Cactus Flower Moduli Spaces

• Cactus flower moduli spaces are moduli of stable genus-0 curves with cactus or flower geometries, defined via Deligne–Mumford and additive compactifications.
• They utilize FI-module methods, operadic models, and polytopal decompositions to compute cohomological invariants and ensure representation stability.
• These spaces unify real algebraic geometry, combinatorial group theory, and Gaudin-model representation theory through degeneration techniques and explicit topological constructions.

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Cactus flower moduli spaces are a family of closely related genus-$0$ moduli spaces in which stable marked curves acquire a distinctly “cactus-like” or “flower-like” geometry. In one foundational incarnation, the real loci $\overline{\mathcal M}_{0,n}(\mathbb R)$ of Deligne–Mumford compactifications parametrize stable rational curves over $\mathbb R$, hence trees of circles with labeled points; their fundamental groups are pure cactus groups, and their topology, cohomology, and representation theory have been studied by $FI$-module methods, operadic models, and geometric group theory (Duque et al., 2015). In a second, additive incarnation, the compactifications $\overline F_n$ of $F_n=(\mathbb C^n\setminus\Delta)/\mathbb C$ parametrize “flower curves,” namely bouquets of framed genus-$0$ components glued at a common distinguished point, and arise as special fibers of one-parameter degenerations whose generic fibers are $\overline M_{n+2}$ (Ilin et al., 2023). Weighted Hassett analogues over $\mathbb R$ interpolate between the usual real Deligne–Mumford spaces and $\mathbb{RP}^{n-2}$, producing weighted cactus groups and polytopal decompositions by products of permutahedra (Levinson et al., 29 Mar 2025). Taken together, these spaces form a nexus between real algebraic geometry, $\overline{\mathcal M}_{0,n}(\mathbb R)$0-topology, operad theory, and the representation theory of Gaudin models (Ilin et al., 2024).

1. Geometric definitions and the “cactus” and “flower” pictures

For $\overline{\mathcal M}_{0,n}(\mathbb R)$1, $\overline{\mathcal M}_{0,n}(\mathbb R)$2 is the smooth projective variety parametrizing stable genus-$\overline{\mathcal M}_{0,n}(\mathbb R)$3 curves with $\overline{\mathcal M}_{0,n}(\mathbb R)$4 labeled marked points. A stable rational curve $\overline{\mathcal M}_{0,n}(\mathbb R)$5 with marked points $\overline{\mathcal M}_{0,n}(\mathbb R)$6 is a finite union of projective lines $\overline{\mathcal M}_{0,n}(\mathbb R)$7 such that each marked point lies on exactly one component, any two components intersect transversely in at most one point, the dual graph is a tree, and each component carries at least three special points, marked or nodal. Over $\overline{\mathcal M}_{0,n}(\mathbb R)$8, each $\overline{\mathcal M}_{0,n}(\mathbb R)$9 is a topological circle, so the real locus $\mathbb R$0 may be visualized as a tree of circles decorated by labeled points; this is the geometric source of the cactus imagery (Duque et al., 2015).

A distinct but related construction begins from

$\mathbb R$1

the configuration space of $\mathbb R$2 distinct ordered points on the affine line modulo simultaneous translation. Its compactification $\mathbb R$3 parametrizes cactus flower curves: bouquets $\mathbb R$4 of framed cactus curves glued together at their distinguished points, collectively carrying $\mathbb R$5 marked points distributed among the petals. Each petal is itself a framed stable nodal $\mathbb R$6 with a nonzero tangent vector at the distinguished point. This framing removes automorphisms of each component, so $\mathbb R$7 is a projective variety with a natural $\mathbb R$8-action permuting the marked points (Ilin et al., 2024).

The additive compactification is controlled by coordinates $\mathbb R$9 on $FI$0, extended projectively so that $FI$1 records points lying on different petals. The compactification $FI$2 is defined by

$FI$3

equivalently by reciprocal coordinates $FI$4 satisfying

$FI$5

The map $FI$6 has fibers that are products of Deligne–Mumford spaces, making precise the idea that a flower curve is an additive configuration together with stable internal structure on each petal (Ilin et al., 2023).

This terminology is not uniform across the literature. In some papers, “cactus flower” refers to the real loci of stable genus-$FI$7 curves, especially when the tree-of-circles picture is emphasized; in others, it refers specifically to the additive compactifications $FI$8. The common feature is the replacement of a single irreducible $FI$9 by a tree or bouquet of components organized around a distinguished attachment pattern.

