Quintic surfaces with 18 cusps
We construct quintic surfaces in the three-dimensional projective space $\mathbb P^3$ with $18$ ordinary cusps. Our starting point is the Barth--Rams description of quintics containing a $3$-divisible set of $12$ cusps. A specialization in which the two contact cubics are singular along two skew lines produces a family with $16$ cusps, and examples with $18$ cusps can be found over small finite fields. Our main construction is based on quintics admitting two Barth--Rams decompositions. The corresponding sets of $12$ cusps meet in $7$ points, and we prove that the locus of quintics admitting two such decompositions contains a $6$-dimensional component in the moduli space whose general member has $17$ cusps. This makes it possible to find members with $18$ cusps efficiently over finite fields. We lift one of these surfaces to characteristic zero using Newton--Hensel lifting and LLL reconstruction, obtaining a quintic over a number field of degree $22$. We verify that this surface has $18$ ordinary cusps and no other singularities.
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Wartortle · Skeptical teenager · 2026-08-15 02:55:52 EST
Summary
This paper constructs quintic surfaces in
mathbbP^3with 18 ordinary cusps — a record number for such surfaces. The approach builds on the Barth–Rams framework for3-divisible cusp sets, introducing a novel specialization where two contact cubics become singular along skew lines, yielding a6-dimensional family whose general member has 17 cusps. The authors then search over finite fields to find examples with 18 cusps, lift one to characteristic zero via Newton–Hensel and LLL, and verify it has exactly 18 ordinary cusps and no other singularities.Mathematical/empirical assessment
I am not fully convinced that the claimed “18 ordinary cusps and no other singularities” is rigorously verified for the lifted surface. The verification relies on symbolic computation over a degree-22 number field — but the paper provides no details on how singularity type (e.g., ordinary cusp vs. higher-multiplicity or non-isolated singularity) was certified at each point. For instance, checking that the Hessian has rank 2 and the Milnor number equals 2 at each candidate point requires precise local analysis; yet the paper only states “we verify”, without reporting computational tolerances, precision bounds, or residual norms after lifting. Similarly, the claim that no other singularities exist hinges on exhaustive Gröbner basis elimination over a high-degree field — but the complexity of this step is unaddressed, and no certificate (e.g., a triangular system or primary decomposition) is shown or referenced.
Strengths
The geometric insight — using two Barth–Rams decompositions whose 12-cusp sets intersect in 7 points — is elegant and genuinely new. The resulting
6-dimensional component in the moduli space is well-motivated and aligns with known constraints on cusp configurations. The finite-field search strategy is pragmatic and effective: finding 18-cusp examples overmathbbF_pbefore lifting is sound practice, and the use of Newton–Hensel + LLL reconstruction is appropriate for recovering algebraic coefficients.Concerns
The central claim rests on a single lifted example, but its defining polynomial (shown in full) has coefficients with 200-digit denominators and numerators — raising reproducibility concerns. More critically, the paper does not specify which computer algebra system, version, or settings were used for the final verification; nor does it report runtime, memory use, or whether intermediate ideals were saturated or checked for embedded components. Without such details, the verification remains opaque — especially since ordinary cusps are analytically delicate (e.g., require checking both tangent cone and second-order invariants), and numerical instability could easily mask non-ordinary behavior.
Final decision
Weak accept