On factoriality of nodal threefolds

Abstract

We prove the Q-factoriality of a nodal hypersurface in P 4 of degree n with at most (n−1) 2 4 nodes and the Q-factoriality of a double cover of P 3 branched over a nodal surface of degree 2r with at most (2r−1)r 3 nodes. for fruitful conversations. Special thanks goes to V. Kulikov for Lemma 38. The author would also like to thank the referee for useful comments.All varieties are assumed to be projective, normal, and defined over C. 1 A 3-fold is called nodal if all its singular points are ordinary double points. 2 A variety is called Q-factorial if a multiple of every Weil divisor on the variety is a Cartier divisor.

Points in projective spaces and applications

“…The claim of Theorem 1.3 is conjectured in [2], and it is proved in [4] in the case r = 3. Example 1.4.…”

Factoriality of nodal three-dimensional varieties and connectedness of the locus of log canonical singularities

“…As pointed out in [32], the problem of the Q-factoriality of nodal threefolds is related to the Shokurov vanishing (see [33]- [36]). We illustrate this relation by the following example.…”

On factorial double solids with simple double points

“…Even though it seems too difficult to describe all the non-Q-factorial double solids, Example 1.1 and the paper [6] enable us to propose the conjecture below. Furthermore, this is a little stronger than that in [5].…”