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Let $X$ be a normal projective complex surface with at worst quotient singularities. Let $\bar{X}\to X$ be the minimal resolution. Further assume that $b_2(X)=1$ and $b_1(X)=b_3(X)=0$. Since quotient singularities are rational, $p_g(\bar{X})=0=q(\bar{X})$. So by the Noether formula we have $K_{\bar{X}}^2=9-n$ where $n$ is the number of exceptional curves of $\bar{X}\to X$. I want to know where $\bar{X}$ belongs according to the Enriques-Kodaira classification of surfaces. Are the followings true?

  1. If $n=9$ and $K_{\bar{X}}^2=0$ then $\bar{X}$ is an Enriques surface.

  2. If $n>9$ and $K_{\bar{X}}^2<0$ then $\bar{X}$ is rational.

  3. If $n<9$ and $K_{\bar{X}}^2>0$ then $\bar{X}$ is either rational or a minimal surface of general type.

If $\bar{X}$ is of general type then must it be minimal? Can $\bar{X}$ be of general type in the case $K_{\bar{X}}^2<0$?

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  • $\begingroup$ I don't think any of these should be true though I don't have counterexamples. Regarding $\overline{X}$ of general type, why can't there be a line in $X$ passing through some of the singular points whose strict transform is a $-1$ curve? $\endgroup$
    – Will Sawin
    Commented Jul 9 at 20:21

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