On Factoriality of Threefolds with Isolated Singularities

Abstract

We investigate the existence of complete intersection threefolds X ⊂ n with only isolated, ordinary multiple points and we provide some sufficient conditions for their factoriality.

Hypersurfaces with defect

“…Cohomological preliminaries. In this section, we relate defect of hypersurfaces to the number of singularities following the ideas presented in [31]. Let K be an algebraically closed field of characteristic p = 2.…”

“…Proof. See [31] for the case of ordinary multiple points and [9], [34] for details on resolving A k singularities.…”

“…Proof. We follow the proof of [31,Proposition 2.15]. Since X is normal and Cohen-Macaulay, the proof of [20, Proposition 2.15] generalizes and gives an exact sequence…”

“…As explained in Subsection 2.11, there are some obstructions to extending the proof of Theorem 1.1. However, for hypersurfaces with very mild singularities, we can use a resolution of singularities approach similar to [31] to prove: Theorem 1.2. Let K be an algebraically closed field of characteristic = 2.…”

Factorial Hypersurfaces

“…Factoriality of algebraic varieties is a very old topic in Algebraic Geometry, with lots of papers written on the subject. We will only point out a few recent papers that demonstrate that the topic is still actively investigated today: [5,6,7,8,9]. Various technical points related to the constructions of this paper were discussed by the author in his talks given in 2009-2014 at Steklov Mathematical Institute.…”