Sesquicuspidal Curves: Properties & Applications

• Sesquicuspidal curves are rational curves defined by a single cusp of type (p,q) with possible nodes, illustrating a clear nexus between algebraic and symplectic geometry.
• Their classification employs explicit existence criteria, including Fibonacci ratios and moduli transitions, which connect toric, tropical, and mirror symmetry methods.
• These curves yield sharp symplectic embedding obstructions and reveal deep links with Gromov–Witten invariants and singularity theory in algebraic geometry.

A sesquicuspidal curve is a rational algebraic or symplectic curve characterized by a unique singularity of cusp type and, possibly, auxiliary ordinary double points (nodes). In algebraic geometry, these typically appear as rational plane curves with precisely one distinguished cusp of type $(p,q)$ and (optionally) ordinary nodes elsewhere. Their classification, existence criteria, moduli, and deep relations to symplectic embedding problems—especially in dimension four—connect algebraic singularity theory, tropical and toric mirror symmetry, and modern Gromov–Witten theory.

1. Definitions and Local–Global Structure

A complex plane curve singularity is of \emph{cusp type} $(p,q)$ if it is locally analytically equivalent to %%%%2%%%%, with $p > q \geq 1$ coprime. A rational irreducible plane curve $C \subset \CP^2$ is termed sesquicuspidal if it possesses exactly one cusp of type $(p,q)$ and otherwise at most ordinary double points as singularities (McDuff et al., 30 Nov 2024, McDuff et al., 23 Apr 2024, Siegel, 17 Nov 2025).

In the symplectic category, a $(p,q)$–sesquicuspidal symplectic curve in a 4-manifold $(M^4,\omega)$ is a smooth embedded symplectic surface except at one point where it is modeled symplectomorphically on the analytic $(p,q)$-cusp, and possibly at finitely many nodes elsewhere (Siegel, 17 Nov 2025).

More generally, a nearly cuspidal (or sesquicuspidal) curve can be defined as an irreducible curve all of whose singularities are cusps except for a single singularity with precisely two branches—this broader class relates to the syzygetic structure of Jacobian modules (Dimca et al., 2018).

2. Classification Theorems and Existence Criteria

The classification of rational sesquicuspidal curves, particularly in the context of algebraic geometry, centers on the following existence theorem for rational index-zero cases in $\CP^2$:

• For pairs $(p, q)$ with $p > q \geq 2$, $p + q \equiv 0 \bmod 3$, a rational degree-$d$ plane curve $C \subset \CP^2$, $d = (p + q)/3$, with exactly one $(p, q)$-cusp and only nodes elsewhere exists if and only if either:
  • $p/q = F_{2k+1}/F_{2k-1}$ for some odd $k \geq 3$, where $F_n$ are Fibonacci numbers (discrete rigid cases), or
  • $p/q > \varphi^4 = (7+3\sqrt{5})/2$ (supercritical regime with continuous moduli) (McDuff et al., 30 Nov 2024, Siegel, 17 Nov 2025).

In particular, for $p/q$ below the threshold $\varphi^4$, only the explicit Fibonacci ratios yield such curves, and then the moduli are discrete. When $p/q > \varphi^4$, there exists a family of sesquicuspidal curves for every such $(p,q)$ and the moduli become continuous.

In the broader class of nearly cuspidal curves studied via syzygies, rational nearly cuspidal curves of even degree $d$ are classified as free, nearly free, or plus-one generated (based on Jacobian syzygy module structure) (Dimca et al., 2018). Odd-degree examples may also exhibit 4-syzygy structure.

For singularities with two Newton pairs, the local analytic type is classified by the pairs $(p_1, q_1), (p_2, q_2)$ and their associated parametrization, delta-invariant, and value semigroup, with explicit realizability except for two cases (Bodnár, 2015).

3. Invariants, Moduli, and Algebraic Structures

Classical invariants for sesquicuspidal curves include:

• Adjunction formula: For a degree-$d$ sesquicuspidal curve with cusp of type $(p, q)$ and only nodes elsewhere,

$$\delta = \frac{1}{2}(d-1)(d-2) - \frac{1}{2}(p-1)(q-1)$$

gives the number of nodes.

• Index: The virtual moduli dimension, or index, associated to a homology class $[C] \in H_2(M)$ is

$\operatorname{ind}(C) = 2(c_1([C]) - (p+q))$

Index-zero curves satisfy $c_1([C]) = p+q$.

• Minimal degree: The minimal possible degree for a plane rational curve with a $(p,q)$–cusp is $d_{\min}(p,q) = (p+q)/3$ whenever this is integral and the existence criterion is met (McDuff et al., 30 Nov 2024, Siegel, 17 Nov 2025).
• Value semigroup and counting function (in two-Newton-pair cusps): The semigroup is generated by $(p_1p_2, q_1p_2, p_1p_2q_1 + q_2)$ with corresponding counting function appearing in the Borodzik–Livingston condition and related conjectures (Bodnár, 2015).

