# Do all Fano threefolds have effective $c_2$?

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)

By a theorem of Miyaoka (Theorem 6.1 in Y. Miyaoka The Chern classes and Kodaira dimension of an algebraic variety', 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.