A Nodal Decic Counterexample to the Generic Projection Incidence Bound
Manuscript 18 October 2026 · Online 18 October 2026
Abstract
We construct a degree-ten hypersurface in complex projective four-space with exactly sixty ordinary double points whose general plane projection lies on a sextic. Thus the projected nodes violate the proposed bound k(n-1), since 60 > 6(10-1) = 54, even though 60 < (10-1)^2. This answers negatively the projection question in Cheltsov's Problem 18 in an AIM workshop list and gives a counterexample to Cheltsov-Park Conjecture 1.6. The node set satisfies the stronger ambient incidence bound 7k for curves of degree k. We use the known sextic-projection property of the projectivized H4 roots, realize those points as nodes of an explicit symmetric decic surface, and remove every unwanted singularity after extending to a threefold. Exact Hessian, Groebner-basis and interpolation certificates accompany the construction. The threefold is factorial; the separate sharp factoriality theorem is not contradicted. This is an AI-assisted, self-audited, unrefereed preprint; it makes no independent-review, formal-certification or absolute-priority claim.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Complete projection counterexample
- Categories
- math.AG
- Manuscript
- 18 October 2026
- Online release
- 18 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “A Nodal Decic Counterexample to the Generic Projection Incidence Bound,” EulerSolve Research Papers, AIM-ALGEBRAIC_GEOMETRY-0318, 2026. https://doi.org/10.5281/zenodo.23453999.
BibTeX
@misc{ferudun2026nodaldecic,
author = {Ferudun, Alper},
title = {A Nodal Decic Counterexample to the Generic Projection Incidence Bound},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/aim-algebraic-geometry-0318/},
doi = {10.5281/zenodo.23453999},
note = {AIM-ALGEBRAIC_GEOMETRY-0318; unrefereed preprint}
}