Let $X$ be a smooth projective surface such that $\chi(O\_X)=1$ and $K\_X^2\geq 3$(or just $K\_X^2>0$). Let $D$ be $(-1)$-class, i.e: $D^2=-1,D^2+D.K\_X=-2$(equivalently, $D^2=-1, \chi(-D)=0$). I wonder whether we can show that $D$ is effective. 


If $X$ is rational surface and $K\_X^2>0$, then such $D$ is effective. I wonder whether this holds in more general settings. Maybe one can add more assumption, say $h^1(O\_X)=h^2(O\_X)=0$. 


Thanks.



Effectivity of $(-1)$-class on smooth projective surface
It is easy to find counterexamples with $X$ of general type. Take $S$ of general type with $\chi(S)=1$, $K^2\_S>1$ and nonzero torsion in $Pic(X)$ (there are plenty of such surfaces, even with $h^1(\mathcal O)=h^1(\mathcal O)=0$). Now blow up $S$ to get $X$ and an effective $-1$ curve $E$ and set $D=E+L$, where $L$ is a non zero torsion element in $Pic(X)$. $D$ is a $-1$-class, but it is not effective: indeed if it were then it would have to contain $E$, since $DE=-1<0$, but then $L=D-E$ would be effective, contradicting the assumption that $L$ is non zero torsion. 


ADDED: for rational surfaces one could try a direct approach, similar to the well known computation for finding $-1$-curves on a Del Pezzo surface of degree 3. $S$ is the blow up of $P^2$ or of a ruled surface $F\_n$ at at most 8 (or 7) points. One knows the Picard group of $S$ and can try to write down explicitly the $-1$-classes and see whether they are all effective. 


