Vol. 105, No. 2, 1983

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Some remarks on algebraic equivalence of cycles

Giuseppe Ceresa and Alberto Collino

Vol. 105 (1983), No. 2, 285–290

Let F P4 be a 3-fold with one ordinary double point p, and let Fbe the proper transform of F under the blowing up of P4 at p. If H Fis 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
Received: 25 September 1981
Published: 1 April 1983
Giuseppe Ceresa
Alberto Collino