Some remarks on algebraic equivalence of cycles (Q795109)

Let \(F\subset {\mathbb{P}}_ 4({\mathbb{C}})\) be a threefold with exactly one singular point, which is an ordinary double point p, and F' be the proper transform of F under the blowing up of \({\mathbb{P}}_ 4({\mathbb{C}})\) at p: F' is smooth, and the inverse image of p in F' is a smooth quadric H with two lines L and M belonging to the two distinct rulings of H. This paper shows that if F is general, of degree at least five, then L and M are not algebraically equivalent in F', although they are homologically equivalent. A result of the same type is obtained for the general non- singular quintic threefolds in \({\mathbb{P}}_ 4({\mathbb{C}})\).