Some remarks on algebraic equivalence of cycles

Giuseppe Ceresa and Alberto Collino

Vol. 105 (1983), No. 2, 285–290

DOI: 10.2140/pjm.1983.105.285

Abstract

Let F ⊆ P⁴ be a 3-fold with one ordinary double point p, and let F′ be the proper transform of F under the blowing up of P⁴ at p. If H ⊆ F′ is the preimage of p on F′, we prove that for F general the algebraic 1-cycle given by the difference of the two generators of the smooth quadric surface H, is not algebraically equivalent to zero on F′. Griffiths has shown this cycle to be homologically equivalent to zero. Also, we show that on a general quintic 3-fold X there are no non-trivial algebraic equivalence relations between the lines of X.

Mathematical Subject Classification 2000

Primary: 14C10

Secondary: 14J30

Milestones

Received: 25 September 1981

Published: 1 April 1983