B$ of FM spaces with a degeneration formula: the special fibre decomposes into components indexed by rigid combinatorial types, each component being a modification of a product of FM spaces on the expanded target’s components. The framework relies on tropicalisation, Artin fans, and refined log structures to provide canonical expansions and a boundary stratification by combinatorial types, with rubber actions encoding isomorphisms within strata. Potential applications include logarithmic unramified Gromov–Witten theory and new intersection-theoretic relations via degeneration formulas, as well as connections to related FM-type constructions (relative curves, weighted configurations, and toric/expanded degenerations). Overall, the work offers a canonical, combinatorially rich compactification and a robust degeneration toolkit for point configurations on degenerations, with broad implications for enumerative geometry in the logarithmic setting.

Abstract

Using techniques in logarithmic geometry, we construct a logarithmic analogue of the Fulton--MacPherson configuration spaces. We similarly construct a logarithmically smooth degeneration of the Fulton--MacPherson configuration spaces. Both constructions parametrise point configurations on certain target degenerations arising from both logarithmic geometry and the original Fulton--MacPherson construction. The degeneration satisfies a "degeneration formula" -- each irreducible component of its special fibre can be described as a proper birational modification of a product of logarithmic Fulton--MacPherson configuration spaces.

Figures (14)

• Figure 1: Combinatorial type of the divisor $V_{k,I}$
• Figure 2: Open stratum of $(\mathbb{P}^2 | D_1 + D_2)^{[2]}$
• Figure 3: The boundary divisor $V_{2,\{2\}}$
• Figure 4: An example of a rubber action
• Figure 5: Planted forest $\nu_{k,I}$

Theorems & Definitions (104)

• Theorem 1.1.1: Theorem \ref{['thm:logFM_fibres']}
• Theorem 1.1.2: Theorem \ref{['thm:FMdegen']}
• Theorem 1.1.3: Theorem \ref{['thm:degen_formula']}
• Definition 2.1.1: Tropicalisation of a pair
• Definition 2.1.2: Artin fan of a pair
• Theorem 2.1.3: CCUW Theorem 6.11
• Definition 2.1.4: Subdivision MR20
• Definition 2.3.1: Marked polyhedral subdivisions of $\mathbb{R}_{\geq 0}$
• Definition 2.3.2: Marked grid subdivisions of $\Sigma_X$
• Lemma 2.3.3