A Nodal Decic Counterexample to the Generic Projection Incidence Bound

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.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-18
DOI
https://doi.org/10.5281/zenodo.23453998
Primary Topic
Algebraic Geometry and Number Theory
Type
preprint
Controls
|||
ALL TIME
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
preprint

A Nodal Decic Counterexample to the Generic Projection Incidence Bound

Alper Ferudun
Zenodo (CERN European Organization for Nuclear Research)
Algebraic Geometry and Number Theory
preprint

A Nodal Decic Counterexample to the Generic Projection Incidence Bound

Alper Ferudun
preprint en

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.

Zenodo (CERN European Organization for Nuclear Research)
Algebraic Geometry and Number Theory
AI Navigator

Ask Laika to Summarize, Analyze, and Connect papers live on the map.

Summarize Papers & Methodologies

Extract key findings, datasets, and comparative methods across publications.

Benchmark Rankings & Visual Analytics

Rank top research institutions, authors, funders, topics, and journals by Field-Weighted Citation Impact (FWCI) and paper volume with instant charts.

Connect Distant Disciplines

Bridge topological clusters on the map to find hidden collaborative intersections.

A Nodal Decic Counterexample to the Generic Projection Incidence Bound — Alper Ferudun · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS