The Ciliberto-Di Gennaro conjecture for $d=5$

TL;DR

This paper proves the Ciliberto-Di Gennaro conjecture for degree 5 hypersurfaces, completing the proof for all degrees, and classifies the structure of nodal hypersurfaces with specific node counts.

Contribution

The paper provides the first proof of the conjecture for degree 5, filling the gap in the classification of nodal hypersurfaces for all degrees.

Findings

Confirmed the conjecture for d=5.

Classified hypersurfaces with at most 2(d-2)(d-1) nodes.

Extended the known cases from d=3,4,6,7+ to d=5.

Abstract

The Ciliberto-Di Gennaro conjecture predicts that a nodal hypersurface of degree $d\geq 3$ with at most $2(d-2)(d-1)$ nodes is either factorial, or contains a plane and has at least $(d-1)^2$ nodes, or contains a quadric surface and has $2(d-2)(d-1)$ nodes. This conjecture is classically known for $d=3,4$. In 2022 the author proved this conjecture for $d\geq 7$ by the author. Kvitko announced a proof for $d=6$ in 2025. In this paper we prove the conjecture for the remaining open value of $d$, namely $d=5$.