Manchester Geometry Seminar 2004/2005

An algebraic 3-fold is called nodal if all its singular points are ordinary double points. Nodal 3-folds arise naturally in many different topics of algebraic geometry. We prove the factoriality of a nodal hypersurface in P⁴ of degree n with at most (n-1)²/4 nodes and the factoriality of a double cover of P³ branched over a nodal surface of degree 2r with at most (2r-1)r/3 nodes. In both cases factoriality simply means that the rank of the 4-th homology group is 1.