Mathematics

In 2004 Ciliberto and Di Gennaro conjectured that a nodal threefold in $\mathbb{P}^4$ with at most $2(d-2)(d-1)$ nodes is either factorial or contains a plane or a quadric surface. In this talk we present a proof for this conjecture. We use Noether-Lefschetz theory for surfaces in $\mathbb{P}^3$ to prove that a non-factorial nodal threefold with at most $2(d-2)(d-1)$ nodes contains a plane or a quadric surface unless it is birationally covered by lines. Then we use the classification of threefolds covered by lines to conclude that we are always in one of the first two cases.