Let $X$ be an Enriques surface. A $(-2)$-curve is an irriducible rational curve on X such that $C^2 = -2$. By Proposition [VIII,16.1] from Barth-Peters-Van de Ven, we have that if $D^2 = -2$, then it is a $(-2)$-curve, but do such curves exist?



Are there (-2)-curves on an Enriques surface?
As explained in J.C. Ottem's answer, the generic Enriques surface contains no smooth rational curves at all.


However, it can happen that some special Enriques surface $X$ contains $(-2)$-curves, and also infinitely many of them (see [this paper](http://www.springerlink.com/content/x175482343514234/) by Cossec and Dolgachev). The maximal number of *disjoint* $(-2)$ curves on $X$ is *eight*, and Enriques surfaces with eight disjoint $(-2)$-curves are classified in the article


Mendes Lopes, Margarida; Pardini, Rita


Enriques surfaces with eight nodes


Math. Z. 241 (2002), no. 4, 673–683. 


The authors first show that, setting $C\_1, \dots,C\_8$ to be the exceptional $(-2)$-curves of $X$, the divisor $C\_1+\dots+C\_8$ is divisible by $2$ in the Picard group of $X$, or equivalently there exists a double cover $\widetilde{X} \to X$ branched exactly over them.


The main theorem then states that an Enriques surface with eight disjoint $(-2)$-curves is isomorphic to $X=D\_1\times D\_2/G$, where $D\_1,D\_2$ are elliptic curves and $G$ is either $\mathbb{Z}\_2^2$ or $\mathbb{Z}\_2^3$.


