Timeline for homologically trivial $1$-cycles and surfaces

8 events

when toggle format	what		by	license	comment
Mar 13, 2018 at 5:36	comment	added	naf		For a projective example, simply take any cycle which is homologically equivalent to zero but not algebraically equivalent to zero. (On a surface, any 1-cycle which is homologically trivial is algebraically trivial.)
Mar 12, 2018 at 15:49	comment	added	Jason Starr		I suspect that this fails already for Hironaka's example, as described in Appendix B of Hartshorne's "Algebraic geometry".
Mar 12, 2018 at 14:55	history	edited	YCor		edited tags
Mar 12, 2018 at 14:39	comment	added	pi_1		Indeed, my question is whether one can choose a class on $\widetilde S$ which is homologically trivial.
Mar 12, 2018 at 14:19	comment	added	byu		If you take any surface containing the curves in the support of $\gamma$, and take a resolution of it, you get a class in $Pic(\widetilde S)$ that pushes forward to $\gamma$ like you say. Is your question whether we can take this to be homologically trivial?
Mar 12, 2018 at 12:38	comment	added	pi_1		Thank you. No, I do mean "curves".
Mar 12, 2018 at 12:26	comment	added	Qfwfq		You mean $\gamma$ in $\mathrm{CH}^1(X)$ (codimension $1$)?
Mar 12, 2018 at 11:53	history	asked	pi_1	CC BY-SA 3.0