A Quintic Hypersurface in $\PP^8(\CC)$ with Many Nodes

TL;DR

This paper constructs a specific degree-5 hypersurface in complex projective 8-space with a record number of 23436 nodes, providing new bounds for the maximum possible nodes on such hypersurfaces.

Contribution

It presents a novel construction of a degree-5 hypersurface with many nodes, improving bounds on the maximum number of nodes in this setting.

Findings

Constructed a hypersurface with 23436 nodes

Established bounds for the maximum number of nodes

Generalized previous construction methods

Abstract

We construct a hypersurface of degree 5 in projective space $\PP^8(\CC)$ which contains exactly 23436 ordinary nodes and no further singularities. This limits the maximum number $\mu_{8}(5)$ of ordinary nodes a hyperquintic in $\PP^8(\CC)$ can have to $23436 \leq \mu_{8}(5) \leq 27876$. Our method generalizes the approach by the $3^{\text{rd}}$ author for the construction of a quintic threefold with 130 nodes in an earlier paper.