Iterated Cuspidal Locus: A Multifaceted Overview

• Iterated Cuspidal Locus is a stratified geometric structure where a primary cusp condition is reimposed to generate secondary layers, seen in settings like hyperbolic dynamics and algebraic geometry.
• It encapsulates multifractal decompositions by encoding repeated cusp excursions, using asymptotic ratios and symbolic dynamics to measure the influence of long cusp windings.
• The concept also models nested degenerations in incidence and eigenvariety frameworks, where cusp-induced strata lead to refined singularity resolutions and higher-derivative bifurcation phenomena.

“Iterated cuspidal locus” is not a standard formal term in the literature represented here. The closest precise realizations are loci defined by repeated cusp excursions in hyperbolic dynamics, nested cusp-induced strata in incidence varieties, derivative hierarchies controlling cusp bifurcation, and canonical filtrations of cusp-type thickenings. This suggests using the term for a locus obtained when a primary cuspidal condition is encoded, stratified, or reimposed at a further level of the construction, so that cusp geometry reappears as a secondary asymptotic, singular, or moduli-theoretic structure (Jaerisch et al., 2016, Boissiere et al., 2022, Waters, 2017).

1. Terminological status and general meaning

Several of the relevant papers explicitly do not use the phrase “iterated cuspidal locus” verbatim, but they isolate objects that are very close to that idea. In the hyperbolic-surface setting, the natural candidates are the level sets

$\mathcal F_\alpha$

of points whose geodesic rays undergo repeated cusp excursions with a prescribed long-term normalized winding rate. In the Fano-variety setting, the nearest analogue is the hierarchy

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$

In the conjugate-locus setting, the relevant hierarchy is

$$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$

And in the eigenvariety setting, the nested sequence

$\operatorname{Ram}(\wt)\subset \mathcal E_0^{\fl}\subset \mathcal E_0\subset \mathcal E$

provides a cuspidal locus with a further distinguished sublocus defined by ramification (Jaerisch et al., 2016, Boissiere et al., 2022, Waters, 2017, Wu, 2020).

A common feature across these contexts is that the first cuspidal condition does not terminate the geometry. Instead, it generates a second structure: a multifractal decomposition, a singular stratum in a moduli space, a higher-derivative bifurcation set, a tower of nilpotent neighborhoods, or a ramified subspace inside a cuspidal eigenvariety. In that restricted but precise sense, the “iterated” aspect is not repetition of the same cusp equation alone; it is repetition of cusp geometry at the level of loci, filtrations, or asymptotic invariants.

2. Hyperbolic surfaces and multifractal cusp iteration

The most direct realization of an iterated cuspidal locus appears in the multifractal analysis of a finitely generated, free, non-elementary Fuchsian group $G$ with parabolic elements, acting on the upper half-plane $(\mathbb H,d)$. The group is written as

$G=H*\Gamma,$

where

$H=\langle h_1\rangle * \cdots * \langle h_u\rangle$

is the free product of finitely many elementary hyperbolic groups, and

$$\Gamma=\langle \gamma_1\rangle * \cdots * \langle \gamma_v\rangle$$

is the free product of finitely many parabolic subgroups, with $v\ge 1$. The quotient $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$0 is a hyperbolic surface with cusps, and the relevant boundary set is the radial limit set $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$1 (Jaerisch et al., 2016).

The coding is Bowen–Series type. For $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$2, the geodesic ray $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$3 from $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$4 to $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$5 yields a reduced infinite word

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$6

which is then decomposed into blocks $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$7. Every hyperbolic letter forms a block of length $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$8, while $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$9 consecutive repetitions of the same parabolic generator form a block of length $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$0. Geometrically, a block of length $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$1 corresponds to a cusp excursion in which the projected geodesic winds exactly $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$2 times around a cusp. The resulting cusp winding process is

$$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$3

Thus $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$4 for a hyperbolic block, $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$5 for a minimal parabolic passage, and $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$6 for a deep cusp excursion.

The central observable is the asymptotic ratio

$$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$7

The associated level sets are

$$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$8

together with the one-sided variants $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$9. These are the multifractal strata of the limit set by asymptotic cusp-iteration rate. The paper states that if one wants an “iterated cuspidal locus,” these $\operatorname{Ram}(\wt)\subset \mathcal E_0^{\fl}\subset \mathcal E_0\subset \mathcal E$0 are the natural candidates.