2. Real Deligne–Mumford loci, asphericity, and pure cactus groups

The real spaces $\overline F_n$0 are compact connected smooth manifolds. In the indexing $\overline F_n$1, one has $\overline F_n$2, with $\overline F_n$3 a point, $\overline F_n$4, and $\overline F_n$5 a connected sum of five real projective planes (Duque et al., 2015). In the shifted indexing $\overline F_n$6, the dimension is $\overline F_n$7, so $\overline F_n$8 and $\overline F_n$9 is a compact connected non-orientable surface with Poincaré polynomial $F_n=(\mathbb C^n\setminus\Delta)/\mathbb C$0 (Khoroshkin et al., 2019).

A basic theorem due to Davis–Januszkiewicz–Scott identifies these spaces as Eilenberg–MacLane spaces $F_n=(\mathbb C^n\setminus\Delta)/\mathbb C$1. Their fundamental groups are pure cactus groups: $F_n=(\mathbb C^n\setminus\Delta)/\mathbb C$2 or, in alternate notation,

$F_n=(\mathbb C^n\setminus\Delta)/\mathbb C$3

Thus

$F_n=(\mathbb C^n\setminus\Delta)/\mathbb C$4

There is also an $F_n=(\mathbb C^n\setminus\Delta)/\mathbb C$5-equivariant fundamental group, the cactus group, fitting into

$F_n=(\mathbb C^n\setminus\Delta)/\mathbb C$6

Its standard generators $F_n=(\mathbb C^n\setminus\Delta)/\mathbb C$7, $F_n=(\mathbb C^n\setminus\Delta)/\mathbb C$8, satisfy involutivity, commutation on disjoint intervals, and the cactus conjugation relation on nested intervals (Levinson et al., 29 Mar 2025).

The relation with configurations on the circle is explicit: $F_n=(\mathbb C^n\setminus\Delta)/\mathbb C$9 A loop reversing the block of labels in an interval $0$0 yields the generator $0$1, and van Kampen relations among such interval-reversal loops recover the cactus relations. This produces a geometric interpretation of $0$2 as the fundamental group of compactified configuration spaces of labeled points on $0$3, and for $0$4 one obtains explicit low-dimensional models: $0$5 with $0$6, and $0$7 with $0$8 (Hama et al., 11 May 2025).

These spaces also carry a canonical cellular decomposition by associahedra. Devadoss’ mosaic operad identifies $0$9 as a quotient of the associahedral cell structure by local $\overline M_{n+2}$0-reflections; equivalently, cells are indexed by planar rooted trees modulo reflections at vertices. This operadic structure is compatible with gluing of stable curves and underlies many later constructions (Khoroshkin et al., 2019).

3. Cohomology rings, operadic models, and representation stability

The cohomology of the real Deligne–Mumford loci is exceptionally explicit. Etingof–Henriques–Kamnitzer–Rains identify $\overline M_{n+2}$1 with a skew-commutative algebra $\overline M_{n+2}$2 generated in degree $\overline M_{n+2}$3 by antisymmetric symbols $\overline M_{n+2}$4, $\overline M_{n+2}$5, subject to

$\overline M_{n+2}$6

and

$\overline M_{n+2}$7

The $\overline M_{n+2}$8-action is induced by permuting indices,

$\overline M_{n+2}$9

A convenient generating set for $\mathbb R$0 is $\mathbb R$1, with no additional linear relations, and

$\mathbb R$2

where $\mathbb R$3 is the standard $\mathbb R$4-dimensional representation of $\mathbb R$5. For $\mathbb R$6,

$\mathbb R$7

and its character is the degree-$\mathbb R$8 character polynomial

$\mathbb R$9

In particular,

$\mathbb{RP}^{n-2}$0

Because the ring is generated multiplicatively by $\mathbb{RP}^{n-2}$1, these degree-$\mathbb{RP}^{n-2}$2 generators control all higher cohomology (Duque et al., 2015).

