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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.

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  • $\begingroup$ Note that that this is a question about surfaces of general type (a non-rational surface with $K^2 > 0 $ Is always Kodiara dimension $2$, see proposition X.1 in Beauville). $\endgroup$ – Nick L May 25 '17 at 12:52
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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.

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Assume $X$ is minimal. Since $S$ is of general type (see my comment), $K_{X}$ is nef ($K_{X}.C \geq 0$ for all curves C). By definition we have $K_{X}.D = -1$, therefore $D$ cannot be effective.

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