The geometric reason is the excursion-length estimate. If $\operatorname{Ram}(\wt)\subset \mathcal E_0^{\fl}\subset \mathcal E_0\subset \mathcal E$1 denotes the geodesic arc corresponding to the $\operatorname{Ram}(\wt)\subset \mathcal E_0^{\fl}\subset \mathcal E_0\subset \mathcal E$2-th block, then

$\operatorname{Ram}(\wt)\subset \mathcal E_0^{\fl}\subset \mathcal E_0\subset \mathcal E$3

and for $\operatorname{Ram}(\wt)\subset \mathcal E_0^{\fl}\subset \mathcal E_0\subset \mathcal E$4,

$\operatorname{Ram}(\wt)\subset \mathcal E_0^{\fl}\subset \mathcal E_0\subset \mathcal E$5

More generally,

$\operatorname{Ram}(\wt)\subset \mathcal E_0^{\fl}\subset \mathcal E_0\subset \mathcal E$6

Accordingly, the defining ratio measures the proportion of total geodesic length spent in repeated cusp winding.

The symbolic dynamics is induced on

$\operatorname{Ram}(\wt)\subset \mathcal E_0^{\fl}\subset \mathcal E_0\subset \mathcal E$7

and the induced map $\operatorname{Ram}(\wt)\subset \mathcal E_0^{\fl}\subset \mathcal E_0\subset \mathcal E$8 is conjugate to a countable-state Markov shift $\operatorname{Ram}(\wt)\subset \mathcal E_0^{\fl}\subset \mathcal E_0\subset \mathcal E$9. On this shift the Hölder potentials

$G$0

and

$G$1

encode cusp winding and geometric contraction. Their Birkhoff sums recover the geometric ratio, and

$G$2

The pressure is defined both geometrically and dynamically: $G$3 and

$G$4

with

$G$5

Moreover,

$G$6

The free energy $G$7 is defined by

$G$8

and $G$9 is real-analytic and strictly convex.

The multifractal spectrum is

$(\mathbb H,d)$0

and the main formula is

$(\mathbb H,d)$1

with

$(\mathbb H,d)$2

The possible exponents are completely classified: $(\mathbb H,d)$3 At the endpoints,

$(\mathbb H,d)$4

where

$(\mathbb H,d)$5

and the Jarník-type set

$(\mathbb H,d)$6

has

$(\mathbb H,d)$7

Thus the iterated cuspidal locus decomposes into a one-parameter family indexed exactly by $(\mathbb H,d)$8, from no asymptotic cusp contribution to maximal cusp dominance.

3. Incidence geometry and nested cusp-induced strata

A distinct realization appears in the Fano variety of lines of a cuspidal cyclic cubic fourfold. The starting point is a nodal cubic threefold

$(\mathbb H,d)$9

with one ordinary double point $G=H*\Gamma,$0, given in suitable coordinates by

$G=H*\Gamma,$1

The associated cuspidal cyclic cubic fourfold is the triple cover of $G=H*\Gamma,$2 branched along $G=H*\Gamma,$3, realized as

$G=H*\Gamma,$4

The node $G=H*\Gamma,$5 of $G=H*\Gamma,$6 produces an isolated singularity of type $G=H*\Gamma,$7 on $G=H*\Gamma,$8 (Boissiere et al., 2022).

Inside the hyperplane $G=H*\Gamma,$9, the paper sets

$H=\langle h_1\rangle * \cdots * \langle h_u\rangle$0

and defines

$H=\langle h_1\rangle * \cdots * \langle h_u\rangle$1

Their intersection

$H=\langle h_1\rangle * \cdots * \langle h_u\rangle$2

is smooth and is a K3 surface. The Fano scheme

$H=\langle h_1\rangle * \cdots * \langle h_u\rangle$3

contains $H=\langle h_1\rangle * \cdots * \langle h_u\rangle$4 as the locus of lines in $H=\langle h_1\rangle * \cdots * \langle h_u\rangle$5 passing through the cusp $H=\langle h_1\rangle * \cdots * \langle h_u\rangle$6. This $H=\langle h_1\rangle * \cdots * \langle h_u\rangle$7 is the primary cusp-induced locus.