The $\mathbb{RP}^{n-2}$3-module formalism packages the sequence $\mathbb{RP}^{n-2}$4 into functorial $\mathbb{RP}^{n-2}$5-representations via forgetful morphisms

$\mathbb{RP}^{n-2}$6

along injections $\mathbb{RP}^{n-2}$7. In this setting, $\mathbb{RP}^{n-2}$8 is generated in degree $\mathbb{RP}^{n-2}$9, has weight $\overline{\mathcal M}_{0,n}(\mathbb R)$00, and has stability degree $\overline{\mathcal M}_{0,n}(\mathbb R)$01. Therefore the Church–Ellenberg–Farb theorem yields uniform representation stability with stable range

$\overline{\mathcal M}_{0,n}(\mathbb R)$02

For $\overline{\mathcal M}_{0,n}(\mathbb R)$03, the character of $\overline{\mathcal M}_{0,n}(\mathbb R)$04 is given by a unique character polynomial $\overline{\mathcal M}_{0,n}(\mathbb R)$05 in $\overline{\mathcal M}_{0,n}(\mathbb R)$06 of degree $\overline{\mathcal M}_{0,n}(\mathbb R)$07, and the Betti number $\overline{\mathcal M}_{0,n}(\mathbb R)$08 is given by a polynomial in $\overline{\mathcal M}_{0,n}(\mathbb R)$09 of degree $\overline{\mathcal M}_{0,n}(\mathbb R)$10 (Duque et al., 2015).

Operadically, the real loci form the mosaic operad. Khoroshkin–Willwacher describe the cellular chains as a cobar construction: $\overline{\mathcal M}_{0,n}(\mathbb R)$11 and identify a homotopy quotient presentation by the $\overline{\mathcal M}_{0,n}(\mathbb R)$12-action on the associative operad. The homology operad is isomorphic to the odd Poisson operad, while the cohomology rings are quadratic Koszul algebras generated by degree-$\overline{\mathcal M}_{0,n}(\mathbb R)$13 classes $\overline{\mathcal M}_{0,n}(\mathbb R)$14 with Arnol'd-type relations. The same work shows that $\overline{\mathcal M}_{0,n}(\mathbb R)$15 is not formal for $\overline{\mathcal M}_{0,n}(\mathbb R)$16, although it remains a rational $\overline{\mathcal M}_{0,n}(\mathbb R)$17-space. Thus asphericity does not imply formality in this setting (Khoroshkin et al., 2019).

A frequent misconception is that the real moduli spaces behave exactly like configuration spaces of points in the plane. The comparison with pure braid groups is strong but incomplete: the forgetful maps $\overline{\mathcal M}_{0,n}(\mathbb R)$18 are not fibrations, and for $\overline{\mathcal M}_{0,n}(\mathbb R)$19 the spaces are not formal. The cactus analogy is therefore geometric and representation-theoretic, not a literal transfer of all braid-group phenomena (Duque et al., 2015).

4. Weighted Hassett spaces and weighted cactus groups

A weighted extension is obtained by replacing ordinary stability with Hassett $\overline{\mathcal M}_{0,n}(\mathbb R)$20-stability. For

$\overline{\mathcal M}_{0,n}(\mathbb R)$21

an $\overline{\mathcal M}_{0,n}(\mathbb R)$22-pointed prestable curve $\overline{\mathcal M}_{0,n}(\mathbb R)$23 is $\overline{\mathcal M}_{0,n}(\mathbb R)$24-stable if for every smooth point $\overline{\mathcal M}_{0,n}(\mathbb R)$25,

$\overline{\mathcal M}_{0,n}(\mathbb R)$26

and for every irreducible component $\overline{\mathcal M}_{0,n}(\mathbb R)$27,

$\overline{\mathcal M}_{0,n}(\mathbb R)$28

The paper on weighted real stable curves specializes to the $\overline{\mathcal M}_{0,n}(\mathbb R)$29-symmetric vector

$\overline{\mathcal M}_{0,n}(\mathbb R)$30

Up to $\overline{\mathcal M}_{0,n}(\mathbb R)$31 of the first $\overline{\mathcal M}_{0,n}(\mathbb R)$32 points may collide, while the last point never collides. When $\overline{\mathcal M}_{0,n}(\mathbb R)$33 or $\overline{\mathcal M}_{0,n}(\mathbb R)$34, one recovers $\overline{\mathcal M}_{0,n}(\mathbb R)$35; when $\overline{\mathcal M}_{0,n}(\mathbb R)$36 and $\overline{\mathcal M}_{0,n}(\mathbb R)$37, one gets $\overline{\mathcal M}_{0,n}(\mathbb R)$38 (Levinson et al., 29 Mar 2025).

