For any rational r in (2,8), a sufficiently high-genus Lefschetz fibration over S^2 exists with slope exactly r.
Korkmaz,Noncomplex smooth4-manifolds with Lefschetz fibrations
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Constructing Lefschetz Fibrations with Arbitrary Slope
For any rational r in (2,8), a sufficiently high-genus Lefschetz fibration over S^2 exists with slope exactly r.