Three-dimensional nodal hypersurfaces and the Ciliberto-Di Gennaro Conjecture

We will talk about three-dimensional hypersurfaces with ordinary double points, commonly known as nodes. Ciliberto and Di Gennaro showed that if the number of singularities falls within a specific range and such a variety does not contain a plane or a quadric, then any smooth surface on it is a complete intersection. Furthermore, they conjectured that this statement holds true for any surface under the specified conditions, implying that the variety is factorial. Later, Cheltsov obtained a lower bound on the number of nodes required for hypersurfaces of a given degree to be non-factorial. Following Klusterman’s paper, we will discuss an alternative proof of this result and prove the conjecture for hypersurfaces of degree 7 or higher.