The corresponding equivariant and ordinary fundamental groups are

$\overline{\mathcal M}_{0,n}(\mathbb R)$39

For $\overline{\mathcal M}_{0,n}(\mathbb R)$40, $\overline{\mathcal M}_{0,n}(\mathbb R)$41 has generators $\overline{\mathcal M}_{0,n}(\mathbb R)$42 for intervals of length $\overline{\mathcal M}_{0,n}(\mathbb R)$43 or at least $\overline{\mathcal M}_{0,n}(\mathbb R)$44, satisfying the cactus relations together with braid relations

$\overline{\mathcal M}_{0,n}(\mathbb R)$45

There is a short exact sequence

$\overline{\mathcal M}_{0,n}(\mathbb R)$46

and the blowdown $\overline{\mathcal M}_{0,n}(\mathbb R)$47 induces a surjection

$\overline{\mathcal M}_{0,n}(\mathbb R)$48

As $\overline{\mathcal M}_{0,n}(\mathbb R)$49 increases, more cactus generators collapse to permutations, producing a chain of quotients

$\overline{\mathcal M}_{0,n}(\mathbb R)$50

At the extreme $\overline{\mathcal M}_{0,n}(\mathbb R)$51, the $\overline{\mathcal M}_{0,n}(\mathbb R)$52-factor is generated by the full reversal $\overline{\mathcal M}_{0,n}(\mathbb R)$53 (Levinson et al., 29 Mar 2025).

The geometry of these weighted spaces is encoded by a dual polytopal decomposition. In the unweighted case, dual cells are cubes

$\overline{\mathcal M}_{0,n}(\mathbb R)$54

indexed by stable trees $\overline{\mathcal M}_{0,n}(\mathbb R)$55. In the weighted case, for an $\overline{\mathcal M}_{0,n}(\mathbb R)$56-stable tree with $\overline{\mathcal M}_{0,n}(\mathbb R)$57 internal edges and leaf labels $\overline{\mathcal M}_{0,n}(\mathbb R)$58, the corresponding dual cell is

$\overline{\mathcal M}_{0,n}(\mathbb R)$59

where $\overline{\mathcal M}_{0,n}(\mathbb R)$60 is the permutahedron. Hexagonal $\overline{\mathcal M}_{0,n}(\mathbb R)$61-cells supply the braid relations; square cells supply commuting and cactus relations. This decomposition generalizes the known dual cube decomposition of $\overline{\mathcal M}_{0,n}(\mathbb R)$62 and makes visible how weighted collisions deform the topology from cubes to products of permutahedra (Levinson et al., 29 Mar 2025).

5. Additive flower curves, compactifications, and degeneration from $\overline{\mathcal M}_{0,n}(\mathbb R)$63

The additive spaces $\overline{\mathcal M}_{0,n}(\mathbb R)$64 are natural compactifications of configurations on the affine line modulo translation. They are defined so that $\overline{\mathcal M}_{0,n}(\mathbb R)$65 is an open dense subset of dimension $\overline{\mathcal M}_{0,n}(\mathbb R)$66, and there is a natural map

$\overline{\mathcal M}_{0,n}(\mathbb R)$67

whose fiber over a point $\overline{\mathcal M}_{0,n}(\mathbb R)$68 is a product $\overline{\mathcal M}_{0,n}(\mathbb R)$69, where $\overline{\mathcal M}_{0,n}(\mathbb R)$70 records which marked points lie on each petal. Boundary strata are combinatorial: codimension-$\overline{\mathcal M}_{0,n}(\mathbb R)$71 strata arise either by separating marked points into two blocks $\overline{\mathcal M}_{0,n}(\mathbb R)$72 with $\overline{\mathcal M}_{0,n}(\mathbb R)$73 for all $\overline{\mathcal M}_{0,n}(\mathbb R)$74, $\overline{\mathcal M}_{0,n}(\mathbb R)$75, or by collapsing a subset $\overline{\mathcal M}_{0,n}(\mathbb R)$76 through the conditions $\overline{\mathcal M}_{0,n}(\mathbb R)$77 for $\overline{\mathcal M}_{0,n}(\mathbb R)$78. Higher codimension strata are obtained by iteration (Ilin et al., 2024).

A central structural result is the existence of a one-parameter degeneration from multiplicative to additive moduli. There is a group scheme over $\overline{\mathcal M}_{0,n}(\mathbb R)$79,

$\overline{\mathcal M}_{0,n}(\mathbb R)$80

with multiplication

$\overline{\mathcal M}_{0,n}(\mathbb R)$81

For $\overline{\mathcal M}_{0,n}(\mathbb R)$82, $\overline{\mathcal M}_{0,n}(\mathbb R)$83, while $\overline{\mathcal M}_{0,n}(\mathbb R)$84. The relative compactification $\overline{\mathcal M}_{0,n}(\mathbb R)$85 has fiber $\overline{\mathcal M}_{0,n}(\mathbb R)$86 for $\overline{\mathcal M}_{0,n}(\mathbb R)$87 and fiber $\overline{\mathcal M}_{0,n}(\mathbb R)$88 for $\overline{\mathcal M}_{0,n}(\mathbb R)$89. Geometrically, two special marked points of a stable genus-$\overline{\mathcal M}_{0,n}(\mathbb R)$90 curve coalesce to form a single distinguished point with a nonzero tangent vector, yielding a flower curve in the limit (Ilin et al., 2024).