In Jacobian module theory, rational nearly cuspidal curves are $m$-syzygy, most notably $m=3$ (plus-one generated or nearly free) or $m=4$ (for certain odd degrees) (Dimca et al., 2018). The global Tjurina number and minimal degree of nontrivial syzygy ($\mathrm{mdr}(f)$) further constrain the possible curves.

4. Construction Techniques and Scattering/Tropical Methods

Explicit constructions exploit birational and toric techniques:

• Toric and Looijenga pair construction: Every index-zero sesquicuspidal curve in $\CP^2$ corresponds, via a “fundamental bijection,” to a toric curve in the suitable blowup of a toric surface. Iterated cluster symmetries (mutations) yield all such curves (McDuff et al., 30 Nov 2024, Siegel, 17 Nov 2025).
• Scattering diagrams and the tropical vertex theorem: The counts of stable maps with specified tangency to boundary divisors in toric varieties (encoded in a scattering diagram) determine the existence and abundance of sesquicuspidal curves. Log Gromov–Witten invariants match coefficients in the tropical vertex expansion (McDuff et al., 30 Nov 2024, Siegel, 17 Nov 2025).
• Orevkov twist and generalizations: Orevkov's birational construction, via iterated blowups and blowdowns along nodal cubics, generates the classical “outer-corner” Fibonacci families, with generalizations to del Pezzo surfaces producing full infinite staircases in symplectic embedding obstructions (McDuff et al., 23 Apr 2024).

Well-placed curves in Looijenga pairs—curves meeting a nodal anticanonical divisor $D$ only at its node, with prescribed multiplicities to each branch—automatically yield sesquicuspidal singularities at the node (Siegel, 17 Nov 2025).

5. Symplectic Applications: Ellipsoid Embedding Obstructions

Sesquicuspidal curves play a decisive role in the paper of symplectic embedding problems, notably in the structure of embedding functions for four-dimensional ellipsoids:

• Obstruction mechanism: The existence of an index-zero $(p,q)$-sesquicuspidal symplectic curve in $M^4$ with class $A$ yields a sharp obstruction: any embedding $E(cq,cp) \hookrightarrow M$ must have

$c \leq \frac{\langle [\omega], A \rangle}{pq}$

Persisting after stabilization, these yield the “Fibonacci outer corners” in embedding function staircases (Siegel, 17 Nov 2025, McDuff et al., 23 Apr 2024).

• Stabilized embedding formula for the ball:

$$E_{B^4}^{N}(1,a) = \begin{cases} E_{B^4}(1,a) & \text{for }a \leq \varphi^4 \ \frac{3a}{a+1} & \text{for } a > \varphi^4 \end{cases}$$

where the value $\varphi^4$ is the phase transition corresponding precisely to the threshold at which the moduli of sesquicuspidal curves jump from discrete to continuous (Siegel, 17 Nov 2025).

On rigid del Pezzo surfaces, infinite staircases in embedding functions are explained and, conjecturally, exhausted by the existence of rational sesquicuspidal curves in the monotone class, constructed via mirror-symmetric or toric degenerations (McDuff et al., 23 Apr 2024).

6. Connections to Syzygies and Singularities

Within the field of plane curve singularity theory and algebraic invariants:

• Rational nearly cuspidal curves are naturally $m$-syzygy for $m=2,3,4$, with tight bounds on the Tjurina number.
• Plus-one generated series and 4-syzygy series furnish explicit algebraic models, as in the families:

$x^{2k+1} + (x^2+y^2)^k z = 0$

and

$x^{2r-1} y^{r-1} z + x^d + y^d = 0$

whose unique singularity has two branches but all other singularities are cuspidal (Dimca et al., 2018).

• Conjectures state that all rational cuspidal curves are either free or nearly free; for nearly cuspidal ones, $\nu(C) \leq 2$, where $\nu(C)$ is the width of the Jacobian module (Dimca et al., 2018).

For singularities with two Newton pairs, topological types and semigroups are classified, and their realizability as rational unicuspidal curves is established apart from two exceptional cases (Bodnár, 2015).

7. Open Questions and Further Developments

Priorities for ongoing research include:

• Extension of the scattering–positivity classification to scattering diagrams with more than two rays.
• Determination of exact Gromov–Witten or log curve counts in the supercritical regime, and explicit description of all such curves' homology classes.
• Generalization of the existence and obstruction correspondence to del Pezzo and further rational surfaces, beyond the monotone rigid case.
• Exploration of virtual vs. actual counts in the multidirectional tangency moduli framework, particularly in the non-semipositive setting (Siegel, 17 Nov 2025).
• Analysis of the limits and extensions of the correspondence for higher genus and more complex singularities (“multicusps”).

A plausible implication is that sesquicuspidal curves, by their unique interaction between singularity theory, symplectic and algebraic geometry, and tropical mirror frameworks, will continue to serve as a unifying structure in both geometry and topology, especially in relation to symplectic embedding rigidity and flexibility (McDuff et al., 30 Nov 2024, Siegel, 17 Nov 2025, McDuff et al., 23 Apr 2024, Dimca et al., 2018, Bodnár, 2015).