## Homological equivalence in $\mathbb{P}^3$ [closed]

Is it true that any smooth irreducible curve in $\mathbb{P}^3_\mathbb{C}$ is homologically equivalent to a union of lines in $\mathbb{P}^3_\mathbb{C}$? If so, can we say something similar for any reduced curve in $\mathbb{P}^3_\mathbb{C}$.

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Yes and yes. The point is that $H^4(\mathbb{P}^2,\mathbb{Z})=\mathbb{Z}$. So the homological equivalence class of a $1$-cycle is determined by its degree. So if it is $d$, then the class is $d$ times the class of a line. – Donu Arapura Apr 22 2012 at 23:28
@Arapura: Thanks a lot. – Naga Venkata Apr 23 2012 at 22:01
I'm closing this question, just because otherwise it will continue to clog up the "unanswered" questions list. (By the way, if anyone is watching, this is a good reason to just post the answer as an answer, rather than beating around the bush with comments.) – Scott Morrison Jul 3 at 0:17