The space $\overline{\mathcal M}_{0,n}(\mathbb R)$91 admits affine charts $\overline{\mathcal M}_{0,n}(\mathbb R)$92 indexed by planar binary forests $\overline{\mathcal M}_{0,n}(\mathbb R)$93, equipped with regular functions $\overline{\mathcal M}_{0,n}(\mathbb R)$94, $\overline{\mathcal M}_{0,n}(\mathbb R)$95, and mixed terms $\overline{\mathcal M}_{0,n}(\mathbb R)$96. It also contains distinguished loci: the equations $\overline{\mathcal M}_{0,n}(\mathbb R)$97 for all $\overline{\mathcal M}_{0,n}(\mathbb R)$98 define a unique $\overline{\mathcal M}_{0,n}(\mathbb R)$99-fixed maximal flower point $\mathbb R$00, while $\mathbb R$01 for all $\mathbb R$02 defines a divisor naturally isomorphic to $\mathbb R$03. More generally, in the total family over $\mathbb R$04, the locus $\mathbb R$05 is a divisor isomorphic to $\mathbb R$06 (Kamnitzer et al., 17 Jul 2025).

Over $\mathbb R$07, the space $\mathbb R$08 has an explicit combinatorial model by a $\mathbb R$09 cube complex $\mathbb R$10 indexed by unordered planar forests, and $\mathbb R$11 is modeled by a quotient $\mathbb R$12 of the permutahedron obtained by identifying parallel faces. Both are nonpositively curved and hence aspherical. Their equivariant fundamental groups are

$\mathbb R$13

the virtual symmetric and virtual cactus groups. Moreover, a twisted real degeneration from $\mathbb R$14 to $\mathbb R$15 induces a natural surjective homomorphism

$\mathbb R$16

from the extended affine cactus group (Ilin et al., 2023).

These additive spaces should not be conflated with the real loci $\mathbb R$17. The former compactify affine configurations modulo translation and carry virtual cactus symmetry; the latter are real Deligne–Mumford spaces with ordinary cactus symmetry. The degeneration $\mathbb R$18 explains their proximity while preserving the distinction.

6. Operadic coverings, Gaudin models, and monodromy

The spaces $\mathbb R$19 and $\mathbb R$20 support compatible gluing maps. Besides the Deligne–Mumford operad structure $\mathbb R$21 on $\mathbb R$22, there is a right module structure

$\mathbb R$23

and an internal flower-gluing operation

$\mathbb R$24

Boundary divisors of $\mathbb R$25 are unions of images of the $\mathbb R$26 and $\mathbb R$27, and $\mathbb R$28 identifies $\mathbb R$29 as the divisor $\mathbb R$30 in $\mathbb R$31. This yields a precise operadic framework in which finite coverings of real loci can be organized by tensor-categorical data (Kamnitzer et al., 17 Jul 2025).

Kamnitzer–Rybnikov show that isomorphism classes of operadic coverings of $\mathbb R$32 are naturally equivalent to equivalence classes of concrete coboundary monoidal categories satisfying semisimplicity and finiteness conditions. A $\mathbb R$33-colored operadic covering consists of $\mathbb R$34-equivariant coverings

$\mathbb R$35

together with covering isomorphisms compatible with $\mathbb R$36, $\mathbb R$37, and $\mathbb R$38. On the categorical side, one has a concrete $\mathbb R$39-colored coboundary monoidal category with multiplicity sets, associator, coboundary commutor, and faithful monoidal functor to sets. The equivalence reconstructs one structure from the other by parallel transport along canonical real paths in $\mathbb R$40 and $\mathbb R$41 (Kamnitzer et al., 17 Jul 2025).

