Minimal Wonderful Model Overview

• Minimal Wonderful Model is a smooth compactification derived from subspace arrangements using minimal building sets that organize irreducible boundary divisors.
• Its construction employs iterated blow-ups ordered by subspace dimension, resulting in disjoint divisors and nested set posets that correspond to combinatorial forests.
• Applications span computing minimal-model cardinalities in Satisfiability Modulo Theories and classifying Weyl invariant arrangements, with case studies like the A2 model illustrating its structure.

A minimal wonderful model is a smooth compactification associated with a subspace arrangement, constructed from the minimal building set in the sense of De Concini–Procesi. It encodes the deep interplay between combinatorial and geometric structures arising from arrangements of subspaces, such as those linked to root systems or generalized Dowling lattices. The defining properties center on the minimal building set—the smallest collection of irreducible subspaces whose intersections govern the boundary divisors of the model, the nested structures among those divisors, and the organization of cohomology and enumerative invariants.

1. Minimal Building Sets and Their Characterization

Given a finite central arrangement $\mathcal{A} = \{H_1, \dots, H_m\}$ of linear subspaces in a complex vector space %%%%1%%%%, the minimal building set $\mathcal{F}(\mathcal{A})$ consists of all irreducible subspaces in the semilattice $\langle\mathcal{A}^*\rangle$ of sums of dual subspaces. A subspace $G \in \langle\mathcal{A}^*\rangle$ is irreducible if it does not admit a non-trivial decomposition as a sum of smaller elements.

For classical root arrangements, explicit descriptions are available:

• Type $A_{n-1}$: Minimal building set $$\mathcal{F}_{A_{n-1}} = \{\ker(e_i - e_j) : 1 \leq i < j \leq n\}$$.
• Type $B_n$ ($C_n$): $$\mathcal{F}_{B_n} = \{\ker(x_i) : 1 \leq i \leq n\} \cup \{\ker(x_i \pm x_j) : 1 \leq i < j \leq n\}$$.
• Type $D_n$: $$\mathcal{F}_{D_n} = \{\ker(x_i \pm x_j) : 1 \leq i < j \leq n\}$$.

The minimal building set is always contained in every other building set for the same arrangement and indexes the irreducible boundary components in the compactification (Gaiffi et al., 2012).

2. Construction of the Minimal Wonderful Model

Let $\mathcal{G} = \mathcal{F}(\mathcal{A})$. The minimal wonderful model $Y_{\mathcal{G}}$ is realized as the closure of the embedding: $$Y_{\mathcal{G}} = \overline{\left\{\,p \mapsto (p, \pi_{G_1}(p), \dots, \pi_{G_r}(p)): p \in V \setminus \bigcup \mathcal{A} \right\}} \subset V \times \prod_{G \in \mathcal{G}} V/G$$ Alternatively, $Y_{\mathcal{G}}$ is constructed via iterated blow-ups: order irreducibles $G \in \mathcal{G}$ by non-decreasing dimension, and blow up $V$ along the smallest ones, then proceed with proper transforms of larger ones.

The boundary divisor is $D = \bigcup_{G \in \mathcal{G}} D_G$, with $D_G$ the exceptional divisor of the blow-up along $G$. Intersections $D_{G_1} \cap \cdots \cap D_{G_k}$ are non-empty if and only if $\{G_1, \dots, G_k\}$ is a nested set, that is, the only non-empty intersections correspond to chains of containment among irreducibles. In the minimal case, all divisors are disjoint—no two distinct irreducibles are nested (Gaiffi et al., 2012, Gaiffi et al., 2018).

3. Nested Set Poset and Forest Representations

The pattern of intersections among boundary divisors is determined by the poset of nested sets. For a generalized Dowling arrangement (triple $(n, G, V)$ with $G$ finite, $V$ faithful), the subspaces $H(i,j,g)$ form the arrangement. The minimal building set $\mathcal{G}$ comprises annihilators of certain subspaces $H^K(i_1^{g_1K}, \dots, i_k^{g_kK})$.

De Concini–Procesi models exhibit a bijection between $\mathcal{G}$-nested sets and combinatorial forests:

• Forests have leaves labeled $1, \dots, n$, partitioned into blocks, each block forming a rooted tree.
• Internal vertices carry closed subgroup labels, edges are labeled by cosets, and multiplication along root-leaf paths matches subspace data.
• For abelian $G$, conjugacy constraints trivialize, and forest enumeration simplifies via generating functions (Gaiffi et al., 2018).

4. Cohomology and Enumerative Invariants

For any building set $\mathcal{G}$, $Y_\mathcal{G}$ is smooth, with boundary $D$ a normal-crossings divisor. In the minimal case (no inclusion among irreducibles), boundary strata are single divisors, all disjoint.