The paper proves that $H=\langle h_1\rangle * \cdots * \langle h_u\rangle$8 is exactly the singular stratum controlling the singularities of $H=\langle h_1\rangle * \cdots * \langle h_u\rangle$9: in the cuspidal cyclic case, $$\Gamma=\langle \gamma_1\rangle * \cdots * \langle \gamma_v\rangle$$0 has transversal $$\Gamma=\langle \gamma_1\rangle * \cdots * \langle \gamma_v\rangle$$1-singularities along $$\Gamma=\langle \gamma_1\rangle * \cdots * \langle \gamma_v\rangle$$2, and in the generic no-plane-through-the-cusp situation,

$$\Gamma=\langle \gamma_1\rangle * \cdots * \langle \gamma_v\rangle$$3

There is also a divisor

$$\Gamma=\langle \gamma_1\rangle * \cdots * \langle \gamma_v\rangle$$4

contracted by the birational map

$$\Gamma=\langle \gamma_1\rangle * \cdots * \langle \gamma_v\rangle$$5

When $$\Gamma=\langle \gamma_1\rangle * \cdots * \langle \gamma_v\rangle$$6 contains no plane through $$\Gamma=\langle \gamma_1\rangle * \cdots * \langle \gamma_v\rangle$$7, $$\Gamma=\langle \gamma_1\rangle * \cdots * \langle \gamma_v\rangle$$8 is regular and contracts $$\Gamma=\langle \gamma_1\rangle * \cdots * \langle \gamma_v\rangle$$9 onto $v\ge 1$0.

The local singularity analysis identifies the transverse surface germ along $v\ge 1$1 as $v\ge 1$2-type. After local elimination, the transverse germ is determined by equations whose lowest-order terms are a rank-$v\ge 1$3 quadratic form plus a cubic term $v\ge 1$4, so the transverse singularity is analytically of $v\ge 1$5-type, equivalently of the form

$v\ge 1$6

up to analytic changes of coordinates. The full singularity along $v\ge 1$7 is locally a product of a smooth $v\ge 1$8-dimensional germ with an $v\ge 1$9 surface singularity.

The blowup

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$00

is a symplectic resolution. Over a point of $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$01, the fiber of $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$02 is, in the cuspidal cyclic case, “the union of two rational curves intersecting transversally.” Thus the cusp of $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$03 induces a surface $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$04 of cusp-lines, and over that surface the exceptional geometry splits in the $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$05 pattern. The nested structure

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$06

is the paper’s closest model for an iterated cuspidal locus.

A further degeneration occurs at the distinguished point

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$07

where the residual conic becomes

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$08

so the line $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$09 is a triple line with tritangent plane $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$10. The paper does not discuss further successive blowups, because the blowup along $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$11 already resolves the symplectic singularity. Accordingly, any iteration beyond this stage is interpretive rather than proved.

4. Derivative hierarchies and bifurcation loci of cusps

In the differential geometry of surfaces, the conjugate locus of a point provides another rigorous hierarchy in which cusp conditions are iterated. Let $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$12 be a point on a smooth surface $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$13, and let

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$14

be the family of unit-speed geodesics issuing from $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$15, where $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$16 is arc length and $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$17 is initial direction. The exponential map is

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$18

The first angular variation

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$19

is a Jacobi field. Writing $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$20 in a parallel orthonormal frame, the Jacobi equation becomes

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$21

with

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$22

Conjugate points are determined by the first zero $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$23 of $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$24, and the conjugate locus is

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$25

A cusp of $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$26 occurs when $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$27 (Waters, 2017).

The second derivative

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$28

satisfies Bażański’s equation, and in the $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$29 frame one obtains

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$30

Since

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$31

the tangential component vanishes automatically on the conjugate locus. Therefore a cusp is detected by

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$32

The third derivative

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$33

produces a further scalar equation

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$34

At a cusp, an $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$35 singularity or arc bifurcation occurs when also

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$36

Hence the paper yields the hierarchy

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$37

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$38

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$39

This derivative stratification is accompanied by a singularity classification. If $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$40 has an $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$41 singularity at $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$42, then the conjugate locus is $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$43 at the corresponding point. Thus $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$44 gives the ordinary cusp $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$45, $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$46 gives the arc-bifurcation model $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$47, and $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$48 gives the symmetric cusp-bifurcation model $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$49.

The tangent-plane contour picture makes the hierarchy especially explicit. In $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$50, the curves

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$51

intersect at the cusps of the conjugate locus. As $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$52 moves, an arc bifurcation occurs when these contours come into tangential contact and then intersect transversely, creating two intersections and hence two new cusps. The paper also notes that the image of the $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$53 contour generally is smooth but develops cusps precisely at bifurcation.

