TL;DR
This paper constructs specific Lefschetz fibrations over the sphere that violate the slope inequality, providing a counterexample to a previously posed question by Hain.
Contribution
It introduces new Lefschetz fibrations with small slope that challenge existing slope inequality assumptions.
Findings
Constructed Lefschetz fibrations over S^2 that do not satisfy the slope inequality.
Provided a negative answer to Hain's question.
Demonstrated the existence of fibrations with small slope values.
Abstract
We construct Lefschetz fibrations over which do not satisfy the slope inequality. This gives a negative answer to a question of Hain.
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