The cohomology ring admits the presentation: $$H^*(Y_\mathcal{G}; \mathbb{Z}) \cong \mathbb{Z}[c_G : G \in \mathcal{G}]/(c_{G_1}c_{G_2} : G_1 \neq G_2)$$ Each $c_G \in H^2$ is the dual of $D_G$. The $\mathbb{Z}$-basis is formed by powers of $c_G$ up to $\dim G - 1$.

The Poincaré polynomial for the minimal model is: $$P_{\mathrm{min}}(q) = 1 + |\mathcal{F}|\,q + \sum_{G \in \mathcal{F}} (\dim G - 1)\, q^2 + \cdots$$ For the case $$Y_{\mathcal{F}_{A_{n-1}}} \cong \overline{M}_{0,n+1}$$,

$$P(\overline{M}_{0,n+1})(q) = 1 + \binom{n}{2}q + \binom{n}{2}\binom{n-1}{2}q^2 + \cdots$$

Tables of Poincaré polynomials for low ranks illustrate the enumerative structure:

Type	$n$	$P_{\min}(q)$
$A_{n-1}$	2	$1$
	3	$1 + q$
	4	$1 + 5q + 5q^2 + q^3$
$B_n$	1	$1$
	2	$1 + 3q$
	3	$1 + 9q + 9q^2 + q^3$
$D_n$	2	$1$
	3	$1 + 3q$
	4	$1 + 6q + 6q^2 + q^3$

(Gaiffi et al., 2012, Gaiffi et al., 2018)

5. Minimal Wonderful Models for Generalized Dowling Arrangements

For $(n, G, V)$, the generalized Dowling arrangement's intersection poset is a generalized Dowling lattice. The minimal wonderful model is $Y_\mathcal{G}$ for the minimal building set, with boundary divisors indexed by irreducibles.

In the abelian case, enumerative formulas for nested sets are given by exponential generating functions: $$\lambda_H(t_H) = \sum_{i \geq 1} \frac{\lambda_{H,i}}{i!} t_H^i$$ with $\lambda_{H,i}$ counting rooted $H$-trees with $i$ labeled leaves. Integral formulas like

$$\overline{\lambda}_H(t) = \int \left[ \prod_{i \geq 2} \exp\left({\frac{1}{r} \frac{(rt)^i}{i!}}\right) - 1 \right] dt$$

describe $H$-tree enumeration, crucial for computing ranks and asymptotics (Gaiffi et al., 2018).

6. Applications in Satisfiability Modulo Theories: Minimal Model Cardinalities

In Satisfiability Modulo Theories, minimal models are studied via the minimal-model cardinalities function $\mu_{T,S}$, mapping quantifier-free formulas to the minimal tuples of cardinalities required to realize a model. The property "computable minimal-model-cardinalities" (CMMF) characterizes when this function is computable.

CMMF relates closely to classical combinatorial properties:

• Finite-Model Property (FMP): All satisfying minimal models are finite.
• Stable-Finiteness (SF): All minimal models may be shrunk to finite size. CMMF, FMP, SF, finite-witnessability, smoothness, and convexity interact as captured by a comprehensive taxonomy (Toledo et al., 2024).

Algorithms compute minimal-model cardinalities by enumerating candidate tuples and extracting minima via Dickson's Lemma. Complexity depends on the underlying theory's decision procedures and model bounds. Notable examples include threshold theories, infinite theories, and noncomputable model cardinalities (e.g., theories linked to Busy Beaver functions) (Toledo et al., 2024).

7. Classification and Structural Properties

For Weyl group invariant arrangements, the minimal building set occupies the minimal (bottom) position in the poset of building sets classified by antichains of "building partitions." This position reflects the strict irreducibility—maximal refinement of the geometric and combinatorial structure. The only nonempty intersections among boundary divisors correspond to totally ordered chains, and the boundary components themselves are disjoint.

For classical arrangements, $Y_{\mathcal{F}_{A_{n-1}}}$ is identified with $\overline{M}_{0,n+1}$, reflecting applications in moduli spaces, while for $B_n$ and $D_n$ types, the classification extends via "singular building partitions" (Gaiffi et al., 2012).

8. Concrete Example: Minimal Wonderful Model for $A_2$

In the $A_2$ case ($n=3$), the arrangement in $V = \{x_1 + x_2 + x_3 = 0\} \subset \mathbb{C}^3$ comprises three lines, corresponding to the minimal building set $\mathcal{F}_{A_2}$. $$Y_{\mathcal{F}} \cong \overline{M}_{0,4} \cong \mathbb{P}^1$$. Each boundary divisor is a point, and all are disjoint. The Poincaré polynomial is $1 + q$, and the cohomology is $H^*(\mathbb{P}^1) \cong \mathbb{Z}[h]/(h^2)$, exemplifying the general features of minimal wonderful models (Gaiffi et al., 2012).

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References:

• Families of building sets and regular wonderful models (Gaiffi et al., 2012)
• Wonderful models for generalized Dowling arrangements (Gaiffi et al., 2018)
• Combining Combination Properties: Minimal Models (Toledo et al., 2024)