The principal representation-theoretic application is to Gaudin algebras. For a simple Lie algebra $\mathbb R$42, the homogeneous Gaudin algebra $\mathbb R$43 extends over $\mathbb R$44, the trigonometric Gaudin algebra $\mathbb R$45 extends over $\mathbb R$46, and the inhomogeneous Gaudin algebra $\mathbb R$47 extends over $\mathbb R$48. These families are unified by

$\mathbb R$49

with

$\mathbb R$50

Thus $\mathbb R$51 is the universal parameter space for the degeneration from trigonometric to inhomogeneous Gaudin subalgebras (Ilin et al., 2024).

Under explicit split or compact real conditions, these commutative algebras act with simple spectrum on tensor products of irreducible $\mathbb R$52-modules. The resulting sets of Bethe eigenlines form equivariant coverings over the relevant real loci. For inhomogeneous Gaudin algebras over $\mathbb R$53, the monodromy group is $\mathbb R$54; for trigonometric Gaudin algebras over the split real form of $\mathbb R$55, the monodromy group is the mirabolic cactus group; for the compact real form, it is the extended affine cactus group. In both trigonometric cases, the monodromy factors through $\mathbb R$56. Using the operadic-covering classification, one recovers Kashiwara $\mathbb R$57-crystals as a concrete coboundary category from these Bethe eigenline coverings, and the resulting $\mathbb R$58-action matches the crystal commutor action on tensor products (Kamnitzer et al., 17 Jul 2025).

In the minuscule case, this provides a combinatorial version of the Bezrukavnikov–Okounkov wall-crossing conjecture for quantum cohomology of minuscule resolutions of affine Grassmannian slices. The point established in the source is precise: the eigenline monodromy of trigonometric Gaudin models over the compact real locus realizes the expected cactus-type wall-crossing on crystals (Kamnitzer et al., 17 Jul 2025).

7. Structural scope, comparisons, and open questions

Several adjacent moduli theories meet here. The real Deligne–Mumford spaces $\mathbb R$59 form the classical cactus landscape: they are smooth aspherical manifolds, mosaic-operadic, and governed by pure cactus groups. The weighted Hassett spaces $\mathbb R$60 interpolate between these and projective space, replacing cubes by products of permutahedra and introducing braid relations into cactus presentations. The additive compactifications $\mathbb R$61 are not merely weighted variants; they arise from a different quotient, $\mathbb R$62, and are tied to virtual cactus symmetry and to the degeneration of $\mathbb R$63 (Levinson et al., 29 Mar 2025).

The comparison with braid-type families is pervasive but quantitatively different. Configuration spaces $\mathbb R$64, pure braid groups, pure virtual and flat braid groups, pure string motions, and $\mathbb R$65 all admit $\mathbb R$66-module approaches when cohomology is generated in degree $\mathbb R$67. The cactus case mirrors this pattern, but its generators have arity $\mathbb R$68, namely the classes $\mathbb R$69, which is why weight and character-degree bounds are $\mathbb R$70 rather than the order-$\mathbb R$71 bounds typical in braid-type examples (Duque et al., 2015).

The main unresolved issues recorded in the cited work concern coefficients, group theory, and sharpness. The $\mathbb R$72-module arguments require characteristic $\mathbb R$73, usually $\mathbb R$74, and do not address torsion in $\mathbb R$75. It is not known whether $\mathbb R$76 is residually nilpotent. The effective stable ranges $\mathbb R$77 and degree bounds $\mathbb R$78 are not claimed to be optimal. On the additive side, $\mathbb R$79 has rich real topology and strong applications to Gaudin monodromy, but its deeper birational and singularity-theoretic features are not the focus of the cited sources (Duque et al., 2015).

A further source of ambiguity is that “cactus flower” has also been used in a different algebro-geometric context for low-socle loci inside Hilbert schemes that control Grassmann cactus varieties. There the relevant moduli functor $\mathbb R$80 parametrizes degree-$\mathbb R$81 finite subschemes with socle dimension at most $\mathbb R$82, and the term denotes parameter spaces of low-socle finite schemes rather than genus-$\mathbb R$83 nodal curves (Buczyńska et al., 29 Jul 2025). This suggests that the phrase has become a broader organizing label for moduli spaces governed by cactus-type incidence phenomena, but the curve-theoretic spaces $\mathbb R$84, $\mathbb R$85, and $\mathbb R$86 remain the central objects in the topology–operad–Gaudin literature.

Within that literature, cactus flower moduli spaces provide a unified geometric language for trees of circles, bouquets of framed rational components, weighted collision patterns, and degenerating Gaudin parameter spaces. Their significance lies in the simultaneous presence of explicit combinatorics, computable cohomology, operadic factorization, and nontrivial monodromy actions by cactus, affine cactus, and virtual cactus groups.