A natural candidate for an iterated cuspidal locus suggested by this framework is

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$54

This formulation is interpretive rather than a formal definition in the paper, but it follows closely from the paper’s higher-derivative criteria.

5. Moduli of cuspidal curves and cusp-type infinitesimal thickenings

In enumerative geometry, a cusp can be realized as the zero locus of a tautological derivative section on a moduli space. For

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$55

and

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$56

the paper defines

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$57

The main moduli space for the cusp problem is

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$58

where the first $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$59 marked points meet generic constraints and the last marked point is free. The cusp condition is imposed by the vanishing of the differential at the last marked point: $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$60 and the cuspidal count is

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$61

After extending the section to the compactification $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$62, the Euler-class identity is

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$63

The paper treats only a single cusp, but it explicitly states that the method “readily applies” to more degenerate singularities, and its bundle-theoretic construction suggests repeated singularity imposition by adding further free marked points and further derivative-vanishing conditions (Biswas et al., 2015).

A more literal iterative structure appears in cuspidal nilpotent multiple structures on a smooth variety $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$64. In characteristic $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$65, with $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$66 smooth connected and $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$67 smooth, a multiple structure $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$68 is a locally Cohen–Macaulay scheme with $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$69. The paper defines $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$70 to be cuspidal of type $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$71 if locally

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$72

The two classes treated in detail are

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$73

and

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$74

(Manolache, 2010).

The construction is organized by three canonical filtrations,

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$75

with associated graded objects

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$76

For $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$77, the local ideals evolve as

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$78

followed by the final cusp closure

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$79

For $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$80, the analogous pattern is

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$81

ending with

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$82

These are genuinely iterative cusp-type thickenings: one builds successive nilpotent neighborhoods and imposes the cusp relation only at the terminal stage. The structure is controlled by line bundles and extensions, with key numerical constraints

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$83

for $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$84 and

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$85

for $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$86. In this setting, an iterated cuspidal locus is not a subset of a parameter space but a tower of infinitesimal cusp layers on fixed support.

6. Parameter loci, cuspidal eigenvarieties, and recurrent structural themes

The phrase also admits more indirect realizations in parameter spaces. For pairs of affine maps in the plane, the connectedness locus in the diagonal case is

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$87

The paper proves that $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$88 has many zero-angle cusp corners. If $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$89 is realized by a unique $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$90 with eventually $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$91 coefficients and

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$92

then there exist $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$93 and $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$94 such that

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$95

This is a parameter-space cusp locus rather than a cuspidal locus in the sense of boundary singularities, but it shows how cusp geometry can arise as a rigid secondary structure in an iterated-function-system setting (Solomyak, 2014).

A closer analogue to a nested cuspidal locus occurs for the cuspidal eigenvariety of $$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$96. The paper defines parabolic cohomology by

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$97

equivalently as the kernel of boundary restriction. The resulting cuspidal eigenvariety is

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$98

and there is a closed immersion

$$\text{cusp of }Y \;\leadsto\; \Sigma \subset F(Y) \;\leadsto\; \text{reducible exceptional fibers over }\Sigma.$$99

where $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$00 is the full eigenvariety. Inside $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$01, the paper isolates the flat locus $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$02 and studies the ramification locus of the weight map

$$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$03

Locally, ramification is cut out by the Noether different $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$04, and at a good point the adjoint $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$05-adic $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$06-function satisfies

$$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$07

Moreover,

$$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$08

This yields a nested sequence

$$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$09

which is a precise cuspidal locus together with a further local-geometric refinement (Wu, 2020).

Taken together, these examples show that the strongest uses of “iterated cuspidal locus” are structural rather than terminological. In the most explicit dynamical case, the locus is a multifractal stratum $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$10 defined by repeated cusp excursions. In the incidence-geometric case, it is the cusp-induced singular surface $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$11 together with its $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$12-exceptional fibers. In the differential-geometric case, it is the higher-derivative bifurcation set where cusp conditions themselves change. In the infinitesimal-algebraic case, it is a filtration of cusp-type nilpotent thickenings. And in the eigenvariety setting, it is a cuspidal subspace with a ramified sublocus detected by an adjoint $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$13-adic $$\text{conjugate point} \;\leadsto\; \text{cusp of }C_p \;\leadsto\; \text{cusp bifurcation locus}.$$14-function. The term is therefore best understood as a convenient designation for mathematically precise situations in which cusp geometry propagates to a second stage of stratification, degeneration, or asymptotic classification.