On a conjecture of Ciliberto

Abstract

We prove that a threefold hypersurface of degree d with at most ordinary double points is factorial if it contains no planes and has at most (d − 1) 2 singular points. Bibliography: 13 titles.

Maximal families of nodal varieties with defect

“…The author is partially supported by DFG-grant KL 2244/2-1. (d − 1) 2 [2] and that if X has positive defect and (d − 1) 2 nodes then X contains a plane [3]. This improves a previously known bound by Ciliberto and Di Gennaro [5].…”

“…k=0 h I H (k) finishes the proof. The ideal I H is very similar to the ideal used by Green [12] to determine the largest component of the Noether-Lefschetz locus of surfaces in P 3 .…”

“…We will use the results from the previous section to reprove the following result by Cheltsov on the minimal number of nodes to have positive defect: Theorem 4.1 (Cheltsov,[2,3]) Let X ⊂ P 4 be a nodal hypersurface of degree d, with d ≥ 3. Assume that h 4 (X ) ≥ 2, i.e., that X has positive defect.…”

“…To illustrate our methods we start by giving a new proof of Cheltsov's theorem Theorem 1.1 (Cheltsov [2,3]) Let X ⊂ P 4 be a nodal hypersurface of degree d, with d ≥ 3.…”

“…We also consider the case of double covers of P 3 . In this case we recover a result by Cheltsov: Theorem 1.3 (Cheltsov [1]) Let f ∈ C[x 0 , x 1 , x 2 , x 3 ] be a homogeneous polynomial of degree 2d such that V ( f ) is a nodal surface.…”