Let $X$ be a smooth complex projective Fano threefold. Then the class $c\_1(X)$ can be realised as an effective divisor in $X$. It is it true that the class $c\_2(X)$ can be realised as an effective curve?


(Note that $c\_3$ cannot, since the Euler characteristics of the cubic Fano $3$-fold is negative) 



Do all Fano threefolds have effective $c_2$?
By a theorem of Miyaoka (Theorem 6.1 in Y. Miyaoka `[The Chern classes and Kodaira dimension of an algebraic variety](http://www.mast.queensu.ca/~mikeroth/NotesForSeminars/Miyaoka-Chern-classes-and-Kodaira-dimension.pdf)', 1987), we have that for a Fano variety $X$, $c\_2(X)\cdot H\ge 0$ for any ample divisor $H$. By duality of the cone of curves and the nef cone, it follows that $c\_2(X)$ is at least *pseudoeffective* (i.e., a limit of effective classes). On the other hand, since $X$ is Fano, the Cone theorem shows that the cone of curves is rational polyhedral, spanned by finitely many classes of rational curves, so $c\_2(X)$ is in fact effective. 


EDIT: As C. Jiang correctly points out, to apply Miyaoka, one needs also the fact that the tangent bundle of a Fano 3-fold is generically nef. This was proved by [Kollár--Miyaoka--Mori--Takagi](https://projecteuclid.org/euclid.pja/1148393517), and later [Peternell](https://eudml.org/doc/277496).


See also Theorem 1.2 in Q. Xie `[On Pseudo-Effectivity of the Second Chern Classes for Terminal Threefolds](https://arxiv.org/pdf/math/0409269.pdf)'.


